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id="toc-Rayleigh's_method" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Rayleigh's_method"> <div class="vector-toc-text"> 1.2 Rayleigh's method </div> </a> <ul id="toc-Rayleigh's_method-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Concrete_numbers_and_base_units" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Concrete_numbers_and_base_units"> <div class="vector-toc-text"> 2 Concrete numbers and base units </div> </a> <button aria-controls="toc-Concrete_numbers_and_base_units-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Concrete numbers and base units subsection </button> <ul id="toc-Concrete_numbers_and_base_units-sublist" class="vector-toc-list"> <li id="toc-Percentages,_derivatives_and_integrals" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Percentages,_derivatives_and_integrals"> <div class="vector-toc-text"> 2.1 Percentages, derivatives and integrals </div> </a> <ul id="toc-Percentages,_derivatives_and_integrals-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Dimensional_homogeneity_(commensurability)" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Dimensional_homogeneity_(commensurability)"> <div class="vector-toc-text"> 3 Dimensional homogeneity (commensurability) </div> </a> <ul id="toc-Dimensional_homogeneity_(commensurability)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Conversion_factor" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Conversion_factor"> <div class="vector-toc-text"> 4 Conversion factor </div> </a> <ul id="toc-Conversion_factor-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> 5 Applications </div> </a> <button aria-controls="toc-Applications-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Applications subsection </button> <ul id="toc-Applications-sublist" class="vector-toc-list"> <li id="toc-Mathematics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Mathematics"> <div class="vector-toc-text"> 5.1 Mathematics </div> </a> <ul id="toc-Mathematics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Finance,_economics,_and_accounting" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Finance,_economics,_and_accounting"> <div class="vector-toc-text"> 5.2 Finance, economics, and accounting </div> </a> <ul id="toc-Finance,_economics,_and_accounting-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Fluid_mechanics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Fluid_mechanics"> <div class="vector-toc-text"> 5.3 Fluid mechanics </div> </a> <ul id="toc-Fluid_mechanics-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> 6 History </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Examples" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Examples"> <div class="vector-toc-text"> 7 Examples </div> </a> <button aria-controls="toc-Examples-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Examples subsection </button> <ul id="toc-Examples-sublist" class="vector-toc-list"> <li id="toc-A_simple_example:_period_of_a_harmonic_oscillator" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#A_simple_example:_period_of_a_harmonic_oscillator"> <div class="vector-toc-text"> 7.1 A simple example: period of a harmonic oscillator </div> </a> <ul id="toc-A_simple_example:_period_of_a_harmonic_oscillator-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-A_more_complex_example:_energy_of_a_vibrating_wire" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#A_more_complex_example:_energy_of_a_vibrating_wire"> <div class="vector-toc-text"> 7.2 A more complex example: energy of a vibrating wire </div> </a> <ul id="toc-A_more_complex_example:_energy_of_a_vibrating_wire-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-A_third_example:_demand_versus_capacity_for_a_rotating_disc" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#A_third_example:_demand_versus_capacity_for_a_rotating_disc"> <div class="vector-toc-text"> 7.3 A third example: demand versus capacity for a rotating disc </div> </a> <ul id="toc-A_third_example:_demand_versus_capacity_for_a_rotating_disc-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Properties" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Properties"> <div class="vector-toc-text"> 8 Properties </div> </a> <button aria-controls="toc-Properties-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Properties subsection </button> <ul id="toc-Properties-sublist" class="vector-toc-list"> <li id="toc-Mathematical_properties" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Mathematical_properties"> <div class="vector-toc-text"> 8.1 Mathematical properties </div> </a> <ul id="toc-Mathematical_properties-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Mechanics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Mechanics"> <div class="vector-toc-text"> 8.2 Mechanics </div> </a> <ul id="toc-Mechanics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_fields_of_physics_and_chemistry" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_fields_of_physics_and_chemistry"> <div class="vector-toc-text"> 8.3 Other fields of physics and chemistry </div> </a> <ul id="toc-Other_fields_of_physics_and_chemistry-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Polynomials_and_transcendental_functions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Polynomials_and_transcendental_functions"> <div class="vector-toc-text"> 8.4 Polynomials and transcendental functions </div> </a> <ul id="toc-Polynomials_and_transcendental_functions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Combining_units_and_numerical_values" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Combining_units_and_numerical_values"> <div class="vector-toc-text"> 8.5 Combining units and numerical values </div> </a> <ul id="toc-Combining_units_and_numerical_values-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Quantity_equations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Quantity_equations"> <div class="vector-toc-text"> 8.6 Quantity equations </div> </a> <ul id="toc-Quantity_equations-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Dimensionless_concepts" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Dimensionless_concepts"> <div class="vector-toc-text"> 9 Dimensionless concepts </div> </a> <button aria-controls="toc-Dimensionless_concepts-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Dimensionless concepts subsection </button> <ul id="toc-Dimensionless_concepts-sublist" class="vector-toc-list"> <li id="toc-Constants" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Constants"> <div class="vector-toc-text"> 9.1 Constants </div> </a> <ul id="toc-Constants-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Formalisms" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Formalisms"> <div class="vector-toc-text"> 9.2 Formalisms </div> </a> <ul id="toc-Formalisms-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Dimensional_equivalences" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Dimensional_equivalences"> <div class="vector-toc-text"> 10 Dimensional equivalences </div> </a> <button aria-controls="toc-Dimensional_equivalences-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Dimensional equivalences subsection </button> <ul id="toc-Dimensional_equivalences-sublist" class="vector-toc-list"> <li id="toc-SI_units" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#SI_units"> <div class="vector-toc-text"> 10.1 SI units </div> </a> <ul id="toc-SI_units-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Programming_languages" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Programming_languages"> <div class="vector-toc-text"> 11 Programming languages </div> </a> <ul id="toc-Programming_languages-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Geometry:_position_vs._displacement" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Geometry:_position_vs._displacement"> <div class="vector-toc-text"> 12 Geometry: position vs. displacement </div> </a> <button aria-controls="toc-Geometry:_position_vs._displacement-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Geometry: position vs. displacement subsection </button> <ul id="toc-Geometry:_position_vs._displacement-sublist" class="vector-toc-list"> <li id="toc-Affine_quantities" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Affine_quantities"> <div class="vector-toc-text"> 12.1 Affine quantities </div> </a> <ul id="toc-Affine_quantities-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Orientation_and_frame_of_reference" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Orientation_and_frame_of_reference"> <div class="vector-toc-text"> 12.2 Orientation and frame of reference </div> </a> <ul id="toc-Orientation_and_frame_of_reference-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Huntley's_extensions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Huntley's_extensions"> <div class="vector-toc-text"> 12.3 Huntley's extensions </div> </a> <ul id="toc-Huntley's_extensions-sublist" class="vector-toc-list"> <li id="toc-Directed_dimensions" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Directed_dimensions"> <div class="vector-toc-text"> 12.3.1 Directed dimensions </div> </a> <ul id="toc-Directed_dimensions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Quantity_of_matter" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Quantity_of_matter"> <div class="vector-toc-text"> 12.3.2 Quantity of matter </div> </a> <ul id="toc-Quantity_of_matter-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Siano's_extension:_orientational_analysis" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Siano's_extension:_orientational_analysis"> <div class="vector-toc-text"> 12.4 Siano's extension: orientational analysis </div> </a> <ul id="toc-Siano's_extension:_orientational_analysis-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 13 See also </div> </a> <button aria-controls="toc-See_also-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle See also subsection </button> <ul id="toc-See_also-sublist" class="vector-toc-list"> <li id="toc-Related_areas_of_mathematics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Related_areas_of_mathematics"> <div class="vector-toc-text"> 13.1 Related areas of mathematics </div> </a> <ul id="toc-Related_areas_of_mathematics-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> 14 Notes </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> 15 References </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> 16 Further reading </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> 17 External links </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Toggle the table of contents </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading">Dimensional analysis</h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 34 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-34" aria-hidden="true" > 34 languages </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AA%D8%AD%D9%84%D9%8A%D9%84_%D8%A8%D8%B9%D8%AF%D9%8A" title="تحليل بعدي – Arabic" lang="ar" hreflang="ar" data-title="تحليل بعدي" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AE%E0%A6%BE%E0%A6%A4%E0%A7%8D%E0%A6%B0%E0%A6%BE_%E0%A6%B8%E0%A6%AE%E0%A7%80%E0%A6%95%E0%A6%B0%E0%A6%A3" title="মাত্রা সমীকরণ – Bangla" lang="bn" hreflang="bn" data-title="মাত্রা সমীকরণ" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target">বাংলা</a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%90%D0%BD%D0%B0%D0%BB%D1%96%D0%B7_%D1%80%D0%B0%D0%B7%D0%BC%D0%B5%D1%80%D0%BD%D0%B0%D1%81%D1%86%D0%B5%D0%B9" title="Аналіз размернасцей – Belarusian" lang="be" hreflang="be" data-title="Аналіз размернасцей" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target">Беларуская</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/An%C3%A0lisi_dimensional" title="Anàlisi dimensional – Catalan" lang="ca" hreflang="ca" data-title="Anàlisi dimensional" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Dimensionsanalyse" title="Dimensionsanalyse – German" lang="de" hreflang="de" data-title="Dimensionsanalyse" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Dimensionaalanal%C3%BC%C3%BCs" title="Dimensionaalanalüüs – Estonian" lang="et" hreflang="et" data-title="Dimensionaalanalüüs" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target">Eesti</a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%94%CE%B9%CE%B1%CF%83%CF%84%CE%B1%CF%84%CE%B9%CE%BA%CE%AE_%CE%B1%CE%BD%CE%AC%CE%BB%CF%85%CF%83%CE%B7" title="Διαστατική ανάλυση – Greek" lang="el" hreflang="el" data-title="Διαστατική ανάλυση" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target">Ελληνικά</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/An%C3%A1lisis_dimensional" title="Análisis dimensional – Spanish" lang="es" hreflang="es" data-title="Análisis dimensional" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D8%AD%D9%84%DB%8C%D9%84_%D8%A7%D8%A8%D8%B9%D8%A7%D8%AF%DB%8C" title="تحلیل ابعادی – Persian" lang="fa" hreflang="fa" data-title="تحلیل ابعادی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Analyse_dimensionnelle" title="Analyse dimensionnelle – French" lang="fr" hreflang="fr" data-title="Analyse dimensionnelle" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Anail%C3%ADs_thoiseach" title="Anailís thoiseach – Irish" lang="ga" hreflang="ga" data-title="Anailís thoiseach" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target">Gaeilge</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%B0%A8%EC%9B%90_%ED%95%B4%EC%84%9D" title="차원 해석 – Korean" lang="ko" hreflang="ko" data-title="차원 해석" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%89%D5%A1%D6%83%D5%A1%D5%B5%D5%B6%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Չափայնություն – Armenian" lang="hy" hreflang="hy" data-title="Չափայնություն" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target">Հայերեն</a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B5%E0%A4%BF%E0%A4%AE%E0%A5%80%E0%A4%AF_%E0%A4%B5%E0%A4%BF%E0%A4%B6%E0%A5%8D%E0%A4%B2%E0%A5%87%E0%A4%B7%E0%A4%A3" title="विमीय विश्लेषण – Hindi" lang="hi" hreflang="hi" data-title="विमीय विश्लेषण" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target">हिन्दी</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Analisis_dimensi" title="Analisis dimensi – Indonesian" lang="id" hreflang="id" data-title="Analisis dimensi" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Analisi_dimensionale" title="Analisi dimensionale – Italian" lang="it" hreflang="it" data-title="Analisi dimensionale" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%90%D7%A0%D7%9C%D7%99%D7%96%D7%94_%D7%9E%D7%9E%D7%93%D7%99%D7%AA" title="אנליזה ממדית – Hebrew" lang="he" hreflang="he" data-title="אנליזה ממדית" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%92%E1%83%90%E1%83%9C%E1%83%96%E1%83%9D%E1%83%9B%E1%83%98%E1%83%9A%E1%83%94%E1%83%91%E1%83%90_(%E1%83%A4%E1%83%98%E1%83%96%E1%83%98%E1%83%99%E1%83%90)" title="განზომილება (ფიზიკა) – Georgian" lang="ka" hreflang="ka" data-title="განზომილება (ფიზიკა)" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target">ქართული</a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D3%A8%D0%BB%D1%88%D0%B5%D0%BC%D0%B4%D1%96%D0%BB%D1%96%D0%BA%D1%82%D0%B5%D1%80%D0%B4%D1%96_%D1%82%D0%B0%D0%BB%D0%B4%D0%B0%D1%83" title="Өлшемділіктерді талдау – Kazakh" lang="kk" hreflang="kk" data-title="Өлшемділіктерді талдау" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target">Қазақша</a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Analiz_dimansyon%C3%A8l" title="Analiz dimansyonèl – Haitian Creole" lang="ht" hreflang="ht" data-title="Analiz dimansyonèl" data-language-autonym="Kreyòl ayisyen" data-language-local-name="Haitian Creole" class="interlanguage-link-target">Kreyòl ayisyen</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%AC%A1%E5%85%83%E8%A7%A3%E6%9E%90" title="次元解析 – Japanese" lang="ja" hreflang="ja" data-title="次元解析" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Dimensjonsanalyse" title="Dimensjonsanalyse – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Dimensjonsanalyse" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target">Norsk nynorsk</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Analiza_wymiarowa" title="Analiza wymiarowa – Polish" lang="pl" hreflang="pl" data-title="Analiza wymiarowa" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/An%C3%A1lise_dimensional" title="Análise dimensional – Portuguese" lang="pt" hreflang="pt" data-title="Análise dimensional" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Analiz%C4%83_dimensional%C4%83" title="Analiză dimensională – Romanian" lang="ro" hreflang="ro" data-title="Analiză dimensională" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%90%D0%BD%D0%B0%D0%BB%D0%B8%D0%B7_%D1%80%D0%B0%D0%B7%D0%BC%D0%B5%D1%80%D0%BD%D0%BE%D1%81%D1%82%D0%B8" title="Анализ размерности – Russian" lang="ru" hreflang="ru" data-title="Анализ размерности" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Dimensional_analysis" title="Dimensional analysis – Simple English" lang="en-simple" hreflang="en-simple" data-title="Dimensional analysis" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target">Simple English</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Razse%C5%BEnostna_analiza" title="Razsežnostna analiza – Slovenian" lang="sl" hreflang="sl" data-title="Razsežnostna analiza" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Dimensioanalyysi" title="Dimensioanalyysi – Finnish" lang="fi" hreflang="fi" data-title="Dimensioanalyysi" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Dimensionsanalys" title="Dimensionsanalys – Swedish" lang="sv" hreflang="sv" data-title="Dimensionsanalys" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Boyut_analizi" title="Boyut analizi – Turkish" lang="tr" hreflang="tr" data-title="Boyut analizi" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target">Türkçe</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9C%D0%B5%D1%82%D0%BE%D0%B4_%D0%B0%D0%BD%D0%B0%D0%BB%D1%96%D0%B7%D1%83_%D1%80%D0%BE%D0%B7%D0%BC%D1%96%D1%80%D0%BD%D0%BE%D1%81%D1%82%D0%B5%D0%B9" title="Метод аналізу розмірностей – Ukrainian" lang="uk" hreflang="uk" data-title="Метод аналізу розмірностей" 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class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Analysis of the relationships between different physical quantities</div> In <a href="/wiki/Engineering" title="Engineering">engineering</a> and <a href="/wiki/Science" title="Science">science</a>, dimensional analysis is the analysis of the relationships between different <a href="/wiki/Physical_quantity" title="Physical quantity">physical quantities</a> by identifying their <a href="/wiki/Base_quantity" class="mw-redirect" title="Base quantity">base quantities</a> (such as <a href="/wiki/Length" title="Length">length</a>, <a href="/wiki/Mass" title="Mass">mass</a>, <a href="/wiki/Time" title="Time">time</a>, and <a href="/wiki/Electric_current" title="Electric current">electric current</a>) and <a href="/wiki/Units_of_measurement" class="mw-redirect" title="Units of measurement">units of measurement</a> (such as metres and grams) and tracking these dimensions as calculations or comparisons are performed. The term dimensional analysis is also used to refer to <a href="/wiki/Conversion_of_units" title="Conversion of units">conversion of units</a> from one dimensional unit to another, which can be used to evaluate scientific formulae. Commensurable physical quantities are of the same <a href="/wiki/Kind_of_quantity" class="mw-redirect" title="Kind of quantity">kind</a> and have the same dimension, and can be directly compared to each other, even if they are expressed in differing units of measurement; e.g., metres and feet, grams and pounds, seconds and years. Incommensurable physical <a href="/wiki/Quantity" title="Quantity">quantities</a> are of different <a href="/wiki/Kind_of_quantity" class="mw-redirect" title="Kind of quantity">kinds</a> and have different dimensions, and can not be directly compared to each other, no matter what <a href="/wiki/Unit_of_measurement" title="Unit of measurement">units</a> they are expressed in, e.g. metres and grams, seconds and grams, metres and seconds. For example, asking whether a gram is larger than an hour is meaningless. Any physically meaningful <a href="/wiki/Equation" title="Equation">equation</a>, or <a href="/wiki/Inequality_(mathematics)" title="Inequality (mathematics)">inequality</a>, must have the same dimensions on its left and right sides, a property known as dimensional homogeneity. Checking for dimensional homogeneity is a common application of dimensional analysis, serving as a plausibility check on <a href="/wiki/Formal_proof" title="Formal proof">derived</a> equations and <a href="/wiki/Computation" title="Computation">computations</a>. It also serves as a guide and constraint in deriving equations that may describe a physical <a href="/wiki/System" title="System">system</a> in the absence of a more rigorous derivation. The concept of physical dimension or quantity dimension, and of dimensional analysis, was introduced by <a href="/wiki/Joseph_Fourier" title="Joseph Fourier">Joseph Fourier</a> in 1822.<a href="#cite_note-Bolster-1">[1]</a>: 42  <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Formulation">Formulation</h2>[<a href="/w/index.php?title=Dimensional_analysis&action=edit&section=1" title="Edit section: Formulation">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">"Dimension (physics)" redirects here. For physical dimensions, see <a href="/wiki/Size" title="Size">Size</a>.</div> The <a href="/wiki/Buckingham_%CF%80_theorem" title="Buckingham π theorem">Buckingham π theorem</a> describes how every physically meaningful equation involving n variables can be equivalently rewritten as an equation of n − m dimensionless parameters, where m is the <a href="/wiki/Rank_of_a_matrix" class="mw-redirect" title="Rank of a matrix">rank</a> of the dimensional <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a>. Furthermore, and most importantly, it provides a method for computing these dimensionless parameters from the given variables. A dimensional equation can have the dimensions reduced or eliminated through <a href="/wiki/Nondimensionalization" title="Nondimensionalization">nondimensionalization</a>, which begins with dimensional analysis, and involves scaling quantities by <a href="/wiki/Characteristic_units" class="mw-redirect" title="Characteristic units">characteristic units</a> of a system or <a href="/wiki/Physical_constant" title="Physical constant">physical constants</a> of nature.<a href="#cite_note-Bolster-1">[1]</a>: 43  This may give insight into the fundamental properties of the system, as illustrated in the examples below. The dimension of a <a href="/wiki/Physical_quantity" title="Physical quantity">physical quantity</a> can be expressed as a product of the base physical dimensions such as length, mass and time, each raised to an integer (and occasionally <a href="/wiki/Rational_number" title="Rational number">rational</a>) <a href="/wiki/Power_(mathematics)" class="mw-redirect" title="Power (mathematics)">power</a>. The dimension of a physical quantity is more fundamental than some scale or <a href="/wiki/Units_of_measurement" class="mw-redirect" title="Units of measurement">unit</a> used to express the amount of that physical quantity. For example, mass is a dimension, while the kilogram is a particular reference quantity chosen to express a quantity of mass. The choice of unit is arbitrary, and its choice is often based on historical precedent. <a href="/wiki/Natural_units" title="Natural units">Natural units</a>, being based on only universal constants, may be thought of as being "less arbitrary". There are many possible choices of base physical dimensions. The <a href="/wiki/International_System_of_Units" title="International System of Units">SI standard</a> selects the following dimensions and corresponding dimension symbols: <dl><dd><a href="/wiki/Time" title="Time">time</a> (T), <a href="/wiki/Length" title="Length">length</a> (L), <a href="/wiki/Mass" title="Mass">mass</a> (M), <a href="/wiki/Electric_current" title="Electric current">electric current</a> (I), <a href="/wiki/Absolute_temperature" class="mw-redirect" title="Absolute temperature">absolute temperature</a> (Θ), <a href="/wiki/Amount_of_substance" title="Amount of substance">amount of substance</a> (N) and <a href="/wiki/Luminous_intensity" title="Luminous intensity">luminous intensity</a> (J).</dd></dl> The symbols are by convention usually written in <a href="/wiki/Roman_type" title="Roman type">roman</a> <a href="/wiki/Sans_serif" class="mw-redirect" title="Sans serif">sans serif</a> typeface.<a href="#cite_note-SIBrochure9th-2">[2]</a> Mathematically, the dimension of the quantity Q is given by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {dim} Q={\mathsf {T}}^{a}{\mathsf {L}}^{b}{\mathsf {M}}^{c}{\mathsf {I}}^{d}{\mathsf {\Theta }}^{e}{\mathsf {N}}^{f}{\mathsf {J}}^{g}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>dim</mi> <mo>⁡</mo> <mi>Q</mi> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">M</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">I</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">Θ</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">N</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">J</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {dim} Q={\mathsf {T}}^{a}{\mathsf {L}}^{b}{\mathsf {M}}^{c}{\mathsf {I}}^{d}{\mathsf {\Theta }}^{e}{\mathsf {N}}^{f}{\mathsf {J}}^{g}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b63e51aa884fe81ae58a4b8c4cf7c3c3448242c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:26.505ex; height:3.009ex;" alt="{\displaystyle \operatorname {dim} Q={\mathsf {T}}^{a}{\mathsf {L}}^{b}{\mathsf {M}}^{c}{\mathsf {I}}^{d}{\mathsf {\Theta }}^{e}{\mathsf {N}}^{f}{\mathsf {J}}^{g}}"></dd></dl> where a, b, c, d, e, f, g are the dimensional exponents. Other physical quantities could be defined as the base quantities, as long as they form a <a href="/wiki/Basis_(linear_algebra)" title="Basis (linear algebra)">basis</a> – for instance, one could replace the dimension (I) of <a href="/wiki/Electric_current" title="Electric current">electric current</a> of the SI basis with a dimension (Q) of <a href="/wiki/Electric_charge" title="Electric charge">electric charge</a>, since Q = TI. A quantity that has only b ≠ 0 (with all other exponents zero) is known as a <a href="/wiki/Geometry" title="Geometry">geometric</a> quantity. A quantity that has only both a ≠ 0 and b ≠ 0 is known as a <a href="/wiki/Kinematics" title="Kinematics">kinematic</a> quantity. A quantity that has only all of a ≠ 0, b ≠ 0, and c ≠ 0 is known as a <a href="/wiki/Dynamics_(mechanics)" class="mw-redirect" title="Dynamics (mechanics)">dynamic</a> quantity.<a href="#cite_note-3">[3]</a> A quantity that has all exponents null is said to have dimension one.<a href="#cite_note-SIBrochure9th-2">[2]</a> The unit chosen to express a physical quantity and its dimension are related, but not identical concepts. The units of a physical quantity are defined by convention and related to some standard; e.g., length may have units of metres, feet, inches, miles or micrometres; but any length always has a dimension of L, no matter what units of length are chosen to express it. Two different units of the same physical quantity have <a href="/wiki/Conversion_of_units" title="Conversion of units">conversion factors</a> that relate them. For example, 1 in = 2.54 cm; in this case 2.54 cm/in is the conversion factor, which is itself dimensionless. Therefore, multiplying by that conversion factor does not change the dimensions of a physical quantity. There are also physicists who have cast doubt on the very existence of incompatible fundamental dimensions of physical quantity,<a href="#cite_note-duff-4">[4]</a> although this does not invalidate the usefulness of dimensional analysis. <div class="mw-heading mw-heading3"><h3 id="Simple_cases">Simple cases</h3>[<a href="/w/index.php?title=Dimensional_analysis&action=edit&section=2" title="Edit section: Simple cases">edit</a>]</div> As examples, the dimension of the physical quantity <a href="/wiki/Speed" title="Speed">speed</a> v is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {dim} v={\frac {\text{length}}{\text{time}}}={\frac {\mathsf {L}}{\mathsf {T}}}={\mathsf {T}}^{-1}{\mathsf {L}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>dim</mi> <mo>⁡</mo> <mi>v</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mtext>length</mtext> <mtext>time</mtext> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">L</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">L</mi> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {dim} v={\frac {\text{length}}{\text{time}}}={\frac {\mathsf {L}}{\mathsf {T}}}={\mathsf {T}}^{-1}{\mathsf {L}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d75f48af42dbe8fc30d7d9a42a1b2bb1220d88b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:30.094ex; height:5.343ex;" alt="{\displaystyle \operatorname {dim} v={\frac {\text{length}}{\text{time}}}={\frac {\mathsf {L}}{\mathsf {T}}}={\mathsf {T}}^{-1}{\mathsf {L}}.}"></dd></dl> The dimension of the physical quantity <a href="/wiki/Acceleration" title="Acceleration">acceleration</a> a is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {dim} a={\frac {\text{speed}}{\text{time}}}={\frac {{\mathsf {T}}^{-1}{\mathsf {L}}}{\mathsf {T}}}={\mathsf {T}}^{-2}{\mathsf {L}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>dim</mi> <mo>⁡</mo> <mi>a</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mtext>speed</mtext> <mtext>time</mtext> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">L</mi> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">L</mi> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {dim} a={\frac {\text{speed}}{\text{time}}}={\frac {{\mathsf {T}}^{-1}{\mathsf {L}}}{\mathsf {T}}}={\mathsf {T}}^{-2}{\mathsf {L}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce331869b778547ce9bb4e96396f997989b0d778" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:33.024ex; height:5.676ex;" alt="{\displaystyle \operatorname {dim} a={\frac {\text{speed}}{\text{time}}}={\frac {{\mathsf {T}}^{-1}{\mathsf {L}}}{\mathsf {T}}}={\mathsf {T}}^{-2}{\mathsf {L}}.}"></dd></dl> The dimension of the physical quantity <a href="/wiki/Force_(physics)" class="mw-redirect" title="Force (physics)">force</a> F is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {dim} F={\text{mass}}\times {\text{acceleration}}={\mathsf {M}}\times {\mathsf {T}}^{-2}{\mathsf {L}}={\mathsf {T}}^{-2}{\mathsf {L}}{\mathsf {M}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>dim</mi> <mo>⁡</mo> <mi>F</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>mass</mtext> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>acceleration</mtext> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">M</mi> </mrow> </mrow> <mo>×</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">L</mi> </mrow> </mrow> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">M</mi> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {dim} F={\text{mass}}\times {\text{acceleration}}={\mathsf {M}}\times {\mathsf {T}}^{-2}{\mathsf {L}}={\mathsf {T}}^{-2}{\mathsf {L}}{\mathsf {M}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a76ad3144e7e52d667c6d2b5e2afe85434d6a203" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:52.995ex; height:2.676ex;" alt="{\displaystyle \operatorname {dim} F={\text{mass}}\times {\text{acceleration}}={\mathsf {M}}\times {\mathsf {T}}^{-2}{\mathsf {L}}={\mathsf {T}}^{-2}{\mathsf {L}}{\mathsf {M}}.}"></dd></dl> The dimension of the physical quantity <a href="/wiki/Pressure" title="Pressure">pressure</a> P is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {dim} P={\frac {\text{force}}{\text{area}}}={\frac {{\mathsf {T}}^{-2}{\mathsf {L}}{\mathsf {M}}}{{\mathsf {L}}^{2}}}={\mathsf {T}}^{-2}{\mathsf {L}}^{-1}{\mathsf {M}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>dim</mi> <mo>⁡</mo> <mi>P</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mtext>force</mtext> <mtext>area</mtext> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">M</mi> </mrow> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">M</mi> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {dim} P={\frac {\text{force}}{\text{area}}}={\frac {{\mathsf {T}}^{-2}{\mathsf {L}}{\mathsf {M}}}{{\mathsf {L}}^{2}}}={\mathsf {T}}^{-2}{\mathsf {L}}^{-1}{\mathsf {M}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca9b0281e6e7b8a6a8b491328afe0c70cf5ee51c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:39.224ex; height:6.176ex;" alt="{\displaystyle \operatorname {dim} P={\frac {\text{force}}{\text{area}}}={\frac {{\mathsf {T}}^{-2}{\mathsf {L}}{\mathsf {M}}}{{\mathsf {L}}^{2}}}={\mathsf {T}}^{-2}{\mathsf {L}}^{-1}{\mathsf {M}}.}"></dd></dl> The dimension of the physical quantity <a href="/wiki/Energy" title="Energy">energy</a> E is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {dim} E={\text{force}}\times {\text{displacement}}={\mathsf {T}}^{-2}{\mathsf {L}}{\mathsf {M}}\times {\mathsf {L}}={\mathsf {T}}^{-2}{\mathsf {L}}^{2}{\mathsf {M}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>dim</mi> <mo>⁡</mo> <mi>E</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>force</mtext> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>displacement</mtext> </mrow> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">M</mi> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">L</mi> </mrow> </mrow> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">M</mi> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {dim} E={\text{force}}\times {\text{displacement}}={\mathsf {T}}^{-2}{\mathsf {L}}{\mathsf {M}}\times {\mathsf {L}}={\mathsf {T}}^{-2}{\mathsf {L}}^{2}{\mathsf {M}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5a5f54aaeba457784c9dd116f1f4159e0cbe9ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:56.432ex; height:3.009ex;" alt="{\displaystyle \operatorname {dim} E={\text{force}}\times {\text{displacement}}={\mathsf {T}}^{-2}{\mathsf {L}}{\mathsf {M}}\times {\mathsf {L}}={\mathsf {T}}^{-2}{\mathsf {L}}^{2}{\mathsf {M}}.}"></dd></dl> The dimension of the physical quantity <a href="/wiki/Power_(physics)" title="Power (physics)">power</a> P is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {dim} P={\frac {\text{energy}}{\text{time}}}={\frac {{\mathsf {T}}^{-2}{\mathsf {L}}^{2}{\mathsf {M}}}{\mathsf {T}}}={\mathsf {T}}^{-3}{\mathsf {L}}^{2}{\mathsf {M}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>dim</mi> <mo>⁡</mo> <mi>P</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mtext>energy</mtext> <mtext>time</mtext> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">M</mi> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">M</mi> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {dim} P={\frac {\text{energy}}{\text{time}}}={\frac {{\mathsf {T}}^{-2}{\mathsf {L}}^{2}{\mathsf {M}}}{\mathsf {T}}}={\mathsf {T}}^{-3}{\mathsf {L}}^{2}{\mathsf {M}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d57a232a36309b729817e6531f599348734433a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:40.808ex; height:5.676ex;" alt="{\displaystyle \operatorname {dim} P={\frac {\text{energy}}{\text{time}}}={\frac {{\mathsf {T}}^{-2}{\mathsf {L}}^{2}{\mathsf {M}}}{\mathsf {T}}}={\mathsf {T}}^{-3}{\mathsf {L}}^{2}{\mathsf {M}}.}"></dd></dl> The dimension of the physical quantity <a href="/wiki/Electric_charge" title="Electric charge">electric charge</a> Q is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {dim} Q={\text{current}}\times {\text{time}}={\mathsf {T}}{\mathsf {I}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>dim</mi> <mo>⁡</mo> <mi>Q</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>current</mtext> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>time</mtext> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">I</mi> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {dim} Q={\text{current}}\times {\text{time}}={\mathsf {T}}{\mathsf {I}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a438dc2f4ac1fb9696bed95071e9cd74a724bb7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:29.912ex; height:2.509ex;" alt="{\displaystyle \operatorname {dim} Q={\text{current}}\times {\text{time}}={\mathsf {T}}{\mathsf {I}}.}"></dd></dl> The dimension of the physical quantity <a href="/wiki/Voltage" title="Voltage">voltage</a> V is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {dim} V={\frac {\text{power}}{\text{current}}}={\frac {{\mathsf {T}}^{-3}{\mathsf {L}}^{2}{\mathsf {M}}}{\mathsf {I}}}={\mathsf {T^{-3}}}{\mathsf {L}}^{2}{\mathsf {M}}{\mathsf {I}}^{-1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>dim</mi> <mo>⁡</mo> <mi>V</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mtext>power</mtext> <mtext>current</mtext> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">M</mi> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">I</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="sans-serif">T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mn mathvariant="sans-serif">3</mn> </mrow> </msup> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">M</mi> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">I</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {dim} V={\frac {\text{power}}{\text{current}}}={\frac {{\mathsf {T}}^{-3}{\mathsf {L}}^{2}{\mathsf {M}}}{\mathsf {I}}}={\mathsf {T^{-3}}}{\mathsf {L}}^{2}{\mathsf {M}}{\mathsf {I}}^{-1}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43aa6a14fe493e84f827e442b7377dea91b7a75e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:44.548ex; height:5.676ex;" alt="{\displaystyle \operatorname {dim} V={\frac {\text{power}}{\text{current}}}={\frac {{\mathsf {T}}^{-3}{\mathsf {L}}^{2}{\mathsf {M}}}{\mathsf {I}}}={\mathsf {T^{-3}}}{\mathsf {L}}^{2}{\mathsf {M}}{\mathsf {I}}^{-1}.}"></dd></dl> The dimension of the physical quantity <a href="/wiki/Capacitance" title="Capacitance">capacitance</a> C is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {dim} C={\frac {\text{electric charge}}{\text{electric potential difference}}}={\frac {{\mathsf {T}}{\mathsf {I}}}{{\mathsf {T}}^{-3}{\mathsf {L}}^{2}{\mathsf {M}}{\mathsf {I}}^{-1}}}={\mathsf {T^{4}}}{\mathsf {L^{-2}}}{\mathsf {M^{-1}}}{\mathsf {I^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>dim</mi> <mo>⁡</mo> <mi>C</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mtext>electric charge</mtext> <mtext>electric potential difference</mtext> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">I</mi> </mrow> </mrow> </mrow> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">M</mi> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">I</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="sans-serif">T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="sans-serif">4</mn> </mrow> </msup> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="sans-serif">L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mn mathvariant="sans-serif">2</mn> </mrow> </msup> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="sans-serif">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="sans-serif">−</mo> <mn mathvariant="sans-serif">1</mn> </mrow> </msup> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="sans-serif">I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="sans-serif">2</mn> </mrow> </msup> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {dim} C={\frac {\text{electric charge}}{\text{electric potential difference}}}={\frac {{\mathsf {T}}{\mathsf {I}}}{{\mathsf {T}}^{-3}{\mathsf {L}}^{2}{\mathsf {M}}{\mathsf {I}}^{-1}}}={\mathsf {T^{4}}}{\mathsf {L^{-2}}}{\mathsf {M^{-1}}}{\mathsf {I^{2}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03cf072064b1eeeecb9379a30f92b826873c6e1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:68.326ex; height:5.843ex;" alt="{\displaystyle \operatorname {dim} C={\frac {\text{electric charge}}{\text{electric potential difference}}}={\frac {{\mathsf {T}}{\mathsf {I}}}{{\mathsf {T}}^{-3}{\mathsf {L}}^{2}{\mathsf {M}}{\mathsf {I}}^{-1}}}={\mathsf {T^{4}}}{\mathsf {L^{-2}}}{\mathsf {M^{-1}}}{\mathsf {I^{2}}}.}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Rayleigh's_method">Rayleigh's method</h3>[<a href="/w/index.php?title=Dimensional_analysis&action=edit&section=3" title="Edit section: Rayleigh's method">edit</a>]</div> In dimensional analysis, Rayleigh's method is a conceptual tool used in <a href="/wiki/Physics" title="Physics">physics</a>, <a href="/wiki/Chemistry" title="Chemistry">chemistry</a>, and <a href="/wiki/Engineering" title="Engineering">engineering</a>. It expresses a <a href="/wiki/Functional_relationship" class="mw-redirect" title="Functional relationship">functional relationship</a> of some <a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">variables</a> in the form of an <a href="/wiki/Exponential_equation" class="mw-redirect" title="Exponential equation">exponential equation</a>. It was named after <a href="/wiki/John_William_Strutt,_3rd_Baron_Rayleigh" title="John William Strutt, 3rd Baron Rayleigh">Lord Rayleigh</a>. The method involves the following steps: <ol><li>Gather all the <a href="/wiki/Independent_variable" class="mw-redirect" title="Independent variable">independent variables</a> that are likely to influence the <a href="/wiki/Dependent_variable" class="mw-redirect" title="Dependent variable">dependent variable</a>.</li> <li>If R is a variable that depends upon independent variables R1, R2, R3, ..., Rn, then the <a href="/wiki/Functional_equation" title="Functional equation">functional equation</a> can be written as R = F(R1, R2, R3, ..., Rn).</li> <li>Write the above equation in the form R = C R1a R2b R3c ... Rnm, where C is a <a href="/wiki/Dimensionless_constant" class="mw-redirect" title="Dimensionless constant">dimensionless constant</a> and a, b, c, ..., m are arbitrary exponents.</li> <li>Express each of the quantities in the equation in some <a href="/wiki/Base_unit_(measurement)" class="mw-redirect" title="Base unit (measurement)">base units</a> in which the solution is required.</li> <li>By using <a href="#Dimensional_homogeneity">dimensional homogeneity</a>, obtain a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> of <a href="/wiki/Simultaneous_equations" class="mw-redirect" title="Simultaneous equations">simultaneous equations</a> involving the exponents a, b, c, ..., m.</li> <li><a href="/wiki/Equation_solving" title="Equation solving">Solve</a> these equations to obtain the values of the exponents a, b, c, ..., m.</li> <li><a href="/wiki/Simultaneous_equations#Substitution_method" class="mw-redirect" title="Simultaneous equations">Substitute</a> the values of exponents in the main equation, and form the <a href="/wiki/Non-dimensional" class="mw-redirect" title="Non-dimensional">non-dimensional</a> <a href="/wiki/Parameter" title="Parameter">parameters</a> by <a href="/wiki/Combining_like_terms" class="mw-redirect" title="Combining like terms">grouping</a> the variables with like exponents.</li></ol> As a drawback, Rayleigh's method does not provide any information regarding number of dimensionless groups to be obtained as a result of dimensional analysis. <div class="mw-heading mw-heading2"><h2 id="Concrete_numbers_and_base_units">Concrete numbers and base units</h2>[<a href="/w/index.php?title=Dimensional_analysis&action=edit&section=4" title="Edit section: Concrete numbers and base units">edit</a>]</div> Many parameters and measurements in the physical sciences and engineering are expressed as a <a href="/wiki/Concrete_number" title="Concrete number">concrete number</a>—a numerical quantity and a corresponding dimensional unit. Often a quantity is expressed in terms of several other quantities; for example, speed is a combination of length and time, e.g. 60 kilometres per hour or 1.4 kilometres per second. Compound relations with "per" are expressed with <a href="/wiki/Division_(mathematics)" title="Division (mathematics)">division</a>, e.g. 60 km/h. Other relations can involve <a href="/wiki/Multiplication" title="Multiplication">multiplication</a> (often shown with a <a href="/wiki/Centered_dot" class="mw-redirect" title="Centered dot">centered dot</a> or <a href="/wiki/Juxtaposition#Mathematics" title="Juxtaposition">juxtaposition</a>), powers (like m2 for square metres), or combinations thereof. A set of <a href="/wiki/Base_unit_(measurement)" class="mw-redirect" title="Base unit (measurement)">base units</a> for a <a href="/wiki/System_of_measurement" class="mw-redirect" title="System of measurement">system of measurement</a> is a conventionally chosen set of units, none of which can be expressed as a combination of the others and in terms of which all the remaining units of the system can be expressed.<a href="#cite_note-5">[5]</a> For example, units for <a href="/wiki/Length" title="Length">length</a> and time are normally chosen as base units. Units for <a href="/wiki/Volume" title="Volume">volume</a>, however, can be factored into the base units of length (m3), thus they are considered derived or compound units. Sometimes the names of units obscure the fact that they are derived units. For example, a <a href="/wiki/Newton_(unit)" title="Newton (unit)">newton</a> (N) is a unit of <a href="/wiki/Force" title="Force">force</a>, which may be expressed as the product of mass (with unit kg) and acceleration (with unit m⋅s−2). The newton is defined as 1 N = 1 kg⋅m⋅s−2. <div class="mw-heading mw-heading3"><h3 id="Percentages,_derivatives_and_integrals">Percentages, derivatives and integrals</h3>[<a href="/w/index.php?title=Dimensional_analysis&action=edit&section=5" title="Edit section: Percentages, derivatives and integrals">edit</a>]</div> Percentages are dimensionless quantities, since they are ratios of two quantities with the same dimensions. In other words, the % sign can be read as "hundredths", since 1% = 1/100. Taking a derivative with respect to a quantity divides the dimension by the dimension of the variable that is differentiated with respect to. Thus: <ul><li>position (x) has the dimension L (length);</li> <li>derivative of position with respect to time (dx/dt, <a href="/wiki/Velocity" title="Velocity">velocity</a>) has dimension T−1L—length from position, time due to the gradient;</li> <li>the second derivative (d2x/dt2 = d(dx/dt) / dt, <a href="/wiki/Acceleration" title="Acceleration">acceleration</a>) has dimension T−2L.</li></ul> Likewise, taking an integral adds the dimension of the variable one is integrating with respect to, but in the numerator. <ul><li><a href="/wiki/Force" title="Force">force</a> has the dimension T−2LM (mass multiplied by acceleration);</li> <li>the integral of force with respect to the distance (s) the object has travelled (⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \int F\ ds}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mo>∫</mo> <mi>F</mi> <mtext> </mtext> <mi>d</mi> <mi>s</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \int F\ ds}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de9a1fdc5398782108b63e2e9287ba5d317b0ded" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.433ex; height:3.176ex;" alt="{\displaystyle \textstyle \int F\ ds}">⁠, <a href="/wiki/Work_(physics)#Mathematical_calculation" title="Work (physics)">work</a>) has dimension T−2L2M.</li></ul> In economics, one distinguishes between <a href="/wiki/Stocks_and_flows" class="mw-redirect" title="Stocks and flows">stocks and flows</a>: a stock has a unit (say, widgets or dollars), while a flow is a derivative of a stock, and has a unit of the form of this unit divided by one of time (say, dollars/year). In some contexts, dimensional quantities are expressed as dimensionless quantities or percentages by omitting some dimensions. For example, <a href="/wiki/Debt-to-GDP_ratio" title="Debt-to-GDP ratio">debt-to-GDP ratios</a> are generally expressed as percentages: total debt outstanding (dimension of currency) divided by annual GDP (dimension of currency)—but one may argue that, in comparing a stock to a flow, annual GDP should have dimensions of currency/time (dollars/year, for instance) and thus debt-to-GDP should have the unit year, which indicates that debt-to-GDP is the number of years needed for a constant GDP to pay the debt, if all GDP is spent on the debt and the debt is otherwise unchanged. <div class="mw-heading mw-heading2"><h2 id="Dimensional_homogeneity_(commensurability)">Dimensional homogeneity (commensurability) </h2>[<a href="/w/index.php?title=Dimensional_analysis&action=edit&section=6" title="Edit section: Dimensional homogeneity (commensurability)">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Apples_and_oranges" title="Apples and oranges">Apples and oranges</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Kind_of_quantity" class="mw-redirect" title="Kind of quantity">Kind of quantity</a></div> The most basic rule of dimensional analysis is that of dimensional homogeneity.<a href="#cite_note-6">[6]</a> <style data-mw-deduplicate="TemplateStyles:r996643573">.mw-parser-output .block-indent{padding-left:3em;padding-right:0;overflow:hidden}</style><div class="block-indent">Only commensurable quantities (physical quantities having the same dimension) may be compared, equated, added, or subtracted.</div> However, the dimensions form an <a href="/wiki/Abelian_group" title="Abelian group">abelian group</a> under multiplication, so: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r996643573"><div class="block-indent">One may take ratios of incommensurable quantities (quantities with different dimensions), and multiply or divide them.</div> For example, it makes no sense to ask whether 1 hour is more, the same, or less than 1 kilometre, as these have different dimensions, nor to add 1 hour to 1 kilometre. However, it makes sense to ask whether 1 mile is more, the same, or less than 1 kilometre, being the same dimension of physical quantity even though the units are different. On the other hand, if an object travels 100 km in 2 hours, one may divide these and conclude that the object's average speed was 50 km/h. The rule implies that in a physically meaningful expression only quantities of the same dimension can be added, subtracted, or compared. For example, if mman, mrat and Lman denote, respectively, the mass of some man, the mass of a rat and the length of that man, the dimensionally homogeneous expression mman + mrat is meaningful, but the heterogeneous expression mman + Lman is meaningless. However, mman/L2man is fine. Thus, dimensional analysis may be used as a <a href="/wiki/Sanity_check" title="Sanity check">sanity check</a> of physical equations: the two sides of any equation must be commensurable or have the same dimensions. Even when two physical quantities have identical dimensions, it may nevertheless be meaningless to compare or add them. For example, although <a href="/wiki/Torque" title="Torque">torque</a> and energy share the dimension T−2L2M, they are fundamentally different physical quantities. To compare, add, or subtract quantities with the same dimensions but expressed in different units, the standard procedure is first to convert them all to the same unit. For example, to compare 32 metres with 35 yards, use 1 yard = 0.9144 m to convert 35 yards to 32.004 m. A related principle is that any physical law that accurately describes the real world must be independent of the units used to measure the physical variables.<a href="#cite_note-7">[7]</a> For example, <a href="/wiki/Newton%27s_laws_of_motion" title="Newton's laws of motion">Newton's laws of motion</a> must hold true whether distance is measured in miles or kilometres. This principle gives rise to the form that a conversion factor between two units that measure the same dimension must take multiplication by a simple constant. It also ensures equivalence; for example, if two buildings are the same height in feet, then they must be the same height in metres. <div class="mw-heading mw-heading2"><h2 id="Conversion_factor">Conversion factor</h2>[<a href="/w/index.php?title=Dimensional_analysis&action=edit&section=7" title="Edit section: Conversion factor">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Conversion_factor" class="mw-redirect" title="Conversion factor">Conversion factor</a></div> In dimensional analysis, a ratio which converts one unit of measure into another without changing the quantity is called a <a href="/wiki/Conversion_factor" class="mw-redirect" title="Conversion factor">conversion factor</a>. For example, kPa and bar are both units of pressure, and 100 kPa = 1 bar. The rules of algebra allow both sides of an equation to be divided by the same expression, so this is equivalent to 100 kPa / 1 bar = 1. Since any quantity can be multiplied by 1 without changing it, the expression "100 kPa / 1 bar" can be used to convert from bars to kPa by multiplying it with the quantity to be converted, including the unit. For example, 5 bar × 100 kPa / 1 bar = 500 kPa because 5 × 100 / 1 = 500, and bar/bar cancels out, so 5 bar = 500 kPa. <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2>[<a href="/w/index.php?title=Dimensional_analysis&action=edit&section=8" title="Edit section: Applications">edit</a>]</div> Dimensional analysis is most often used in physics and chemistry – and in the mathematics thereof – but finds some applications outside of those fields as well. <div class="mw-heading mw-heading3"><h3 id="Mathematics">Mathematics</h3>[<a href="/w/index.php?title=Dimensional_analysis&action=edit&section=9" title="Edit section: Mathematics">edit</a>]</div> A simple application of dimensional analysis to mathematics is in computing the form of the <a href="/wiki/N-sphere#Volume_of_the_n-ball" title="N-sphere">volume of an n-ball</a> (the solid ball in n dimensions), or the area of its surface, the <a href="/wiki/N-sphere" title="N-sphere">n-sphere</a>: being an n-dimensional figure, the volume scales as xn, while the surface area, being (n − 1)-dimensional, scales as xn−1. Thus the volume of the n-ball in terms of the radius is Cnrn, for some constant Cn. Determining the constant takes more involved mathematics, but the form can be deduced and checked by dimensional analysis alone. <div class="mw-heading mw-heading3"><h3 id="Finance,_economics,_and_accounting">Finance, economics, and accounting</h3>[<a href="/w/index.php?title=Dimensional_analysis&action=edit&section=10" title="Edit section: Finance, economics, and accounting">edit</a>]</div> In finance, economics, and accounting, dimensional analysis is most commonly referred to in terms of the <a href="/wiki/Stock_and_flow" title="Stock and flow">distinction between stocks and flows</a>. More generally, dimensional analysis is used in interpreting various <a href="/wiki/Financial_ratios" class="mw-redirect" title="Financial ratios">financial ratios</a>, economics ratios, and accounting ratios. <ul><li>For example, the <a href="/wiki/P/E_ratio" class="mw-redirect" title="P/E ratio">P/E ratio</a> has dimensions of time (unit: year), and can be interpreted as "years of earnings to earn the price paid".</li> <li>In economics, <a href="/wiki/Debt-to-GDP_ratio" title="Debt-to-GDP ratio">debt-to-GDP ratio</a> also has the unit year (debt has a unit of currency, GDP has a unit of currency/year).</li> <li><a href="/wiki/Velocity_of_money" title="Velocity of money">Velocity of money</a> has a unit of 1/years (GDP/money supply has a unit of currency/year over currency): how often a unit of currency circulates per year.</li> <li>Annual continuously compounded interest rates and simple interest rates are often expressed as a percentage (adimensional quantity) while time is expressed as an adimensional quantity consisting of the number of years. However, if the time includes year as the unit of measure, the dimension of the rate is 1/year. Of course, there is nothing special (apart from the usual convention) about using year as a unit of time: any other time unit can be used. Furthermore, if rate and time include their units of measure, the use of different units for each is not problematic. In contrast, rate and time need to refer to a common period if they are adimensional. (Note that effective interest rates can only be defined as adimensional quantities.)</li> <li>In financial analysis, <a href="/wiki/Bond_duration" class="mw-redirect" title="Bond duration">bond duration</a> can be defined as (dV/dr)/V, where V is the value of a bond (or portfolio), r is the continuously compounded interest rate and dV/dr is a derivative. From the previous point, the dimension of r is 1/time. Therefore, the dimension of duration is time (usually expressed in years) because dr is in the "denominator" of the derivative.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Fluid_mechanics">Fluid mechanics</h3>[<a href="/w/index.php?title=Dimensional_analysis&action=edit&section=11" title="Edit section: Fluid mechanics">edit</a>]</div> In <a href="/wiki/Fluid_mechanics" title="Fluid mechanics">fluid mechanics</a>, dimensional analysis is performed to obtain dimensionless <a href="/wiki/Buckingham_%CF%80_theorem" title="Buckingham π theorem">pi terms</a> or groups. According to the principles of dimensional analysis, any prototype can be described by a series of these terms or groups that describe the behaviour of the system. Using suitable pi terms or groups, it is possible to develop a similar set of pi terms for a model that has the same dimensional relationships.<a href="#cite_note-8">[8]</a> In other words, pi terms provide a shortcut to developing a model representing a certain prototype. Common dimensionless groups in fluid mechanics include: <ul><li><a href="/wiki/Reynolds_number" title="Reynolds number">Reynolds number</a> (Re), generally important in all types of fluid problems: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Re} ={\frac {\rho \,ud}{\mu }}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> <mi mathvariant="normal">e</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ρ</mi> <mspace width="thinmathspace" /> <mi>u</mi> <mi>d</mi> </mrow> <mi>μ</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Re} ={\frac {\rho \,ud}{\mu }}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2bb62f29538eb15474ddbbb1923b10c09b061705" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:11.459ex; height:5.843ex;" alt="{\displaystyle \mathrm {Re} ={\frac {\rho \,ud}{\mu }}.}"></li> <li><a href="/wiki/Froude_number" title="Froude number">Froude number</a> (Fr), modeling flow with a free surface: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Fr} ={\frac {u}{\sqrt {g\,L}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">F</mi> <mi mathvariant="normal">r</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>u</mi> <msqrt> <mi>g</mi> <mspace width="thinmathspace" /> <mi>L</mi> </msqrt> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Fr} ={\frac {u}{\sqrt {g\,L}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57700331f119ecfd90ea77b0238e484dff5d6dcb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:12.421ex; height:6.009ex;" alt="{\displaystyle \mathrm {Fr} ={\frac {u}{\sqrt {g\,L}}}.}"></li> <li><a href="/wiki/Euler_number_(physics)" title="Euler number (physics)">Euler number</a> (Eu), used in problems in which pressure is of interest: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Eu} ={\frac {\Delta p}{\rho u^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> <mi mathvariant="normal">u</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mi>p</mi> </mrow> <mrow> <mi>ρ</mi> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Eu} ={\frac {\Delta p}{\rho u^{2}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe36061df39f64561eba660adaba0c0b3ca4bc4f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:11.043ex; height:6.009ex;" alt="{\displaystyle \mathrm {Eu} ={\frac {\Delta p}{\rho u^{2}}}.}"></li> <li><a href="/wiki/Mach_number" title="Mach number">Mach number</a> (Ma), important in high speed flows where the velocity approaches or exceeds the local speed of sound: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Ma} ={\frac {u}{c}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> <mi mathvariant="normal">a</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>u</mi> <mi>c</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Ma} ={\frac {u}{c}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc6be14ef82592d3b433dc91a7f3a43c81be89e6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.205ex; height:4.676ex;" alt="{\displaystyle \mathrm {Ma} ={\frac {u}{c}},}"> where c is the local speed of sound.</li></ul> <div class="mw-heading mw-heading2"><h2 id="History">History</h2>[<a href="/w/index.php?title=Dimensional_analysis&action=edit&section=12" title="Edit section: History">edit</a>]</div> The origins of dimensional analysis have been disputed by historians.<a href="#cite_note-9">[9]</a><a href="#cite_note-Martins_1981-10">[10]</a> The first written application of dimensional analysis has been credited to <a href="/wiki/Fran%C3%A7ois_Daviet_de_Foncenex" title="François Daviet de Foncenex">François Daviet</a>, a student of <a href="/wiki/Joseph-Louis_Lagrange" title="Joseph-Louis Lagrange">Joseph-Louis Lagrange</a>, in a 1799 article at the <a href="/wiki/Turin" title="Turin">Turin</a> Academy of Science.<a href="#cite_note-Martins_1981-10">[10]</a> This led to the conclusion that meaningful laws must be homogeneous equations in their various units of measurement, a result which was eventually later formalized in the <a href="/wiki/Buckingham_%CF%80_theorem" title="Buckingham π theorem">Buckingham π theorem</a>. <a href="/wiki/Simeon_Poisson" class="mw-redirect" title="Simeon Poisson">Simeon Poisson</a> also treated the same problem of the <a href="/wiki/Parallelogram_law" title="Parallelogram law">parallelogram law</a> by Daviet, in his treatise of 1811 and 1833 (vol I, p. 39).<a href="#cite_note-11">[11]</a> In the second edition of 1833, Poisson explicitly introduces the term dimension instead of the Daviet homogeneity. In 1822, the important Napoleonic scientist <a href="/wiki/Joseph_Fourier" title="Joseph Fourier">Joseph Fourier</a> made the first credited important contributions<a href="#cite_note-12">[12]</a> based on the idea that physical laws like <a href="/wiki/Newton%27s_second_law" class="mw-redirect" title="Newton's second law">F = ma</a> should be independent of the units employed to measure the physical variables. <a href="/wiki/James_Clerk_Maxwell" title="James Clerk Maxwell">James Clerk Maxwell</a> played a major role in establishing modern use of dimensional analysis by distinguishing mass, length, and time as fundamental units, while referring to other units as derived.<a href="#cite_note-maxwell-13">[13]</a> Although Maxwell defined length, time and mass to be "the three fundamental units", he also noted that gravitational mass can be derived from length and time by assuming a form of <a href="/wiki/Newton%27s_law_of_universal_gravitation" title="Newton's law of universal gravitation">Newton's law of universal gravitation</a> in which the <a href="/wiki/Gravitational_constant" title="Gravitational constant">gravitational constant</a> G is taken as <a href="/wiki/1" title="1">unity</a>, thereby defining M = T−2L3.<a href="#cite_note-maxwell2-14">[14]</a> By assuming a form of <a href="/wiki/Coulomb%27s_law" title="Coulomb's law">Coulomb's law</a> in which the <a href="/wiki/Coulomb_constant" class="mw-redirect" title="Coulomb constant">Coulomb constant</a> ke is taken as unity, Maxwell then determined that the dimensions of an electrostatic unit of charge were Q = T−1L3/2M1/2,<a href="#cite_note-maxwell3-15">[15]</a> which, after substituting his M = T−2L3 equation for mass, results in charge having the same dimensions as mass, viz. Q = T−2L3. Dimensional analysis is also used to derive relationships between the physical quantities that are involved in a particular phenomenon that one wishes to understand and characterize. It was used for the first time in this way in 1872 by <a href="/wiki/Lord_Rayleigh" class="mw-redirect" title="Lord Rayleigh">Lord Rayleigh</a>, who was trying to understand why the sky is blue.<a href="#cite_note-16">[16]</a> Rayleigh first published the technique in his 1877 book The Theory of Sound.<a href="#cite_note-17">[17]</a> The original meaning of the word dimension, in Fourier's Theorie de la Chaleur, was the numerical value of the exponents of the base units. For example, acceleration was considered to have the dimension 1 with respect to the unit of length, and the dimension −2 with respect to the unit of time.<a href="#cite_note-FOOTNOTEFourier1822[httpsbooksgooglecombooksidTDQJAAAAIAAJpgPA156_156]-18">[18]</a> This was slightly changed by Maxwell, who said the dimensions of acceleration are T−2L, instead of just the exponents.<a href="#cite_note-maxwell4-19">[19]</a> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2>[<a href="/w/index.php?title=Dimensional_analysis&action=edit&section=13" title="Edit section: Examples">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="A_simple_example:_period_of_a_harmonic_oscillator">A simple example: period of a harmonic oscillator</h3>[<a href="/w/index.php?title=Dimensional_analysis&action=edit&section=14" title="Edit section: A simple example: period of a harmonic oscillator">edit</a>]</div> What is the period of <a href="/wiki/Harmonic_oscillator" title="Harmonic oscillator">oscillation</a> T of a mass m attached to an ideal linear spring with spring constant k suspended in gravity of strength g? That period is the solution for T of some dimensionless equation in the variables T, m, k, and g. The four quantities have the following dimensions: T [T]; m [M]; k [M/T2]; and g [L/T2]. From these we can form only one dimensionless product of powers of our chosen variables, G1 = T2k/m [T2 · M/T2 / M = 1], and putting G1 = C for some dimensionless constant C gives the dimensionless equation sought. The dimensionless product of powers of variables is sometimes referred to as a dimensionless group of variables; here the term "group" means "collection" rather than mathematical <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a>. They are often called <a href="/wiki/Dimensionless_number" class="mw-redirect" title="Dimensionless number">dimensionless numbers</a> as well. The variable g does not occur in the group. It is easy to see that it is impossible to form a dimensionless product of powers that combines g with k, m, and T, because g is the only quantity that involves the dimension L. This implies that in this problem the g is irrelevant. Dimensional analysis can sometimes yield strong statements about the irrelevance of some quantities in a problem, or the need for additional parameters. If we have chosen enough variables to properly describe the problem, then from this argument we can conclude that the period of the mass on the spring is independent of g: it is the same on the earth or the moon. The equation demonstrating the existence of a product of powers for our problem can be written in an entirely equivalent way: ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T=\kappa {\sqrt {\tfrac {m}{k}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>=</mo> <mi>κ</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>m</mi> <mi>k</mi> </mfrac> </mstyle> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T=\kappa {\sqrt {\tfrac {m}{k}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5ebb262b2a45b53961263dbad0baed52c4b3ee9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:10.676ex; height:4.843ex;" alt="{\displaystyle T=\kappa {\sqrt {\tfrac {m}{k}}}}">⁠, for some dimensionless constant κ (equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {C}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>C</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {C}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2c4a9f5aa72990db3f302b96fb9c389ff151f9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.702ex; height:3.009ex;" alt="{\displaystyle {\sqrt {C}}}"> from the original dimensionless equation). When faced with a case where dimensional analysis rejects a variable (g, here) that one intuitively expects to belong in a physical description of the situation, another possibility is that the rejected variable is in fact relevant, but that some other relevant variable has been omitted, which might combine with the rejected variable to form a dimensionless quantity. That is, however, not the case here. When dimensional analysis yields only one dimensionless group, as here, there are no unknown functions, and the solution is said to be "complete" – although it still may involve unknown dimensionless constants, such as κ. <div class="mw-heading mw-heading3"><h3 id="A_more_complex_example:_energy_of_a_vibrating_wire">A more complex example: energy of a vibrating wire</h3>[<a href="/w/index.php?title=Dimensional_analysis&action=edit&section=15" title="Edit section: A more complex example: energy of a vibrating wire">edit</a>]</div> Consider the case of a vibrating wire of <a href="/wiki/Length" title="Length">length</a> ℓ (L) vibrating with an <a href="/wiki/Amplitude" title="Amplitude">amplitude</a> A (L). The wire has a <a href="/wiki/Linear_density" title="Linear density">linear density</a> ρ (M/L) and is under <a href="/wiki/Tension_(physics)" title="Tension (physics)">tension</a> s (LM/T2), and we want to know the energy E (L2M/T2) in the wire. Let π1 and π2 be two dimensionless products of <a href="/wiki/Power_(mathematics)" class="mw-redirect" title="Power (mathematics)">powers</a> of the variables chosen, given by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\pi _{1}&={\frac {E}{As}}\\\pi _{2}&={\frac {\ell }{A}}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>E</mi> <mrow> <mi>A</mi> <mi>s</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ℓ</mi> <mi>A</mi> </mfrac> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\pi _{1}&={\frac {E}{As}}\\\pi _{2}&={\frac {\ell }{A}}.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b16f0518441549e8e5b49cba80ba81ecd86b999" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.675ex; margin-bottom: -0.329ex; width:9.899ex; height:11.176ex;" alt="{\displaystyle {\begin{aligned}\pi _{1}&={\frac {E}{As}}\\\pi _{2}&={\frac {\ell }{A}}.\end{aligned}}}"></dd></dl> The linear density of the wire is not involved. The two groups found can be combined into an equivalent form as an equation <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F\left({\frac {E}{As}},{\frac {\ell }{A}}\right)=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>E</mi> <mrow> <mi>A</mi> <mi>s</mi> </mrow> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ℓ</mi> <mi>A</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F\left({\frac {E}{As}},{\frac {\ell }{A}}\right)=0,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/160ea2b194ff1cf5693f462eb534ac281b63d9e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:17.74ex; height:6.176ex;" alt="{\displaystyle F\left({\frac {E}{As}},{\frac {\ell }{A}}\right)=0,}"></dd></dl> where F is some unknown function, or, equivalently as <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=Asf\left({\frac {\ell }{A}}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mi>A</mi> <mi>s</mi> <mi>f</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ℓ</mi> <mi>A</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=Asf\left({\frac {\ell }{A}}\right),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4abcbfd2075735058dd91c478886c2403001ef5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.408ex; height:6.176ex;" alt="{\displaystyle E=Asf\left({\frac {\ell }{A}}\right),}"></dd></dl> where f is some other unknown function. Here the unknown function implies that our solution is now incomplete, but dimensional analysis has given us something that may not have been obvious: the energy is proportional to the first power of the tension. Barring further analytical analysis, we might proceed to experiments to discover the form for the unknown function f. But our experiments are simpler than in the absence of dimensional analysis. We'd perform none to verify that the energy is proportional to the tension. Or perhaps we might guess that the energy is proportional to ℓ, and so infer that E = ℓs. The power of dimensional analysis as an aid to experiment and forming hypotheses becomes evident. The power of dimensional analysis really becomes apparent when it is applied to situations, unlike those given above, that are more complicated, the set of variables involved are not apparent, and the underlying equations hopelessly complex. Consider, for example, a small pebble sitting on the bed of a river. If the river flows fast enough, it will actually raise the pebble and cause it to flow along with the water. At what critical velocity will this occur? Sorting out the guessed variables is not so easy as before. But dimensional analysis can be a powerful aid in understanding problems like this, and is usually the very first tool to be applied to complex problems where the underlying equations and constraints are poorly understood. In such cases, the answer may depend on a <a href="/wiki/Dimensionless_number" class="mw-redirect" title="Dimensionless number">dimensionless number</a> such as the <a href="/wiki/Reynolds_number" title="Reynolds number">Reynolds number</a>, which may be interpreted by dimensional analysis. <div class="mw-heading mw-heading3"><h3 id="A_third_example:_demand_versus_capacity_for_a_rotating_disc">A third example: demand versus capacity for a rotating disc</h3>[<a href="/w/index.php?title=Dimensional_analysis&action=edit&section=16" title="Edit section: A third example: demand versus capacity for a rotating disc">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Dimensional_analysis_01.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/88/Dimensional_analysis_01.jpg/330px-Dimensional_analysis_01.jpg" decoding="async" width="330" height="226" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/88/Dimensional_analysis_01.jpg/495px-Dimensional_analysis_01.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/88/Dimensional_analysis_01.jpg/660px-Dimensional_analysis_01.jpg 2x" data-file-width="1067" data-file-height="732" /></a><figcaption>Dimensional analysis and numerical experiments for a rotating disc</figcaption></figure> Consider the case of a thin, solid, parallel-sided rotating disc of axial thickness t (L) and radius R (L). The disc has a density ρ (M/L3), rotates at an angular velocity ω (T−1) and this leads to a stress S (T−2L−1M) in the material. There is a theoretical linear elastic solution, given by Lame, to this problem when the disc is thin relative to its radius, the faces of the disc are free to move axially, and the plane stress constitutive relations can be assumed to be valid. As the disc becomes thicker relative to the radius then the plane stress solution breaks down. If the disc is restrained axially on its free faces then a state of plane strain will occur. However, if this is not the case then the state of stress may only be determined though consideration of three-dimensional elasticity and there is no known theoretical solution for this case. An engineer might, therefore, be interested in establishing a relationship between the five variables. Dimensional analysis for this case leads to the following (5 − 3 = 2) non-dimensional groups: <dl><dd>demand/capacity = ρR2ω2/S</dd> <dd>thickness/radius or aspect ratio = t/R</dd></dl> Through the use of numerical experiments using, for example, the <a href="/wiki/Finite_element_method" title="Finite element method">finite element method</a>, the nature of the relationship between the two non-dimensional groups can be obtained as shown in the figure. As this problem only involves two non-dimensional groups, the complete picture is provided in a single plot and this can be used as a design/assessment chart for rotating discs.<a href="#cite_note-20">[20]</a> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2>[<a href="/w/index.php?title=Dimensional_analysis&action=edit&section=17" title="Edit section: Properties">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Mathematical_properties">Mathematical properties</h3>[<a href="/w/index.php?title=Dimensional_analysis&action=edit&section=18" title="Edit section: Mathematical properties">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Buckingham_%CF%80_theorem" title="Buckingham π theorem">Buckingham π theorem</a></div> The dimensions that can be formed from a given collection of basic physical dimensions, such as T, L, and M, form an <a href="/wiki/Abelian_group" title="Abelian group">abelian group</a>: The <a href="/wiki/Identity_element" title="Identity element">identity</a> is written as 1;[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] L0 = 1, and the inverse of L is 1/L or L−1. L raised to any integer power p is a member of the group, having an inverse of L−p or 1/Lp. The operation of the group is multiplication, having the usual rules for handling exponents (Ln × Lm = Ln+m). Physically, 1/L can be interpreted as <a href="/wiki/Reciprocal_length" title="Reciprocal length">reciprocal length</a>, and 1/T as reciprocal time (see <a href="/wiki/Reciprocal_second" class="mw-redirect" title="Reciprocal second">reciprocal second</a>). An abelian group is equivalent to a <a href="/wiki/Module_(mathematics)" title="Module (mathematics)">module</a> over the integers, with the dimensional symbol TiLjMk corresponding to the tuple (i, j, k). When physical measured quantities (be they like-dimensioned or unlike-dimensioned) are multiplied or divided by one other, their dimensional units are likewise multiplied or divided; this corresponds to addition or subtraction in the module. When measurable quantities are raised to an integer power, the same is done to the dimensional symbols attached to those quantities; this corresponds to <a href="/wiki/Scalar_multiplication" title="Scalar multiplication">scalar multiplication</a> in the module. A basis for such a module of dimensional symbols is called a set of <a href="/wiki/Base_quantity" class="mw-redirect" title="Base quantity">base quantities</a>, and all other vectors are called derived units. As in any module, one may choose different <a href="/wiki/Basis_(linear_algebra)" title="Basis (linear algebra)">bases</a>, which yields different systems of units (e.g., <a href="/wiki/Ampere#Proposed_future_definition" title="Ampere">choosing</a> whether the unit for charge is derived from the unit for current, or vice versa). The group identity, the dimension of dimensionless quantities, corresponds to the origin in this module, (0, 0, 0). In certain cases, one can define fractional dimensions, specifically by formally defining fractional powers of one-dimensional vector spaces, like VL1/2.<sup id="cite_ref-FOOTNOTETao2012"With_a_bit_of_additional_effort_(and_taking_full_advantage_of_the_one-dimensionality_of_the_vector_spaces),_one_can_also_define_spaces_with_fractional_exponents&nbsp;..."_21-0" class="reference"><a href="#cite_note-FOOTNOTETao2012"With_a_bit_of_additional_effort_(and_taking_full_advantage_of_the_one-dimensionality_of_the_vector_spaces),_one_can_also_define_spaces_with_fractional_exponents&nbsp;..."-21">[21]</a> However, it is not possible to take arbitrary fractional powers of units, due to <a href="/wiki/Representation_theory" title="Representation theory">representation-theoretic</a> obstructions.<sup id="cite_ref-FOOTNOTETao2012"However,_when_working_with_vector-valued_quantities_in_two_and_higher_dimensions,_there_are_representation-theoretic_obstructions_to_taking_arbitrary_fractional_powers_of_units&nbsp;..."_22-0" class="reference"><a href="#cite_note-FOOTNOTETao2012"However,_when_working_with_vector-valued_quantities_in_two_and_higher_dimensions,_there_are_representation-theoretic_obstructions_to_taking_arbitrary_fractional_powers_of_units&nbsp;..."-22">[22]</a> One can work with vector spaces with given dimensions without needing to use units (corresponding to coordinate systems of the vector spaces). For example, given dimensions M and L, one has the vector spaces VM and VL, and can define VML := VM ⊗ VL as the <a href="/wiki/Tensor_product" title="Tensor product">tensor product</a>. Similarly, the dual space can be interpreted as having "negative" dimensions.<a href="#cite_note-23">[23]</a> This corresponds to the fact that under the <a href="/wiki/Natural_pairing" class="mw-redirect" title="Natural pairing">natural pairing</a> between a vector space and its dual, the dimensions cancel, leaving a <a href="/wiki/Dimensionless" class="mw-redirect" title="Dimensionless">dimensionless</a> scalar. The set of units of the physical quantities involved in a problem correspond to a set of vectors (or a matrix). The <a href="/wiki/Kernel_(linear_algebra)#nullity" title="Kernel (linear algebra)">nullity</a> describes some number (e.g., m) of ways in which these vectors can be combined to produce a zero vector. These correspond to producing (from the measurements) a number of dimensionless quantities, {π1, ..., πm}. (In fact these ways completely span the null subspace of another different space, of powers of the measurements.) Every possible way of multiplying (and <a href="/wiki/Exponent_(mathematics)" class="mw-redirect" title="Exponent (mathematics)">exponentiating</a>) together the measured quantities to produce something with the same unit as some derived quantity X can be expressed in the general form <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=\prod _{i=1}^{m}(\pi _{i})^{k_{i}}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>=</mo> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msup> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X=\prod _{i=1}^{m}(\pi _{i})^{k_{i}}\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad65a9fa94cc730508862772b02da652c4f7002d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:14.729ex; height:6.843ex;" alt="{\displaystyle X=\prod _{i=1}^{m}(\pi _{i})^{k_{i}}\,.}"></dd></dl> Consequently, every possible <a href="#Commensurability">commensurate</a> equation for the physics of the system can be rewritten in the form <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\pi _{1},\pi _{2},...,\pi _{m})=0\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(\pi _{1},\pi _{2},...,\pi _{m})=0\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56210f927e04df5c215a2f43167f2f8bb8dba8cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.345ex; height:2.843ex;" alt="{\displaystyle f(\pi _{1},\pi _{2},...,\pi _{m})=0\,.}"></dd></dl> Knowing this restriction can be a powerful tool for obtaining new insight into the system. <div class="mw-heading mw-heading3"><h3 id="Mechanics">Mechanics</h3>[<a href="/w/index.php?title=Dimensional_analysis&action=edit&section=19" title="Edit section: Mechanics">edit</a>]</div> The dimension of physical quantities of interest in <a href="/wiki/Mechanics" title="Mechanics">mechanics</a> can be expressed in terms of base dimensions T, L, and M – these form a 3-dimensional vector space. This is not the only valid choice of base dimensions, but it is the one most commonly used. For example, one might choose force, length and mass as the base dimensions (as some have done), with associated dimensions F, L, M; this corresponds to a different basis, and one may convert between these representations by a <a href="/wiki/Change_of_basis" title="Change of basis">change of basis</a>. The choice of the base set of dimensions is thus a convention, with the benefit of increased utility and familiarity. The choice of base dimensions is not entirely arbitrary, because they must form a <a href="/wiki/Basis_(linear_algebra)" title="Basis (linear algebra)">basis</a>: they must <a href="/wiki/Linear_span" title="Linear span">span</a> the space, and be <a href="/wiki/Linearly_independent" class="mw-redirect" title="Linearly independent">linearly independent</a>. For example, F, L, M form a set of fundamental dimensions because they form a basis that is equivalent to T, L, M: the former can be expressed as [F = LM/T2], L, M, while the latter can be expressed as [T = (LM/F)1/2], L, M. On the other hand, length, velocity and time (T, L, V) do not form a set of base dimensions for mechanics, for two reasons: <ul><li>There is no way to obtain mass – or anything derived from it, such as force – without introducing another base dimension (thus, they do not span the space).</li> <li>Velocity, being expressible in terms of length and time (V = L/T), is redundant (the set is not linearly independent).</li></ul> <div class="mw-heading mw-heading3"><h3 id="Other_fields_of_physics_and_chemistry">Other fields of physics and chemistry</h3>[<a href="/w/index.php?title=Dimensional_analysis&action=edit&section=20" title="Edit section: Other fields of physics and chemistry">edit</a>]</div> Depending on the field of physics, it may be advantageous to choose one or another extended set of dimensional symbols. In electromagnetism, for example, it may be useful to use dimensions of T, L, M and Q, where Q represents the dimension of <a href="/wiki/Electric_charge" title="Electric charge">electric charge</a>. In <a href="/wiki/Thermodynamics" title="Thermodynamics">thermodynamics</a>, the base set of dimensions is often extended to include a dimension for temperature, Θ. In chemistry, the <a href="/wiki/Amount_of_substance" title="Amount of substance">amount of substance</a> (the number of molecules divided by the <a href="/wiki/Avogadro_constant" title="Avogadro constant">Avogadro constant</a>, ≈ 6.02×1023 mol−1) is also defined as a base dimension, N. In the interaction of <a href="/wiki/Relativistic_plasma" title="Relativistic plasma">relativistic plasma</a> with strong laser pulses, a dimensionless <a href="/wiki/Relativistic_similarity_parameter" title="Relativistic similarity parameter">relativistic similarity parameter</a>, connected with the symmetry properties of the collisionless <a href="/wiki/Vlasov_equation" title="Vlasov equation">Vlasov equation</a>, is constructed from the plasma-, electron- and critical-densities in addition to the electromagnetic vector potential. The choice of the dimensions or even the number of dimensions to be used in different fields of physics is to some extent arbitrary, but consistency in use and ease of communications are common and necessary features. <div class="mw-heading mw-heading3"><h3 id="Polynomials_and_transcendental_functions">Polynomials and transcendental functions</h3>[<a href="/w/index.php?title=Dimensional_analysis&action=edit&section=21" title="Edit section: Polynomials and transcendental functions">edit</a>]</div> Bridgman's theorem restricts the type of function that can be used to define a physical quantity from general (dimensionally compounded) quantities to only products of powers of the quantities, unless some of the independent quantities are algebraically combined to yield dimensionless groups, whose functions are grouped together in the dimensionless numeric multiplying factor.<a href="#cite_note-24">[24]</a><a href="#cite_note-25">[25]</a> This excludes polynomials of more than one term or transcendental functions not of that form. <a href="/wiki/Scalar_(physics)" title="Scalar (physics)">Scalar</a> arguments to <a href="/wiki/Transcendental_function" title="Transcendental function">transcendental functions</a> such as <a href="/wiki/Exponential_function" title="Exponential function">exponential</a>, <a href="/wiki/Trigonometric_function" class="mw-redirect" title="Trigonometric function">trigonometric</a> and <a href="/wiki/Logarithm" title="Logarithm">logarithmic</a> functions, or to <a href="/wiki/Inhomogeneous_polynomial" class="mw-redirect" title="Inhomogeneous polynomial">inhomogeneous polynomials</a>, must be <a href="/wiki/Dimensionless_number" class="mw-redirect" title="Dimensionless number">dimensionless quantities</a>. (Note: this requirement is somewhat relaxed in Siano's orientational analysis described below, in which the square of certain dimensioned quantities are dimensionless.) While most mathematical identities about dimensionless numbers translate in a straightforward manner to dimensional quantities, care must be taken with logarithms of ratios: the identity log(a/b) = log a − log b, where the logarithm is taken in any base, holds for dimensionless numbers a and b, but it does not hold if a and b are dimensional, because in this case the left-hand side is well-defined but the right-hand side is not.<a href="#cite_note-26">[26]</a> Similarly, while one can evaluate <a href="/wiki/Monomials" class="mw-redirect" title="Monomials">monomials</a> (xn) of dimensional quantities, one cannot evaluate polynomials of mixed degree with dimensionless coefficients on dimensional quantities: for x2, the expression (3 m)2 = 9 m2 makes sense (as an area), while for x2 + x, the expression (3 m)2 + 3 m = 9 m2 + 3 m does not make sense. However, polynomials of mixed degree can make sense if the coefficients are suitably chosen physical quantities that are not dimensionless. For example, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{2}}\cdot (\mathrm {-9.8~m/s^{2}} )\cdot t^{2}+(\mathrm {500~m/s} )\cdot t.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>9.8</mn> <mtext> </mtext> <mi mathvariant="normal">m</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi mathvariant="normal">s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>500</mn> <mtext> </mtext> <mi mathvariant="normal">m</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">s</mi> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>t</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{2}}\cdot (\mathrm {-9.8~m/s^{2}} )\cdot t^{2}+(\mathrm {500~m/s} )\cdot t.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00f068315e8b0b4d644f847699c8aa45c532b82c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:35.047ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{2}}\cdot (\mathrm {-9.8~m/s^{2}} )\cdot t^{2}+(\mathrm {500~m/s} )\cdot t.}"></dd></dl> This is the height to which an object rises in time t if the acceleration of <a href="/wiki/Gravity" title="Gravity">gravity</a> is 9.8 metres per second per second and the initial upward speed is 500 metres per second. It is not necessary for t to be in seconds. For example, suppose t = 0.01 minutes. Then the first term would be <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&{\tfrac {1}{2}}\cdot (\mathrm {-9.8~m/s^{2}} )\cdot (\mathrm {0.01~min} )^{2}\\[10pt]={}&{\tfrac {1}{2}}\cdot -9.8\cdot \left(0.01^{2}\right)(\mathrm {min/s} )^{2}\cdot \mathrm {m} \\[10pt]={}&{\tfrac {1}{2}}\cdot -9.8\cdot \left(0.01^{2}\right)\cdot 60^{2}\cdot \mathrm {m} .\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="1.3em 1.3em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>9.8</mn> <mtext> </mtext> <mi mathvariant="normal">m</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi mathvariant="normal">s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0.01</mn> <mtext> </mtext> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>⋅</mo> <mo>−</mo> <mn>9.8</mn> <mo>⋅</mo> <mrow> <mo>(</mo> <msup> <mn>0.01</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>)</mo> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">s</mi> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>⋅</mo> <mo>−</mo> <mn>9.8</mn> <mo>⋅</mo> <mrow> <mo>(</mo> <msup> <mn>0.01</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>)</mo> </mrow> <mo>⋅</mo> <msup> <mn>60</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&{\tfrac {1}{2}}\cdot (\mathrm {-9.8~m/s^{2}} )\cdot (\mathrm {0.01~min} )^{2}\\[10pt]={}&{\tfrac {1}{2}}\cdot -9.8\cdot \left(0.01^{2}\right)(\mathrm {min/s} )^{2}\cdot \mathrm {m} \\[10pt]={}&{\tfrac {1}{2}}\cdot -9.8\cdot \left(0.01^{2}\right)\cdot 60^{2}\cdot \mathrm {m} .\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69b5bd672290410cfcc7876ecb6d84b36b7b2421" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.338ex; width:33.139ex; height:15.843ex;" alt="{\displaystyle {\begin{aligned}&{\tfrac {1}{2}}\cdot (\mathrm {-9.8~m/s^{2}} )\cdot (\mathrm {0.01~min} )^{2}\\[10pt]={}&{\tfrac {1}{2}}\cdot -9.8\cdot \left(0.01^{2}\right)(\mathrm {min/s} )^{2}\cdot \mathrm {m} \\[10pt]={}&{\tfrac {1}{2}}\cdot -9.8\cdot \left(0.01^{2}\right)\cdot 60^{2}\cdot \mathrm {m} .\end{aligned}}}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Combining_units_and_numerical_values">Combining units and numerical values</h3>[<a href="/w/index.php?title=Dimensional_analysis&action=edit&section=22" title="Edit section: Combining units and numerical values">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Physical_quantity#Components" title="Physical quantity">Physical quantity § Components</a></div> The value of a dimensional physical quantity Z is written as the product of a <a href="/wiki/Unit_of_measurement" title="Unit of measurement">unit</a> [Z] within the dimension and a dimensionless numerical value or numerical factor, n.<a href="#cite_note-Pisanty13-27">[27]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z=n\times [Z]=n[Z]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mo>=</mo> <mi>n</mi> <mo>×</mo> <mo stretchy="false">[</mo> <mi>Z</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mi>n</mi> <mo stretchy="false">[</mo> <mi>Z</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z=n\times [Z]=n[Z]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a301c02a2b018e24efb3c614727ead393565b8b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.455ex; height:2.843ex;" alt="{\displaystyle Z=n\times [Z]=n[Z]}"></dd></dl> When like-dimensioned quantities are added or subtracted or compared, it is convenient to express them in the same unit so that the numerical values of these quantities may be directly added or subtracted. But, in concept, there is no problem adding quantities of the same dimension expressed in different units. For example, 1 metre added to 1 foot is a length, but one cannot derive that length by simply adding 1 and 1. A <a href="/wiki/Conversion_of_units" title="Conversion of units">conversion factor</a>, which is a ratio of like-dimensioned quantities and is equal to the dimensionless unity, is needed: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {1\,ft} =\mathrm {0.3048\,m} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mspace width="thinmathspace" /> <mi mathvariant="normal">f</mi> <mi mathvariant="normal">t</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0.3048</mn> <mspace width="thinmathspace" /> <mi mathvariant="normal">m</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {1\,ft} =\mathrm {0.3048\,m} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9880ba71e05ee42612ad7450e2edc24c5398f547" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.2ex; height:2.176ex;" alt="{\displaystyle \mathrm {1\,ft} =\mathrm {0.3048\,m} }"> is identical to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1={\frac {\mathrm {0.3048\,m} }{\mathrm {1\,ft} }}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mn>0.3048</mn> <mspace width="thinmathspace" /> <mi mathvariant="normal">m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mspace width="thinmathspace" /> <mi mathvariant="normal">f</mi> <mi mathvariant="normal">t</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1={\frac {\mathrm {0.3048\,m} }{\mathrm {1\,ft} }}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52e1f20475d7427b2eab7a17381cdfc1c9acc674" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.526ex; height:5.343ex;" alt="{\displaystyle 1={\frac {\mathrm {0.3048\,m} }{\mathrm {1\,ft} }}.}"></dd></dl> The factor 0.3048 m/ft is identical to the dimensionless 1, so multiplying by this conversion factor changes nothing. Then when adding two quantities of like dimension, but expressed in different units, the appropriate conversion factor, which is essentially the dimensionless 1, is used to convert the quantities to the same unit so that their numerical values can be added or subtracted. Only in this manner is it meaningful to speak of adding like-dimensioned quantities of differing units. <div class="mw-heading mw-heading3"><h3 id="Quantity_equations">Quantity equations</h3>[<a href="/w/index.php?title=Dimensional_analysis&action=edit&section=23" title="Edit section: Quantity equations">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="/wiki/Quantity_theory_of_money" title="Quantity theory of money">Quantity theory of money</a>.</div> A quantity equation, also sometimes called a complete equation, is an equation that remains valid independently of the <a href="/wiki/Unit_of_measurement" title="Unit of measurement">unit of measurement</a> used when expressing the <a href="/wiki/Physical_quantity" title="Physical quantity">physical quantities</a>.<a href="#cite_note-nist-28">[28]</a> In contrast, in a numerical-value equation, just the numerical values of the quantities occur, without units. Therefore, it is only valid when each numerical values is referenced to a specific unit. For example, a quantity equation for <a href="/wiki/Displacement_(geometry)" title="Displacement (geometry)">displacement</a> d as <a href="/wiki/Speed" title="Speed">speed</a> s multiplied by time difference t would be: <dl><dd>d = s t</dd></dl> for s = 5 m/s, where t and d may be expressed in any units, <a href="/wiki/Conversion_of_units" title="Conversion of units">converted</a> if necessary. In contrast, a corresponding numerical-value equation would be: <dl><dd>D = 5 T</dd></dl> where T is the numeric value of t when expressed in seconds and D is the numeric value of d when expressed in metres. Generally, the use of numerical-value equations is discouraged.<a href="#cite_note-nist-28">[28]</a> <div class="mw-heading mw-heading2"><h2 id="Dimensionless_concepts">Dimensionless concepts</h2>[<a href="/w/index.php?title=Dimensional_analysis&action=edit&section=24" title="Edit section: Dimensionless concepts">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Constants">Constants</h3>[<a href="/w/index.php?title=Dimensional_analysis&action=edit&section=25" title="Edit section: Constants">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Dimensionless_quantity" title="Dimensionless quantity">Dimensionless quantity</a></div> The dimensionless constants that arise in the results obtained, such as the C in the Poiseuille's Law problem and the κ in the spring problems discussed above, come from a more detailed analysis of the underlying physics and often arise from integrating some differential equation. Dimensional analysis itself has little to say about these constants, but it is useful to know that they very often have a magnitude of order unity. This observation can allow one to sometimes make "<a href="/wiki/Back_of_the_envelope" class="mw-redirect" title="Back of the envelope">back of the envelope</a>" calculations about the phenomenon of interest, and therefore be able to more efficiently design experiments to measure it, or to judge whether it is important, etc. <div class="mw-heading mw-heading3"><h3 id="Formalisms">Formalisms</h3>[<a href="/w/index.php?title=Dimensional_analysis&action=edit&section=26" title="Edit section: Formalisms">edit</a>]</div> Paradoxically, dimensional analysis can be a useful tool even if all the parameters in the underlying theory are dimensionless, e.g., lattice models such as the <a href="/wiki/Ising_model" title="Ising model">Ising model</a> can be used to study phase transitions and critical phenomena. Such models can be formulated in a purely dimensionless way. As we approach the critical point closer and closer, the distance over which the variables in the lattice model are correlated (the so-called correlation length, χ) becomes larger and larger. Now, the correlation length is the relevant length scale related to critical phenomena, so one can, e.g., surmise on "dimensional grounds" that the non-analytical part of the free energy per lattice site should be ~ 1/χd, where d is the dimension of the lattice. It has been argued by some physicists, e.g., <a href="/wiki/Michael_Duff_(physicist)" title="Michael Duff (physicist)">Michael J. Duff</a>,<a href="#cite_note-duff-4">[4]</a><a href="#cite_note-29">[29]</a> that the laws of physics are inherently dimensionless. The fact that we have assigned incompatible dimensions to Length, Time and Mass is, according to this point of view, just a matter of convention, borne out of the fact that before the advent of modern physics, there was no way to relate mass, length, and time to each other. The three independent dimensionful constants: <a href="/wiki/Speed_of_light" title="Speed of light">c</a>, <a href="/wiki/Planck_constant" title="Planck constant">ħ</a>, and <a href="/wiki/Gravitational_constant" title="Gravitational constant">G</a>, in the fundamental equations of physics must then be seen as mere conversion factors to convert Mass, Time and Length into each other. Just as in the case of critical properties of lattice models, one can recover the results of dimensional analysis in the appropriate scaling limit; e.g., dimensional analysis in mechanics can be derived by reinserting the constants ħ, c, and G (but we can now consider them to be dimensionless) and demanding that a nonsingular relation between quantities exists in the limit c → ∞, ħ → 0 and G → 0. In problems involving a gravitational field the latter limit should be taken such that the field stays finite. <div class="mw-heading mw-heading2"><h2 id="Dimensional_equivalences">Dimensional equivalences</h2>[<a href="/w/index.php?title=Dimensional_analysis&action=edit&section=27" title="Edit section: Dimensional equivalences">edit</a>]</div> Following are tables of commonly occurring expressions in physics, related to the dimensions of energy, momentum, and force.<a href="#cite_note-30">[30]</a><a href="#cite_note-31">[31]</a><a href="#cite_note-Martin08-32">[32]</a> <div class="mw-heading mw-heading3"><h3 id="SI_units">SI units</h3>[<a href="/w/index.php?title=Dimensional_analysis&action=edit&section=28" title="Edit section: SI units">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/International_System_of_Units" title="International System of Units">International System of Units</a></div> <table class="wikitable"> <tbody><tr> <th scope="col" style="width:100px;">Energy, E T−2L2M </th> <th scope="col" style="width:100px;">Expression </th> <th scope="col" style="width:350px;">Nomenclature </th></tr> <tr> <td rowspan="4">Mechanical </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Fd}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Fd}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4341fc3d6f7042fe530b3e46f827478a3e1aa3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.957ex; height:2.176ex;" alt="{\displaystyle Fd}"> </td> <td>F = <a href="/wiki/Force" title="Force">force</a>, d = <a href="/wiki/Distance" title="Distance">distance</a> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S/t\equiv Pt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>t</mi> <mo>≡</mo> <mi>P</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S/t\equiv Pt}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fcfdae8d8f8494479da8e6d28f4cd7ca003299b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.185ex; height:2.843ex;" alt="{\displaystyle S/t\equiv Pt}"> </td> <td>S = <a href="/wiki/Action_(physics)" title="Action (physics)">action</a>, t = time, P = <a href="/wiki/Power_(physics)" title="Power (physics)">power</a> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle mv^{2}\equiv pv\equiv p^{2}/m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>≡</mo> <mi>p</mi> <mi>v</mi> <mo>≡</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle mv^{2}\equiv pv\equiv p^{2}/m}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc4fe64c19cf0d7ef4aa36dbbb56f11e9a6014d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.143ex; height:3.176ex;" alt="{\displaystyle mv^{2}\equiv pv\equiv p^{2}/m}"> </td> <td>m = <a href="/wiki/Mass" title="Mass">mass</a>, v = <a href="/wiki/Velocity" title="Velocity">velocity</a>, p = <a href="/wiki/Momentum" title="Momentum">momentum</a> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I\omega ^{2}\equiv L\omega \equiv L^{2}/I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>≡</mo> <mi>L</mi> <mi>ω</mi> <mo>≡</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I\omega ^{2}\equiv L\omega \equiv L^{2}/I}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/221126104fe8f34f17a1f3e02cfc2a615e6b3c44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.869ex; height:3.176ex;" alt="{\displaystyle I\omega ^{2}\equiv L\omega \equiv L^{2}/I}"> </td> <td>L = <a href="/wiki/Angular_momentum" title="Angular momentum">angular momentum</a>, I = <a href="/wiki/Moment_of_inertia" title="Moment of inertia">moment of inertia</a>, ω = <a href="/wiki/Angular_velocity" title="Angular velocity">angular velocity</a> </td></tr> <tr> <td>Ideal gases </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle pV\equiv NT}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mi>V</mi> <mo>≡</mo> <mi>N</mi> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle pV\equiv NT}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4663267c271c99cabd8d2e47b052fe5bec973c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:9.844ex; height:2.509ex;" alt="{\displaystyle pV\equiv NT}"> </td> <td>p = pressure, V = volume, T = temperature, N = <a href="/wiki/Amount_of_substance" title="Amount of substance">amount of substance</a> </td></tr> <tr> <td>Waves </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle AIt\equiv ASt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>I</mi> <mi>t</mi> <mo>≡</mo> <mi>A</mi> <mi>S</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle AIt\equiv ASt}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0f6510d54783c2ded7fb4bee5851676489a9d28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.935ex; height:2.176ex;" alt="{\displaystyle AIt\equiv ASt}"> </td> <td>A = <a href="/wiki/Area" title="Area">area</a> of <a href="/wiki/Huygens%E2%80%93Fresnel_principle" title="Huygens–Fresnel principle">wave front</a>, I = wave <a href="/wiki/Intensity_(physics)" title="Intensity (physics)">intensity</a>, t = <a href="/wiki/Time" title="Time">time</a>, S = <a href="/wiki/Poynting_vector" title="Poynting vector">Poynting vector</a> </td></tr> <tr> <td rowspan="3">Electromagnetic </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q\phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q\phi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/788fd1628f0f3ecf23ad1e662119c00043f1f322" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.455ex; height:2.509ex;" alt="{\displaystyle q\phi }"> </td> <td>q = <a href="/wiki/Electric_charge" title="Electric charge">electric charge</a>, ϕ = <a href="/wiki/Electric_potential" title="Electric potential">electric potential</a> (for changes this is <a href="/wiki/Voltage" title="Voltage">voltage</a>) </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon E^{2}V\equiv B^{2}V/\mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>V</mi> <mo>≡</mo> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon E^{2}V\equiv B^{2}V/\mu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39c658c66f158c728a1aea97161eabb5c12633ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.987ex; height:3.176ex;" alt="{\displaystyle \varepsilon E^{2}V\equiv B^{2}V/\mu }"> </td> <td>E = <a href="/wiki/Electric_field" title="Electric field">electric field</a>, B = <a href="/wiki/Magnetic_field" title="Magnetic field">magnetic field</a>, ε = <a href="/wiki/Permittivity" title="Permittivity">permittivity</a>, μ = <a href="/wiki/Permeability_(electromagnetism)" title="Permeability (electromagnetism)">permeability</a>, V = 3d <a href="/wiki/Volume" title="Volume">volume</a> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle pE\equiv mB\equiv IAB}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mi>E</mi> <mo>≡</mo> <mi>m</mi> <mi>B</mi> <mo>≡</mo> <mi>I</mi> <mi>A</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle pE\equiv mB\equiv IAB}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a02e8e5ebc7db1187a1651d4957365c9888a5631" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:17.715ex; height:2.509ex;" alt="{\displaystyle pE\equiv mB\equiv IAB}"> </td> <td>p = <a href="/wiki/Electric_dipole_moment" title="Electric dipole moment">electric dipole moment</a>, m = magnetic moment, A = area (bounded by a current loop), I = <a href="/wiki/Electric_current" title="Electric current">electric current</a> in loop </td></tr></tbody></table> <table class="wikitable"> <tbody><tr> <th scope="col" style="width:100px;">Momentum, p T−1LM </th> <th scope="col" style="width:100px;">Expression </th> <th scope="col" style="width:350px;">Nomenclature </th></tr> <tr> <td rowspan="2">Mechanical </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle mv\equiv Ft}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mi>v</mi> <mo>≡</mo> <mi>F</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle mv\equiv Ft}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2cfaa0a56cff383603db63a441d85041a48a313c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.847ex; height:2.176ex;" alt="{\displaystyle mv\equiv Ft}"> </td> <td>m = mass, v = velocity, F = force, t = time </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S/r\equiv L/r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>r</mi> <mo>≡</mo> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S/r\equiv L/r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/525943bd549c2065cb15857a227643e81d129e92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.603ex; height:2.843ex;" alt="{\displaystyle S/r\equiv L/r}"> </td> <td>S = action, L = angular momentum, r = <a href="/wiki/Displacement_(vector)" class="mw-redirect" title="Displacement (vector)">displacement</a> </td></tr> <tr> <td>Thermal </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m{\sqrt {\left\langle v^{2}\right\rangle }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow> <mo>⟨</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⟩</mo> </mrow> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m{\sqrt {\left\langle v^{2}\right\rangle }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab78cfbf5be9137387006a6de82b01aa572a656b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.741ex; height:4.843ex;" alt="{\displaystyle m{\sqrt {\left\langle v^{2}\right\rangle }}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\left\langle v^{2}\right\rangle }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow> <mo>⟨</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⟩</mo> </mrow> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\left\langle v^{2}\right\rangle }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f77c4f373c76bce15f45faa9dc4ad87da3977056" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.7ex; height:4.843ex;" alt="{\displaystyle {\sqrt {\left\langle v^{2}\right\rangle }}}"> = <a href="/wiki/Root_mean_square_velocity" class="mw-redirect" title="Root mean square velocity">root mean square velocity</a>, m = mass (of a molecule) </td></tr> <tr> <td>Waves </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho Vv}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mi>V</mi> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho Vv}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94bb528cde8928d89ff558ffcf1ee8270a5a65c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.117ex; height:2.676ex;" alt="{\displaystyle \rho Vv}"> </td> <td>ρ = <a href="/wiki/Density" title="Density">density</a>, V = <a href="/wiki/Volume" title="Volume">volume</a>, v = <a href="/wiki/Phase_velocity" title="Phase velocity">phase velocity</a> </td></tr> <tr> <td>Electromagnetic </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle qA}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle qA}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/657b35ea84812416259a0be0fd54980d5e209a50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.813ex; height:2.509ex;" alt="{\displaystyle qA}"> </td> <td>A = <a href="/wiki/Magnetic_vector_potential" title="Magnetic vector potential">magnetic vector potential</a> </td></tr></tbody></table> <table class="wikitable"> <tbody><tr> <th scope="col" style="width:100px;">Force, F T−2LM </th> <th scope="col" style="width:100px;">Expression </th> <th scope="col" style="width:350px;">Nomenclature </th></tr> <tr> <td>Mechanical </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ma\equiv p/t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mi>a</mi> <mo>≡</mo> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ma\equiv p/t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a623c77b846e5702838c98061ad6c05c54f42364" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.54ex; height:2.843ex;" alt="{\displaystyle ma\equiv p/t}"> </td> <td>m = mass, a = acceleration </td></tr> <tr> <td>Thermal </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T\delta S/\delta r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mi>δ</mi> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>δ</mi> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T\delta S/\delta r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8c4b6c235c95f506897802e37000652273e290a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.444ex; height:2.843ex;" alt="{\displaystyle T\delta S/\delta r}"> </td> <td>S = entropy, T = temperature, r = displacement (see <a href="/wiki/Entropic_force" title="Entropic force">entropic force</a>) </td></tr> <tr> <td>Electromagnetic </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Eq\equiv Bqv}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mi>q</mi> <mo>≡</mo> <mi>B</mi> <mi>q</mi> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Eq\equiv Bqv}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e7b8ee629533fe114953c7f0b90663b8d29a666" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.905ex; height:2.509ex;" alt="{\displaystyle Eq\equiv Bqv}"> </td> <td>E = electric field, B = magnetic field, v = velocity, q = charge </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Programming_languages">Programming languages</h2>[<a href="/w/index.php?title=Dimensional_analysis&action=edit&section=29" title="Edit section: Programming languages">edit</a>]</div> Dimensional correctness as part of <a href="/wiki/Type_system#Type_checking" title="Type system">type checking</a> has been studied since 1977.<a href="#cite_note-33">[33]</a> Implementations for Ada<a href="#cite_note-34">[34]</a> and C++<a href="#cite_note-35">[35]</a> were described in 1985 and 1988. Kennedy's 1996 thesis describes an implementation in <a href="/wiki/Standard_ML" title="Standard ML">Standard ML</a>,<a href="#cite_note-36">[36]</a> and later in <a href="/wiki/F_Sharp_(programming_language)" title="F Sharp (programming language)">F#</a>.<a href="#cite_note-37">[37]</a> There are implementations for <a href="/wiki/Haskell" title="Haskell">Haskell</a>,<a href="#cite_note-38">[38]</a> <a href="/wiki/OCaml" title="OCaml">OCaml</a>,<a href="#cite_note-39">[39]</a> and <a href="/wiki/Rust_(programming_language)" title="Rust (programming language)">Rust</a>,<a href="#cite_note-40">[40]</a> Python,<a href="#cite_note-41">[41]</a> and a code checker for <a href="/wiki/Fortran" title="Fortran">Fortran</a>.<a href="#cite_note-42">[42]</a><a href="#cite_note-43">[43]</a> Griffioen's 2019 thesis extended Kennedy's <a href="/wiki/Hindley%E2%80%93Milner_type_system" title="Hindley–Milner type system">Hindley–Milner type system</a> to support Hart's matrices.<a href="#cite_note-44">[44]</a><a href="#cite_note-45">[45]</a> McBride and Nordvall-Forsberg show how to use <a href="/wiki/Dependent_type" title="Dependent type">dependent types</a> to extend type systems for units of measure.<a href="#cite_note-46">[46]</a> Mathematica 13.2 has a function for transformations with quantities named NondimensionalizationTransform that applies a nondimensionalization transform to an equation.<a href="#cite_note-47">[47]</a> Mathematica also has a function to find the dimensions of a unit such as 1 J named UnitDimensions.<a href="#cite_note-48">[48]</a> Mathematica also has a function that will find dimensionally equivalent combinations of a subset of physical quantities named DimensionalCombations.<a href="#cite_note-49">[49]</a> Mathematica can also factor out certain dimension with UnitDimensions by specifying an argument to the function UnityDimensions.<a href="#cite_note-:0-50">[50]</a> For example, you can use UnityDimensions to factor out angles.<a href="#cite_note-:0-50">[50]</a> In addition to UnitDimensions, Mathematica can find the dimensions of a QuantityVariable with the function QuantityVariableDimensions.<a href="#cite_note-51">[51]</a> <div class="mw-heading mw-heading2"><h2 id="Geometry:_position_vs._displacement">Geometry: position vs. displacement</h2>[<a href="/w/index.php?title=Dimensional_analysis&action=edit&section=30" title="Edit section: Geometry: position vs. displacement">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Affine_quantities">Affine quantities</h3>[<a href="/w/index.php?title=Dimensional_analysis&action=edit&section=31" title="Edit section: Affine quantities">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Affine_space" title="Affine space">Affine space</a></div> Some discussions of dimensional analysis implicitly describe all quantities as mathematical vectors. In mathematics scalars are considered a special case of vectors;[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] vectors can be added to or subtracted from other vectors, and, inter alia, multiplied or divided by scalars. If a vector is used to define a position, this assumes an implicit point of reference: an <a href="/wiki/Origin_(mathematics)" title="Origin (mathematics)">origin</a>. While this is useful and often perfectly adequate, allowing many important errors to be caught, it can fail to model certain aspects of physics. A more rigorous approach requires distinguishing between position and displacement (or moment in time versus duration, or absolute temperature versus temperature change). Consider points on a line, each with a position with respect to a given origin, and distances among them. Positions and displacements all have units of length, but their meaning is not interchangeable: <ul><li>adding two displacements should yield a new displacement (walking ten paces then twenty paces gets you thirty paces forward),</li> <li>adding a displacement to a position should yield a new position (walking one block down the street from an intersection gets you to the next intersection),</li> <li>subtracting two positions should yield a displacement,</li> <li>but one may not add two positions.</li></ul> This illustrates the subtle distinction between affine quantities (ones modeled by an <a href="/wiki/Affine_space" title="Affine space">affine space</a>, such as position) and vector quantities (ones modeled by a <a href="/wiki/Vector_space" title="Vector space">vector space</a>, such as displacement). <ul><li>Vector quantities may be added to each other, yielding a new vector quantity, and a vector quantity may be added to a suitable affine quantity (a vector space <a href="/wiki/Group_action_(mathematics)" class="mw-redirect" title="Group action (mathematics)">acts on</a> an affine space), yielding a new affine quantity.</li> <li>Affine quantities cannot be added, but may be subtracted, yielding relative quantities which are vectors, and these relative differences may then be added to each other or to an affine quantity.</li></ul> Properly then, positions have dimension of affine length, while displacements have dimension of vector length. To assign a number to an affine unit, one must not only choose a unit of measurement, but also a <a href="/wiki/Origin_(mathematics)" title="Origin (mathematics)">point of reference</a>, while to assign a number to a vector unit only requires a unit of measurement. Thus some physical quantities are better modeled by vectorial quantities while others tend to require affine representation, and the distinction is reflected in their dimensional analysis. This distinction is particularly important in the case of temperature, for which the numeric value of <a href="/wiki/Absolute_zero" title="Absolute zero">absolute zero</a> is not the origin 0 in some scales. For absolute zero, <dl><dd>−273.15 °C ≘ 0 K = 0 °R ≘ −459.67 °F,</dd></dl> where the symbol ≘ means corresponds to, since although these values on the respective temperature scales correspond, they represent distinct quantities in the same way that the distances from distinct starting points to the same end point are distinct quantities, and cannot in general be equated. For temperature differences, <dl><dd>1 K = 1 °C ≠ 1 °F = 1 °R.</dd></dl> (Here °R refers to the <a href="/wiki/Rankine_scale" title="Rankine scale">Rankine scale</a>, not the <a href="/wiki/R%C3%A9aumur_scale" title="Réaumur scale">Réaumur scale</a>). Unit conversion for temperature differences is simply a matter of multiplying by, e.g., 1 °F / 1 K (although the ratio is not a constant value). But because some of these scales have origins that do not correspond to absolute zero, conversion from one temperature scale to another requires accounting for that. As a result, simple dimensional analysis can lead to errors if it is ambiguous whether 1 K means the absolute temperature equal to −272.15 °C, or the temperature difference equal to 1 °C. <div class="mw-heading mw-heading3"><h3 id="Orientation_and_frame_of_reference">Orientation and frame of reference</h3>[<a href="/w/index.php?title=Dimensional_analysis&action=edit&section=32" title="Edit section: Orientation and frame of reference">edit</a>]</div> Similar to the issue of a point of reference is the issue of orientation: a displacement in 2 or 3 dimensions is not just a length, but is a length together with a direction. (In 1 dimension, this issue is equivalent to the distinction between positive and negative.) Thus, to compare or combine two dimensional quantities in multi-dimensional Euclidean space, one also needs a bearing: they need to be compared to a <a href="/wiki/Frame_of_reference" title="Frame of reference">frame of reference</a>. This leads to the <a href="#Extensions">extensions</a> discussed below, namely Huntley's directed dimensions and Siano's orientational analysis. <div class="mw-heading mw-heading3"><h3 id="Huntley's_extensions">Huntley's extensions</h3>[<a href="/w/index.php?title=Dimensional_analysis&action=edit&section=33" title="Edit section: Huntley's extensions">edit</a>]</div> Huntley has pointed out that a dimensional analysis can become more powerful by discovering new independent dimensions in the quantities under consideration, thus increasing the rank <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"> of the dimensional matrix.<a href="#cite_note-52">[52]</a> He introduced two approaches: <ul><li>The magnitudes of the components of a vector are to be considered dimensionally independent. For example, rather than an undifferentiated length dimension L, we may have Lx represent dimension in the x-direction, and so forth. This requirement stems ultimately from the requirement that each component of a physically meaningful equation (scalar, vector, or tensor) must be dimensionally consistent.</li> <li>Mass as a measure of the quantity of matter is to be considered dimensionally independent from mass as a measure of inertia.</li></ul> <div class="mw-heading mw-heading4"><h4 id="Directed_dimensions">Directed dimensions</h4>[<a href="/w/index.php?title=Dimensional_analysis&action=edit&section=34" title="Edit section: Directed dimensions">edit</a>]</div> As an example of the usefulness of the first approach, suppose we wish to calculate the <a href="/wiki/Trajectory#Range_and_height" title="Trajectory">distance a cannonball travels</a> when fired with a vertical velocity component <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{\text{y}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>y</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{\text{y}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0f36a81a82eeb0ce3efa733e5017794d8e4220e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.228ex; height:2.343ex;" alt="{\displaystyle v_{\text{y}}}"> and a horizontal velocity component ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{\text{x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>x</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{\text{x}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b3e262033b4eebdbb4568c08311ca11ad013028" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.228ex; height:2.009ex;" alt="{\displaystyle v_{\text{x}}}">⁠, assuming it is fired on a flat surface. Assuming no use of directed lengths, the quantities of interest are then R, the distance travelled, with dimension L, ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{\text{x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>x</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{\text{x}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b3e262033b4eebdbb4568c08311ca11ad013028" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.228ex; height:2.009ex;" alt="{\displaystyle v_{\text{x}}}">⁠, ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{\text{y}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>y</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{\text{y}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0f36a81a82eeb0ce3efa733e5017794d8e4220e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.228ex; height:2.343ex;" alt="{\displaystyle v_{\text{y}}}">⁠, both dimensioned as T−1L, and g the downward acceleration of gravity, with dimension T−2L. With these four quantities, we may conclude that the equation for the range R may be written: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\propto v_{\text{x}}^{a}\,v_{\text{y}}^{b}\,g^{c}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>∝</mo> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>x</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msubsup> <mspace width="thinmathspace" /> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>y</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mspace width="thinmathspace" /> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R\propto v_{\text{x}}^{a}\,v_{\text{y}}^{b}\,g^{c}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52a799c5039f2c298544c83cf4dcab4e623debf1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.803ex; height:3.009ex;" alt="{\displaystyle R\propto v_{\text{x}}^{a}\,v_{\text{y}}^{b}\,g^{c}.}"></dd></dl> Or dimensionally <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {L}}=\left({\mathsf {T}}^{-1}{\mathsf {L}}\right)^{a+b}\left({\mathsf {T}}^{-2}{\mathsf {L}}\right)^{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">L</mi> </mrow> </mrow> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">L</mi> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>+</mo> <mi>b</mi> </mrow> </msup> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">L</mi> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {L}}=\left({\mathsf {T}}^{-1}{\mathsf {L}}\right)^{a+b}\left({\mathsf {T}}^{-2}{\mathsf {L}}\right)^{c}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4816433cb2ceafb2eb9cb824002b27caf059a89d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.999ex; height:3.843ex;" alt="{\displaystyle {\mathsf {L}}=\left({\mathsf {T}}^{-1}{\mathsf {L}}\right)^{a+b}\left({\mathsf {T}}^{-2}{\mathsf {L}}\right)^{c}}"></dd></dl> from which we may deduce that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+b+c=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>+</mo> <mi>c</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+b+c=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c0020e7d6928791bd699105650f592c7646a1d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.176ex; height:2.343ex;" alt="{\displaystyle a+b+c=1}"> and ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+b+2c=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>+</mo> <mn>2</mn> <mi>c</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+b+2c=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3692070878cb1ba5581ba5a6fc383efa357ce328" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.338ex; height:2.343ex;" alt="{\displaystyle a+b+2c=0}">⁠, which leaves one exponent undetermined. This is to be expected since we have two fundamental dimensions T and L, and four parameters, with one equation. However, if we use directed length dimensions, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{\mathrm {x} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">x</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{\mathrm {x} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3578b981b6a3ca532e8b41e4ce4b539c6a8140cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.228ex; height:2.009ex;" alt="{\displaystyle v_{\mathrm {x} }}"> will be dimensioned as T−1Lx, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{\mathrm {y} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">y</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{\mathrm {y} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4dd81310a6f5b82062901c1ca2e895bc31f928ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.228ex; height:2.343ex;" alt="{\displaystyle v_{\mathrm {y} }}"> as T−1Ly, R as Lx and g as T−2Ly. The dimensional equation becomes: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {L}}_{\mathrm {x} }=\left({{\mathsf {T}}^{-1}}{{\mathsf {L}}_{\mathrm {x} }}\right)^{a}\left({{\mathsf {T}}^{-1}}{{\mathsf {L}}_{\mathrm {y} }}\right)^{b}\left({{\mathsf {T}}^{-2}}{{\mathsf {L}}_{\mathrm {y} }}\right)^{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">x</mi> </mrow> </mrow> </msub> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">x</mi> </mrow> </mrow> </msub> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> <msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">y</mi> </mrow> </mrow> </msub> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msup> <msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">y</mi> </mrow> </mrow> </msub> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {L}}_{\mathrm {x} }=\left({{\mathsf {T}}^{-1}}{{\mathsf {L}}_{\mathrm {x} }}\right)^{a}\left({{\mathsf {T}}^{-1}}{{\mathsf {L}}_{\mathrm {y} }}\right)^{b}\left({{\mathsf {T}}^{-2}}{{\mathsf {L}}_{\mathrm {y} }}\right)^{c}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58537db8ab43c0b798e808a05a2e0e9735102b56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:33.659ex; height:3.843ex;" alt="{\displaystyle {\mathsf {L}}_{\mathrm {x} }=\left({{\mathsf {T}}^{-1}}{{\mathsf {L}}_{\mathrm {x} }}\right)^{a}\left({{\mathsf {T}}^{-1}}{{\mathsf {L}}_{\mathrm {y} }}\right)^{b}\left({{\mathsf {T}}^{-2}}{{\mathsf {L}}_{\mathrm {y} }}\right)^{c}}"></dd></dl> and we may solve completely as a = 1, b = 1 and c = −1. The increase in deductive power gained by the use of directed length dimensions is apparent. Huntley's concept of directed length dimensions however has some serious limitations: <ul><li>It does not deal well with vector equations involving the <a href="/wiki/Cross_product" title="Cross product">cross product</a>,</li> <li>nor does it handle well the use of angles as physical variables.</li></ul> It also is often quite difficult to assign the L, Lx, Ly, Lz, symbols to the physical variables involved in the problem of interest. He invokes a procedure that involves the "symmetry" of the physical problem. This is often very difficult to apply reliably: It is unclear as to what parts of the problem that the notion of "symmetry" is being invoked. Is it the symmetry of the physical body that forces are acting upon, or to the points, lines or areas at which forces are being applied? What if more than one body is involved with different symmetries? Consider the spherical bubble attached to a cylindrical tube, where one wants the flow rate of air as a function of the pressure difference in the two parts. What are the Huntley extended dimensions of the viscosity of the air contained in the connected parts? What are the extended dimensions of the pressure of the two parts? Are they the same or different? These difficulties are responsible for the limited application of Huntley's directed length dimensions to real problems. <div class="mw-heading mw-heading4"><h4 id="Quantity_of_matter">Quantity of matter</h4>[<a href="/w/index.php?title=Dimensional_analysis&action=edit&section=35" title="Edit section: Quantity of matter">edit</a>]</div> In Huntley's second approach, he holds that it is sometimes useful (e.g., in fluid mechanics and thermodynamics) to distinguish between mass as a measure of inertia (inertial mass), and mass as a measure of the quantity of matter. Quantity of matter is defined by Huntley as a quantity only proportional to inertial mass, while not implicating inertial properties. No further restrictions are added to its definition. For example, consider the derivation of <a href="/wiki/Poiseuille%27s_Law" class="mw-redirect" title="Poiseuille's Law">Poiseuille's Law</a>. We wish to find the rate of mass flow of a viscous fluid through a circular pipe. Without drawing distinctions between inertial and substantial mass, we may choose as the relevant variables: <table class="wikitable"> <tbody><tr> <th>Symbol</th> <th>Variable</th> <th>Dimension </th></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {m}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>m</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {m}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad59b9876301e8fb75b9ddbf08de594b87251d3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:2.176ex;" alt="{\displaystyle {\dot {m}}}"></td> <td>mass flow rate</td> <td>T−1M </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{\text{x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>x</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{\text{x}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23ad30e48e8d6c3fd04cbcf1e5aac5440efb3306" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.359ex; height:2.009ex;" alt="{\displaystyle p_{\text{x}}}"></td> <td>pressure gradient along the pipe</td> <td>T−2L−2M </td></tr> <tr> <td>ρ</td> <td>density</td> <td>L−3M </td></tr> <tr> <td>η</td> <td>dynamic fluid viscosity</td> <td>T−1L−1M </td></tr> <tr> <td>r</td> <td>radius of the pipe</td> <td>L </td></tr></tbody></table> There are three fundamental variables, so the above five equations will yield two independent dimensionless variables: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{1}={\frac {\dot {m}}{\eta r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>m</mi> <mo>˙</mo> </mover> </mrow> <mrow> <mi>η</mi> <mi>r</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi _{1}={\frac {\dot {m}}{\eta r}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98540ff5f962f3280bf6c0801845ef9bf2c98600" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:8.532ex; height:5.676ex;" alt="{\displaystyle \pi _{1}={\frac {\dot {m}}{\eta r}}}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{2}={\frac {p_{\mathrm {x} }\rho r^{5}}{{\dot {m}}^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">x</mi> </mrow> </mrow> </msub> <mi>ρ</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>m</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi _{2}={\frac {p_{\mathrm {x} }\rho r^{5}}{{\dot {m}}^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b652e2d64ee66874cc1c6f46a6cbf7d585b7fad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:11.888ex; height:6.176ex;" alt="{\displaystyle \pi _{2}={\frac {p_{\mathrm {x} }\rho r^{5}}{{\dot {m}}^{2}}}}"></dd></dl> If we distinguish between inertial mass with dimension <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{\text{i}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>i</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{\text{i}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82e4fc40496ff3c8d792e636ec74e3ba2a45ea6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.944ex; height:2.509ex;" alt="{\displaystyle M_{\text{i}}}"> and quantity of matter with dimension <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{\text{m}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>m</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{\text{m}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a37e0b48800b793082f7c0904989473cb4dbb598" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.855ex; height:2.509ex;" alt="{\displaystyle M_{\text{m}}}">, then mass flow rate and density will use quantity of matter as the mass parameter, while the pressure gradient and coefficient of viscosity will use inertial mass. We now have four fundamental parameters, and one dimensionless constant, so that the dimensional equation may be written: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C={\frac {p_{\mathrm {x} }\rho r^{4}}{\eta {\dot {m}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">x</mi> </mrow> </mrow> </msub> <mi>ρ</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> <mrow> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>m</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C={\frac {p_{\mathrm {x} }\rho r^{4}}{\eta {\dot {m}}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71911d472bceefc6681a6c38f3544bcb64b7d4d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:11.275ex; height:6.343ex;" alt="{\displaystyle C={\frac {p_{\mathrm {x} }\rho r^{4}}{\eta {\dot {m}}}}}"></dd></dl> where now only C is an undetermined constant (found to be equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi /8}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>8</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi /8}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5560bd24fb121d12a76b93236ac084e4c5844770" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.657ex; height:2.843ex;" alt="{\displaystyle \pi /8}"> by methods outside of dimensional analysis). This equation may be solved for the mass flow rate to yield <a href="/wiki/Poiseuille%27s_law" class="mw-redirect" title="Poiseuille's law">Poiseuille's law</a>. Huntley's recognition of quantity of matter as an independent quantity dimension is evidently successful in the problems where it is applicable, but his definition of quantity of matter is open to interpretation, as it lacks specificity beyond the two requirements he postulated for it. For a given substance, the SI dimension <a href="/wiki/Amount_of_substance" title="Amount of substance">amount of substance</a>, with unit <a href="/wiki/Mole_(unit)" title="Mole (unit)">mole</a>, does satisfy Huntley's two requirements as a measure of quantity of matter, and could be used as a quantity of matter in any problem of dimensional analysis where Huntley's concept is applicable. <div class="mw-heading mw-heading3"><h3 id="Siano's_extension:_orientational_analysis">Siano's extension: orientational analysis</h3>[<a href="/w/index.php?title=Dimensional_analysis&action=edit&section=36" title="Edit section: Siano's extension: orientational analysis">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Angle#Dimensional_analysis" title="Angle">Angle § Dimensional analysis</a></div> <a href="/wiki/Angle" title="Angle">Angles</a> are, by convention, considered to be dimensionless quantities (although the wisdom of this is contested <a href="#cite_note-53">[53]</a>) . As an example, consider again the projectile problem in which a point mass is launched from the origin (x, y) = (0, 0) at a speed v and angle θ above the x-axis, with the force of gravity directed along the negative y-axis. It is desired to find the range R, at which point the mass returns to the x-axis. Conventional analysis will yield the dimensionless variable π = R g/v2, but offers no insight into the relationship between R and θ. Siano has suggested that the directed dimensions of Huntley be replaced by using orientational symbols 1x 1y 1z to denote vector directions, and an orientationless symbol 10.<a href="#cite_note-54">[54]</a> Thus, Huntley's Lx becomes L1x with L specifying the dimension of length, and 1x specifying the orientation. Siano further shows that the orientational symbols have an algebra of their own. Along with the requirement that 1i−1 = 1i, the following multiplication table for the orientation symbols results: <table class="wikitable"> <tbody><tr> <th></th> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {1_{0}} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mn mathvariant="bold">1</mn> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {1_{0}} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60946c01b1a16761213276e6153b6192d310a3dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.514ex; height:2.509ex;" alt="{\displaystyle \mathbf {1_{0}} }"></th> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {1_{\text{x}}} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mn mathvariant="bold">1</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">x</mtext> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {1_{\text{x}}} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/daf6d2297c066b12add1b30c1fc6077e33f50e4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.567ex; height:2.509ex;" alt="{\displaystyle \mathbf {1_{\text{x}}} }"></th> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {1_{\text{y}}} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mn mathvariant="bold">1</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">y</mtext> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {1_{\text{y}}} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7545bbd6d06311aaf74da6ea62935584336b870" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.567ex; height:2.843ex;" alt="{\displaystyle \mathbf {1_{\text{y}}} }"></th> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {1_{\text{z}}} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mn mathvariant="bold">1</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">z</mtext> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {1_{\text{z}}} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/331365873e0e7a389539ebad41a431147efc01cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.409ex; height:2.509ex;" alt="{\displaystyle \mathbf {1_{\text{z}}} }"> </th></tr> <tr> <th scope="col"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {1_{0}} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mn mathvariant="bold">1</mn> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {1_{0}} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60946c01b1a16761213276e6153b6192d310a3dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.514ex; height:2.509ex;" alt="{\displaystyle \mathbf {1_{0}} }"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e5120fdb64146fb2d84f979ec73a8e0e2932eea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.217ex; height:2.509ex;" alt="{\displaystyle 1_{0}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1_{\text{x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext>x</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1_{\text{x}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c546de465b5fea97b52d56a6c548407d0b782e34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.263ex; height:2.509ex;" alt="{\displaystyle 1_{\text{x}}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1_{\text{y}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext>y</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1_{\text{y}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc82f359b641be89de30f5a826d4beb8feb798e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.263ex; height:2.843ex;" alt="{\displaystyle 1_{\text{y}}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1_{\text{z}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext>z</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1_{\text{z}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1117a4db60559b9ee6daabcebd32a6f686435b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.125ex; height:2.509ex;" alt="{\displaystyle 1_{\text{z}}}"> </td></tr> <tr> <th scope="col"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {1_{\text{x}}} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mn mathvariant="bold">1</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">x</mtext> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {1_{\text{x}}} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/daf6d2297c066b12add1b30c1fc6077e33f50e4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.567ex; height:2.509ex;" alt="{\displaystyle \mathbf {1_{\text{x}}} }"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1_{\text{x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext>x</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1_{\text{x}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c546de465b5fea97b52d56a6c548407d0b782e34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.263ex; height:2.509ex;" alt="{\displaystyle 1_{\text{x}}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e5120fdb64146fb2d84f979ec73a8e0e2932eea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.217ex; height:2.509ex;" alt="{\displaystyle 1_{0}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1_{\text{z}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext>z</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1_{\text{z}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1117a4db60559b9ee6daabcebd32a6f686435b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.125ex; height:2.509ex;" alt="{\displaystyle 1_{\text{z}}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1_{\text{y}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext>y</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1_{\text{y}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc82f359b641be89de30f5a826d4beb8feb798e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.263ex; height:2.843ex;" alt="{\displaystyle 1_{\text{y}}}"> </td></tr> <tr> <th scope="col"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {1_{\text{y}}} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mn mathvariant="bold">1</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">y</mtext> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {1_{\text{y}}} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7545bbd6d06311aaf74da6ea62935584336b870" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.567ex; height:2.843ex;" alt="{\displaystyle \mathbf {1_{\text{y}}} }"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1_{\text{y}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext>y</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1_{\text{y}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc82f359b641be89de30f5a826d4beb8feb798e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.263ex; height:2.843ex;" alt="{\displaystyle 1_{\text{y}}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1_{\text{z}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext>z</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1_{\text{z}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1117a4db60559b9ee6daabcebd32a6f686435b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.125ex; height:2.509ex;" alt="{\displaystyle 1_{\text{z}}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e5120fdb64146fb2d84f979ec73a8e0e2932eea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.217ex; height:2.509ex;" alt="{\displaystyle 1_{0}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1_{\text{x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext>x</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1_{\text{x}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c546de465b5fea97b52d56a6c548407d0b782e34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.263ex; height:2.509ex;" alt="{\displaystyle 1_{\text{x}}}"> </td></tr> <tr> <th scope="col"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {1_{\text{z}}} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mn mathvariant="bold">1</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="bold">z</mtext> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {1_{\text{z}}} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/331365873e0e7a389539ebad41a431147efc01cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.409ex; height:2.509ex;" alt="{\displaystyle \mathbf {1_{\text{z}}} }"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1_{\text{z}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext>z</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1_{\text{z}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1117a4db60559b9ee6daabcebd32a6f686435b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.125ex; height:2.509ex;" alt="{\displaystyle 1_{\text{z}}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1_{\text{y}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext>y</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1_{\text{y}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc82f359b641be89de30f5a826d4beb8feb798e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.263ex; height:2.843ex;" alt="{\displaystyle 1_{\text{y}}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1_{\text{x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext>x</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1_{\text{x}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c546de465b5fea97b52d56a6c548407d0b782e34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.263ex; height:2.509ex;" alt="{\displaystyle 1_{\text{x}}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e5120fdb64146fb2d84f979ec73a8e0e2932eea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.217ex; height:2.509ex;" alt="{\displaystyle 1_{0}}"> </td></tr></tbody></table> The orientational symbols form a group (the <a href="/wiki/Klein_four-group" title="Klein four-group">Klein four-group</a> or "Viergruppe"). In this system, scalars always have the same orientation as the identity element, independent of the "symmetry of the problem". Physical quantities that are vectors have the orientation expected: a force or a velocity in the z-direction has the orientation of 1z. For angles, consider an angle θ that lies in the z-plane. Form a right triangle in the z-plane with θ being one of the acute angles. The side of the right triangle adjacent to the angle then has an orientation 1x and the side opposite has an orientation 1y. Since (using ~ to indicate orientational equivalence) tan(θ) = θ + ... ~ 1y/1x we conclude that an angle in the xy-plane must have an orientation 1y/1x = 1z, which is not unreasonable. Analogous reasoning forces the conclusion that sin(θ) has orientation 1z while cos(θ) has orientation 10. These are different, so one concludes (correctly), for example, that there are no solutions of physical equations that are of the form a cos(θ) + b sin(θ), where a and b are real scalars. An expression such as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(\theta +\pi /2)=\cos(\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>+</mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(\theta +\pi /2)=\cos(\theta )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5fd33535f8ef0b98c867b3a3f627cfc81a59754" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.362ex; height:2.843ex;" alt="{\displaystyle \sin(\theta +\pi /2)=\cos(\theta )}"> is not dimensionally inconsistent since it is a special case of the sum of angles formula and should properly be written: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin \left(a\,1_{\text{z}}+b\,1_{\text{z}}\right)=\sin \left(a\,1_{\text{z}})\cos(b\,1_{\text{z}}\right)+\sin \left(b\,1_{\text{z}})\cos(a\,1_{\text{z}}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>a</mi> <mspace width="thinmathspace" /> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext>z</mtext> </mrow> </msub> <mo>+</mo> <mi>b</mi> <mspace width="thinmathspace" /> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext>z</mtext> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>a</mi> <mspace width="thinmathspace" /> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext>z</mtext> </mrow> </msub> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mspace width="thinmathspace" /> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext>z</mtext> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>b</mi> <mspace width="thinmathspace" /> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext>z</mtext> </mrow> </msub> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mspace width="thinmathspace" /> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext>z</mtext> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin \left(a\,1_{\text{z}}+b\,1_{\text{z}}\right)=\sin \left(a\,1_{\text{z}})\cos(b\,1_{\text{z}}\right)+\sin \left(b\,1_{\text{z}})\cos(a\,1_{\text{z}}\right),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e5956429b00de5508b610127664b79d852196b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:55.789ex; height:2.843ex;" alt="{\displaystyle \sin \left(a\,1_{\text{z}}+b\,1_{\text{z}}\right)=\sin \left(a\,1_{\text{z}})\cos(b\,1_{\text{z}}\right)+\sin \left(b\,1_{\text{z}})\cos(a\,1_{\text{z}}\right),}"></dd></dl> which for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=\theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=\theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/066be9fc9a748456a8198b286d97c4ec84bb87d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.419ex; height:2.176ex;" alt="{\displaystyle a=\theta }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=\pi /2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=\pi /2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eba796bff329a22705673036509fcd2184aa022c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.753ex; height:2.843ex;" alt="{\displaystyle b=\pi /2}"> yields ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(\theta \,1_{\text{z}}+[\pi /2]\,1_{\text{z}})=1_{\text{z}}\cos(\theta \,1_{\text{z}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mspace width="thinmathspace" /> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext>z</mtext> </mrow> </msub> <mo>+</mo> <mo stretchy="false">[</mo> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">]</mo> <mspace width="thinmathspace" /> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext>z</mtext> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext>z</mtext> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mspace width="thinmathspace" /> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext>z</mtext> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(\theta \,1_{\text{z}}+[\pi /2]\,1_{\text{z}})=1_{\text{z}}\cos(\theta \,1_{\text{z}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02f99f85aa2a4dffc37a8a2fd4ad17446fd4b443" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.703ex; height:2.843ex;" alt="{\displaystyle \sin(\theta \,1_{\text{z}}+[\pi /2]\,1_{\text{z}})=1_{\text{z}}\cos(\theta \,1_{\text{z}})}">⁠. Siano distinguishes between geometric angles, which have an orientation in 3-dimensional space, and phase angles associated with time-based oscillations, which have no spatial orientation, i.e. the orientation of a phase angle is ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e5120fdb64146fb2d84f979ec73a8e0e2932eea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.217ex; height:2.509ex;" alt="{\displaystyle 1_{0}}">⁠. The assignment of orientational symbols to physical quantities and the requirement that physical equations be orientationally homogeneous can actually be used in a way that is similar to dimensional analysis to derive more information about acceptable solutions of physical problems. In this approach, one solves the dimensional equation as far as one can. If the lowest power of a physical variable is fractional, both sides of the solution is raised to a power such that all powers are integral, putting it into <a href="/wiki/Canonical_form" title="Canonical form">normal form</a>. The orientational equation is then solved to give a more restrictive condition on the unknown powers of the orientational symbols. The solution is then more complete than the one that dimensional analysis alone gives. Often, the added information is that one of the powers of a certain variable is even or odd. As an example, for the projectile problem, using orientational symbols, θ, being in the xy-plane will thus have dimension 1z and the range of the projectile R will be of the form: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R=g^{a}\,v^{b}\,\theta ^{c}{\text{ which means }}{\mathsf {L}}\,1_{\mathrm {x} }\sim \left({\frac {{\mathsf {L}}\,1_{\text{y}}}{{\mathsf {T}}^{2}}}\right)^{a}\left({\frac {\mathsf {L}}{\mathsf {T}}}\right)^{b}\,1_{\mathsf {z}}^{c}.\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>=</mo> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> <mspace width="thinmathspace" /> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msup> <mspace width="thinmathspace" /> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mtext> which means </mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">L</mi> </mrow> </mrow> <mspace width="thinmathspace" /> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">x</mi> </mrow> </mrow> </msub> <mo>∼</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">L</mi> </mrow> </mrow> <mspace width="thinmathspace" /> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext>y</mtext> </mrow> </msub> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">L</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msup> <mspace width="thinmathspace" /> <msubsup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">z</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msubsup> <mo>.</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R=g^{a}\,v^{b}\,\theta ^{c}{\text{ which means }}{\mathsf {L}}\,1_{\mathrm {x} }\sim \left({\frac {{\mathsf {L}}\,1_{\text{y}}}{{\mathsf {T}}^{2}}}\right)^{a}\left({\frac {\mathsf {L}}{\mathsf {T}}}\right)^{b}\,1_{\mathsf {z}}^{c}.\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a12e838d15bc98f7785efe718044aa5d3ebf37b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:52.566ex; height:6.676ex;" alt="{\displaystyle R=g^{a}\,v^{b}\,\theta ^{c}{\text{ which means }}{\mathsf {L}}\,1_{\mathrm {x} }\sim \left({\frac {{\mathsf {L}}\,1_{\text{y}}}{{\mathsf {T}}^{2}}}\right)^{a}\left({\frac {\mathsf {L}}{\mathsf {T}}}\right)^{b}\,1_{\mathsf {z}}^{c}.\,}"></dd></dl> Dimensional homogeneity will now correctly yield a = −1 and b = 2, and orientational homogeneity requires that ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1_{x}/(1_{y}^{a}1_{z}^{c})=1_{z}^{c+1}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <msubsup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msubsup> <msubsup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1_{x}/(1_{y}^{a}1_{z}^{c})=1_{z}^{c+1}=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/530572b120d22c1c8e417c3823bb16a24e6beb0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:21.302ex; height:3.509ex;" alt="{\displaystyle 1_{x}/(1_{y}^{a}1_{z}^{c})=1_{z}^{c+1}=1}">⁠. In other words, that c must be an odd integer. In fact, the required function of theta will be sin(θ)cos(θ) which is a series consisting of odd powers of θ. It is seen that the Taylor series of sin(θ) and cos(θ) are orientationally homogeneous using the above multiplication table, while expressions like cos(θ) + sin(θ) and exp(θ) are not, and are (correctly) deemed unphysical. Siano's orientational analysis is compatible with the conventional conception of angular quantities as being dimensionless, and within orientational analysis, the <a href="/wiki/Radian" title="Radian">radian</a> may still be considered a dimensionless unit. The orientational analysis of a quantity equation is carried out separately from the ordinary dimensional analysis, yielding information that supplements the dimensional analysis. <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Dimensional_analysis&action=edit&section=37" title="Edit section: See also">edit</a>]</div> <ul><li><a href="/wiki/Buckingham_%CF%80_theorem" title="Buckingham π theorem">Buckingham π theorem</a></li> <li><a href="/wiki/Dimensionless_numbers_in_fluid_mechanics" title="Dimensionless numbers in fluid mechanics">Dimensionless numbers in fluid mechanics</a></li> <li><a href="/wiki/Fermi_problem" title="Fermi problem">Fermi estimate</a> – used to teach dimensional analysis</li> <li><a href="/wiki/Numerical-value_equation" class="mw-redirect" title="Numerical-value equation">Numerical-value equation</a></li> <li><a href="/wiki/Rayleigh%27s_method_of_dimensional_analysis" class="mw-redirect" title="Rayleigh's method of dimensional analysis">Rayleigh's method of dimensional analysis</a></li> <li><a href="/wiki/Similitude" title="Similitude">Similitude</a> – an application of dimensional analysis</li> <li><a href="/wiki/System_of_measurement" class="mw-redirect" title="System of measurement">System of measurement</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Related_areas_of_mathematics">Related areas of mathematics</h3>[<a href="/w/index.php?title=Dimensional_analysis&action=edit&section=38" title="Edit section: Related areas of mathematics">edit</a>]</div> <ul><li><a href="/wiki/Covariance_and_contravariance_of_vectors" title="Covariance and contravariance of vectors">Covariance and contravariance of vectors</a></li> <li><a href="/wiki/Exterior_algebra" title="Exterior algebra">Exterior algebra</a></li> <li><a href="/wiki/Geometric_algebra" title="Geometric algebra">Geometric algebra</a></li> <li><a href="/wiki/Quantity_calculus" title="Quantity calculus">Quantity calculus</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2>[<a href="/w/index.php?title=Dimensional_analysis&action=edit&section=39" title="Edit section: Notes">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-Bolster-1">^ <a href="#cite_ref-Bolster_1-0">a</a> <a href="#cite_ref-Bolster_1-1">b</a> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFBolsterHershbergerDonnelly2011" class="citation journal cs1">Bolster, Diogo; Hershberger, Robert E.; Donnelly, Russell E. (September 2011). <a rel="nofollow" class="external text" href="https://pubs.aip.org/physicstoday/article-abstract/64/9/42/413713/Dynamic-similarity-the-dimensionless">"Dynamic similarity, the dimensionless science"</a>. Physics Today. 64 (9): 42–47. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2FPT.3.1258">10.1063/PT.3.1258</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physics+Today&rft.atitle=Dynamic+similarity%2C+the+dimensionless+science&rft.volume=64&rft.issue=9&rft.pages=42-47&rft.date=2011-09&rft_id=info%3Adoi%2F10.1063%2FPT.3.1258&rft.aulast=Bolster&rft.aufirst=Diogo&rft.au=Hershberger%2C+Robert+E.&rft.au=Donnelly%2C+Russell+E.&rft_id=https%3A%2F%2Fpubs.aip.org%2Fphysicstoday%2Farticle-abstract%2F64%2F9%2F42%2F413713%2FDynamic-similarity-the-dimensionless&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988"> </li> <li id="cite_note-SIBrochure9th-2">^ <a href="#cite_ref-SIBrochure9th_2-0">a</a> <a href="#cite_ref-SIBrochure9th_2-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBIPM2019" class="citation book cs1 cs1-prop-foreign-lang-source"><a href="/wiki/BIPM" class="mw-redirect" title="BIPM">BIPM</a> (2019). "2.3.3 Dimensions of quantities". <a rel="nofollow" class="external text" href="https://www.bipm.org/en/publications/si-brochure">SI Brochure: The International System of Units (SI)</a> (PDF) (in English and French) (v. 1.08, 9th ed.). pp. 136–137. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-92-822-2272-0" title="Special:BookSources/978-92-822-2272-0"><bdi>978-92-822-2272-0</bdi></a>. Retrieved 1 September 2021.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=2.3.3+Dimensions+of+quantities&rft.btitle=SI+Brochure%3A+The+International+System+of+Units+%28SI%29&rft.pages=136-137&rft.edition=v.+1.08%2C+9th&rft.date=2019&rft.isbn=978-92-822-2272-0&rft.au=BIPM&rft_id=https%3A%2F%2Fwww.bipm.org%2Fen%2Fpublications%2Fsi-brochure&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988"> </li> <li id="cite_note-3"><a href="#cite_ref-3">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFYalin1971" class="citation book cs1">Yalin, M. Selim (1971). <a rel="nofollow" class="external text" href="https://link.springer.com/chapter/10.1007/978-1-349-00245-0_1">"Principles of the Theory of Dimensions"</a>. Theory of Hydraulic Models. pp. 1–34. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-1-349-00245-0_1">10.1007/978-1-349-00245-0_1</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-349-00247-4" title="Special:BookSources/978-1-349-00247-4"><bdi>978-1-349-00247-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Principles+of+the+Theory+of+Dimensions&rft.btitle=Theory+of+Hydraulic+Models&rft.pages=1-34&rft.date=1971&rft_id=info%3Adoi%2F10.1007%2F978-1-349-00245-0_1&rft.isbn=978-1-349-00247-4&rft.aulast=Yalin&rft.aufirst=M.+Selim&rft_id=https%3A%2F%2Flink.springer.com%2Fchapter%2F10.1007%2F978-1-349-00245-0_1&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988"> </li> <li id="cite_note-duff-4">^ <a href="#cite_ref-duff_4-0">a</a> <a href="#cite_ref-duff_4-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDuffOkunVeneziano2002" class="citation cs2">Duff, M.J.; Okun, L.B.; Veneziano, G. (September 2002), "Trialogue on the number of fundamental constants", Journal of High Energy Physics, 2002 (3): 023, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/physics/0110060">physics/0110060</a>, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2002JHEP...03..023D">2002JHEP...03..023D</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F1126-6708%2F2002%2F03%2F023">10.1088/1126-6708/2002/03/023</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:15806354">15806354</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+High+Energy+Physics&rft.atitle=Trialogue+on+the+number+of+fundamental+constants&rft.volume=2002&rft.issue=3&rft.pages=023&rft.date=2002-09&rft_id=info%3Aarxiv%2Fphysics%2F0110060&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A15806354%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1088%2F1126-6708%2F2002%2F03%2F023&rft_id=info%3Abibcode%2F2002JHEP...03..023D&rft.aulast=Duff&rft.aufirst=M.J.&rft.au=Okun%2C+L.B.&rft.au=Veneziano%2C+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988"> </li> <li id="cite_note-5"><a href="#cite_ref-5">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJCGM2012" class="citation cs2"><a href="/wiki/Joint_Committee_for_Guides_in_Metrology" title="Joint Committee for Guides in Metrology">JCGM</a> (2012), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150923224356/http://www.bipm.org/utils/common/documents/jcgm/JCGM_200_2012.pdf">JCGM 200:2012 – International vocabulary of metrology – Basic and general concepts and associated terms (VIM)</a> (PDF) (3rd ed.), archived from <a rel="nofollow" class="external text" href="https://www.bipm.org/utils/common/documents/jcgm/JCGM_200_2012.pdf">the original</a> (PDF) on 23 September 2015, retrieved 2 June 2015</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=JCGM+200%3A2012+%E2%80%93+International+vocabulary+of+metrology+%E2%80%93+Basic+and+general+concepts+and+associated+terms+%28VIM%29&rft.edition=3rd&rft.date=2012&rft.au=JCGM&rft_id=https%3A%2F%2Fwww.bipm.org%2Futils%2Fcommon%2Fdocuments%2Fjcgm%2FJCGM_200_2012.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988"> </li> <li id="cite_note-6"><a href="#cite_ref-6">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCimbalaÇengel2006" class="citation book cs1">Cimbala, John; Çengel, Yunus (2006). <a rel="nofollow" class="external text" href="http://highered.mcgraw-hill.com/sites/0073138355/student_view0/chapter7/">"§7-2 Dimensional homogeneity"</a>. Essential of Fluid Mechanics: Fundamentals and Applications. McGraw-Hill. p. 203–. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780073138350" title="Special:BookSources/9780073138350"><bdi>9780073138350</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=%C2%A77-2+Dimensional+homogeneity&rft.btitle=Essential+of+Fluid+Mechanics%3A+Fundamentals+and+Applications&rft.pages=203-&rft.pub=McGraw-Hill&rft.date=2006&rft.isbn=9780073138350&rft.aulast=Cimbala&rft.aufirst=John&rft.au=%C3%87engel%2C+Yunus&rft_id=http%3A%2F%2Fhighered.mcgraw-hill.com%2Fsites%2F0073138355%2Fstudent_view0%2Fchapter7%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988"> </li> <li id="cite_note-7"><a href="#cite_ref-7">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFde_JongQuade1967" class="citation book cs1">de Jong, Frits J.; Quade, Wilhelm (1967). <a rel="nofollow" class="external text" href="https://archive.org/details/dimensionalanaly0000jong">Dimensional analysis for economists</a>. North Holland. p. <a rel="nofollow" class="external text" href="https://archive.org/details/dimensionalanaly0000jong/page/28">28</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Dimensional+analysis+for+economists&rft.pages=28&rft.pub=North+Holland&rft.date=1967&rft.aulast=de+Jong&rft.aufirst=Frits+J.&rft.au=Quade%2C+Wilhelm&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fdimensionalanaly0000jong&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988"> </li> <li id="cite_note-8"><a href="#cite_ref-8">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWaiteFine2007" class="citation book cs1">Waite, Lee; Fine, Jerry (2007). <a rel="nofollow" class="external text" href="https://archive.org/details/appliedbiofluidm00wait">Applied Biofluid Mechanics</a>. New York: McGraw-Hill. p. <a rel="nofollow" class="external text" href="https://archive.org/details/appliedbiofluidm00wait/page/n278">260</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-147217-3" title="Special:BookSources/978-0-07-147217-3"><bdi>978-0-07-147217-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Applied+Biofluid+Mechanics&rft.place=New+York&rft.pages=260&rft.pub=McGraw-Hill&rft.date=2007&rft.isbn=978-0-07-147217-3&rft.aulast=Waite&rft.aufirst=Lee&rft.au=Fine%2C+Jerry&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fappliedbiofluidm00wait&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988"> </li> <li id="cite_note-9"><a href="#cite_ref-9">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMacagno1971" class="citation journal cs1">Macagno, Enzo O. (1971). "Historico-critical review of dimensional analysis". Journal of the Franklin Institute. 292 (6): 391–340. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0016-0032%2871%2990160-8">10.1016/0016-0032(71)90160-8</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+the+Franklin+Institute&rft.atitle=Historico-critical+review+of+dimensional+analysis&rft.volume=292&rft.issue=6&rft.pages=391-340&rft.date=1971&rft_id=info%3Adoi%2F10.1016%2F0016-0032%2871%2990160-8&rft.aulast=Macagno&rft.aufirst=Enzo+O.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988"> </li> <li id="cite_note-Martins_1981-10">^ <a href="#cite_ref-Martins_1981_10-0">a</a> <a href="#cite_ref-Martins_1981_10-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMartins1981" class="citation journal cs1">Martins, Roberto De A. (1981). "The origin of dimensional analysis". Journal of the Franklin Institute. 311 (5): 331–337. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0016-0032%2881%2990475-0">10.1016/0016-0032(81)90475-0</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+the+Franklin+Institute&rft.atitle=The+origin+of+dimensional+analysis&rft.volume=311&rft.issue=5&rft.pages=331-337&rft.date=1981&rft_id=info%3Adoi%2F10.1016%2F0016-0032%2881%2990475-0&rft.aulast=Martins&rft.aufirst=Roberto+De+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988"> </li> <li id="cite_note-11"><a href="#cite_ref-11">^</a> Martins, p. 403 in the Proceedings book containing his article </li> <li id="cite_note-12"><a href="#cite_ref-12">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMason1962" class="citation cs2">Mason, Stephen Finney (1962), A history of the sciences, New York: Collier Books, p. 169, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-02-093400-4" title="Special:BookSources/978-0-02-093400-4"><bdi>978-0-02-093400-4</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+history+of+the+sciences&rft.place=New+York&rft.pages=169&rft.pub=Collier+Books&rft.date=1962&rft.isbn=978-0-02-093400-4&rft.aulast=Mason&rft.aufirst=Stephen+Finney&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988"> </li> <li id="cite_note-maxwell-13"><a href="#cite_ref-maxwell_13-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRoche1998" class="citation cs2">Roche, John J (1998), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=eiQOqS-Q6EkC&pg=PA203">The Mathematics of Measurement: A Critical History</a>, Springer, p. 203, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-91581-4" title="Special:BookSources/978-0-387-91581-4"><bdi>978-0-387-91581-4</bdi></a>, <q>Beginning apparently with Maxwell, mass, length and time began to be interpreted as having a privileged fundamental character and all other quantities as derivative, not merely with respect to measurement, but with respect to their physical status as well.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Mathematics+of+Measurement%3A+A+Critical+History&rft.pages=203&rft.pub=Springer&rft.date=1998&rft.isbn=978-0-387-91581-4&rft.aulast=Roche&rft.aufirst=John+J&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DeiQOqS-Q6EkC%26pg%3DPA203&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988"> </li> <li id="cite_note-maxwell2-14"><a href="#cite_ref-maxwell2_14-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMaxwell1873" class="citation cs2">Maxwell, James Clerk (1873), A Treatise on Electricity and Magnetism, p. 4</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Treatise+on+Electricity+and+Magnetism&rft.pages=4&rft.date=1873&rft.aulast=Maxwell&rft.aufirst=James+Clerk&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988"> </li> <li id="cite_note-maxwell3-15"><a href="#cite_ref-maxwell3_15-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMaxwell1873" class="citation cs2">Maxwell, James Clerk (1873), A Treatise on Electricity and Magnetism, Clarendon Press series, Oxford, p. 45, <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/2027%2Fuc1.l0065867749">2027/uc1.l0065867749</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Treatise+on+Electricity+and+Magnetism&rft.series=Clarendon+Press+series&rft.pages=45&rft.pub=Oxford&rft.date=1873&rft_id=info%3Ahdl%2F2027%2Fuc1.l0065867749&rft.aulast=Maxwell&rft.aufirst=James+Clerk&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988"> </li> <li id="cite_note-16"><a href="#cite_ref-16">^</a> (<a href="#CITEREFPesic2005">Pesic 2005</a>) </li> <li id="cite_note-17"><a href="#cite_ref-17">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRayleigh1877" class="citation cs2">Rayleigh, Baron John William Strutt (1877), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=kvxYAAAAYAAJ">The Theory of Sound</a>, Macmillan</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Theory+of+Sound&rft.pub=Macmillan&rft.date=1877&rft.aulast=Rayleigh&rft.aufirst=Baron+John+William+Strutt&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DkvxYAAAAYAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988"> </li> <li id="cite_note-FOOTNOTEFourier1822[httpsbooksgooglecombooksidTDQJAAAAIAAJpgPA156_156]-18"><a href="#cite_ref-FOOTNOTEFourier1822[httpsbooksgooglecombooksidTDQJAAAAIAAJpgPA156_156]_18-0">^</a> <a href="#CITEREFFourier1822">Fourier (1822)</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=TDQJAAAAIAAJ&pg=PA156">156</a>. </li> <li id="cite_note-maxwell4-19"><a href="#cite_ref-maxwell4_19-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMaxwell1873" class="citation cs2">Maxwell, James Clerk (1873), <a rel="nofollow" class="external text" href="https://archive.org/stream/electricandmagne01maxwrich#page/n41/mode/2up">A Treatise on Electricity and Magnetism, volume 1</a>, p. 5</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Treatise+on+Electricity+and+Magnetism%2C+volume+1&rft.pages=5&rft.date=1873&rft.aulast=Maxwell&rft.aufirst=James+Clerk&rft_id=https%3A%2F%2Farchive.org%2Fstream%2Felectricandmagne01maxwrich%23page%2Fn41%2Fmode%2F2up&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988"> </li> <li id="cite_note-20"><a href="#cite_ref-20">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRamsay" class="citation web cs1">Ramsay, Angus. <a rel="nofollow" class="external text" href="http://www.ramsay-maunder.co.uk/knowledge-base/technical-notes/dimensional-analysis--numerical-experiments-for-a-rotating-disc/">"Dimensional Analysis and Numerical Experiments for a Rotating Disc"</a>. Ramsay Maunder Associates. Retrieved 15 April 2017.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Ramsay+Maunder+Associates&rft.atitle=Dimensional+Analysis+and+Numerical+Experiments+for+a+Rotating+Disc&rft.aulast=Ramsay&rft.aufirst=Angus&rft_id=http%3A%2F%2Fwww.ramsay-maunder.co.uk%2Fknowledge-base%2Ftechnical-notes%2Fdimensional-analysis--numerical-experiments-for-a-rotating-disc%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988"> </li> <li id="cite_note-FOOTNOTETao2012"With_a_bit_of_additional_effort_(and_taking_full_advantage_of_the_one-dimensionality_of_the_vector_spaces),_one_can_also_define_spaces_with_fractional_exponents&nbsp;..."-21"><a href="#cite_ref-FOOTNOTETao2012"With_a_bit_of_additional_effort_(and_taking_full_advantage_of_the_one-dimensionality_of_the_vector_spaces),_one_can_also_define_spaces_with_fractional_exponents&nbsp;..."_21-0">^</a> <a href="#CITEREFTao2012">Tao 2012</a>, "With a bit of additional effort (and taking full advantage of the one-dimensionality of the vector spaces), one can also define spaces with fractional exponents ...". </li> <li id="cite_note-FOOTNOTETao2012"However,_when_working_with_vector-valued_quantities_in_two_and_higher_dimensions,_there_are_representation-theoretic_obstructions_to_taking_arbitrary_fractional_powers_of_units&nbsp;..."-22"><a href="#cite_ref-FOOTNOTETao2012"However,_when_working_with_vector-valued_quantities_in_two_and_higher_dimensions,_there_are_representation-theoretic_obstructions_to_taking_arbitrary_fractional_powers_of_units&nbsp;..."_22-0">^</a> <a href="#CITEREFTao2012">Tao 2012</a>, "However, when working with vector-valued quantities in two and higher dimensions, there are representation-theoretic obstructions to taking arbitrary fractional powers of units ...". </li> <li id="cite_note-23"><a href="#cite_ref-23">^</a> <a href="#CITEREFTao2012">Tao 2012</a> "Similarly, one can define VT−1 as the dual space to VT ..." </li> <li id="cite_note-24"><a href="#cite_ref-24">^</a> <a href="#CITEREFBridgman1922">Bridgman 1922</a>, 2. Dimensional Formulas pp. 17–27 </li> <li id="cite_note-25"><a href="#cite_ref-25">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBerberan-SantosPogliani1999" class="citation journal cs1">Berberan-Santos, Mário N.; Pogliani, Lionello (1999). <a rel="nofollow" class="external text" href="https://core.ac.uk/download/pdf/22873054.pdf">"Two alternative derivations of Bridgman's theorem"</a> (PDF). Journal of Mathematical Chemistry. 26: 255–261, See §5 General Results p. 259. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1023%2FA%3A1019102415633">10.1023/A:1019102415633</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:14833238">14833238</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Mathematical+Chemistry&rft.atitle=Two+alternative+derivations+of+Bridgman%27s+theorem&rft.volume=26&rft.pages=255-261%2C+See+%C2%A75+General+Results+p.+259&rft.date=1999&rft_id=info%3Adoi%2F10.1023%2FA%3A1019102415633&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A14833238%23id-name%3DS2CID&rft.aulast=Berberan-Santos&rft.aufirst=M%C3%A1rio+N.&rft.au=Pogliani%2C+Lionello&rft_id=https%3A%2F%2Fcore.ac.uk%2Fdownload%2Fpdf%2F22873054.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988"> </li> <li id="cite_note-26"><a href="#cite_ref-26">^</a> <a href="#CITEREFBerberan-SantosPogliani1999">Berberan-Santos & Pogliani 1999</a>, p. 256 </li> <li id="cite_note-Pisanty13-27"><a href="#cite_ref-Pisanty13_27-0">^</a> For a review of the different conventions in use see: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPisanty2013" class="citation web cs1">Pisanty, E (17 September 2013). <a rel="nofollow" class="external text" href="http://physics.stackexchange.com/q/77690">"Square bracket notation for dimensions and units: usage and conventions"</a>. Physics Stack Exchange. Retrieved 15 July 2014.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Physics+Stack+Exchange&rft.atitle=Square+bracket+notation+for+dimensions+and+units%3A+usage+and+conventions&rft.date=2013-09-17&rft.aulast=Pisanty&rft.aufirst=E&rft_id=http%3A%2F%2Fphysics.stackexchange.com%2Fq%2F77690&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988"> </li> <li id="cite_note-nist-28">^ <a href="#cite_ref-nist_28-0">a</a> <a href="#cite_ref-nist_28-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFThompson2009" class="citation book cs1">Thompson, Ambler (November 2009). <a rel="nofollow" class="external text" href="https://physics.nist.gov/cuu/pdf/sp811.pdf">Guide for the Use of the International System of Units (SI): The Metric System</a> (PDF). DIANE Publishing. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781437915594" title="Special:BookSources/9781437915594"><bdi>9781437915594</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Guide+for+the+Use+of+the+International+System+of+Units+%28SI%29%3A+The+Metric+System&rft.pub=DIANE+Publishing&rft.date=2009-11&rft.isbn=9781437915594&rft.aulast=Thompson&rft.aufirst=Ambler&rft_id=https%3A%2F%2Fphysics.nist.gov%2Fcuu%2Fpdf%2Fsp811.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988"> </li> <li id="cite_note-29"><a href="#cite_ref-29">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDuff2004" class="citation arxiv cs1">Duff, Michael James (July 2004). "Comment on time-variation of fundamental constants". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/hep-th/0208093v3">hep-th/0208093v3</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=Comment+on+time-variation+of+fundamental+constants&rft.date=2004-07&rft_id=info%3Aarxiv%2Fhep-th%2F0208093v3&rft.aulast=Duff&rft.aufirst=Michael+James&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988"> </li> <li id="cite_note-30"><a href="#cite_ref-30">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWoan2010" class="citation cs2">Woan, G. (2010), <a rel="nofollow" class="external text" href="https://archive.org/details/cambridgehandboo0000woan">The Cambridge Handbook of Physics Formulas</a>, Cambridge University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-57507-2" title="Special:BookSources/978-0-521-57507-2"><bdi>978-0-521-57507-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Cambridge+Handbook+of+Physics+Formulas&rft.pub=Cambridge+University+Press&rft.date=2010&rft.isbn=978-0-521-57507-2&rft.aulast=Woan&rft.aufirst=G.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fcambridgehandboo0000woan&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988"> </li> <li id="cite_note-31"><a href="#cite_ref-31">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMoscaTipler2007" class="citation cs2">Mosca, Gene; Tipler, Paul Allen (2007), Physics for Scientists and Engineers – with Modern Physics (6th ed.), San Francisco: W. H. Freeman, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7167-8964-2" title="Special:BookSources/978-0-7167-8964-2"><bdi>978-0-7167-8964-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Physics+for+Scientists+and+Engineers+%E2%80%93+with+Modern+Physics&rft.place=San+Francisco&rft.edition=6th&rft.pub=W.+H.+Freeman&rft.date=2007&rft.isbn=978-0-7167-8964-2&rft.aulast=Mosca&rft.aufirst=Gene&rft.au=Tipler%2C+Paul+Allen&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988"> </li> <li id="cite_note-Martin08-32"><a href="#cite_ref-Martin08_32-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMartinShawManchester_Physics2008" class="citation cs2">Martin, B.R.; Shaw, G.; Manchester Physics (2008), Particle Physics (2nd ed.), Wiley, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-470-03294-7" title="Special:BookSources/978-0-470-03294-7"><bdi>978-0-470-03294-7</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Particle+Physics&rft.edition=2nd&rft.pub=Wiley&rft.date=2008&rft.isbn=978-0-470-03294-7&rft.aulast=Martin&rft.aufirst=B.R.&rft.au=Shaw%2C+G.&rft.au=Manchester+Physics&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988"> </li> <li id="cite_note-33"><a href="#cite_ref-33">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGehani1977" class="citation journal cs1">Gehani, N. (1977). "Units of measure as a data attribute". Comput. Lang. 2 (3): 93–111. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0096-0551%2877%2990010-8">10.1016/0096-0551(77)90010-8</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Comput.+Lang.&rft.atitle=Units+of+measure+as+a+data+attribute&rft.volume=2&rft.issue=3&rft.pages=93-111&rft.date=1977&rft_id=info%3Adoi%2F10.1016%2F0096-0551%2877%2990010-8&rft.aulast=Gehani&rft.aufirst=N.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988"> </li> <li id="cite_note-34"><a href="#cite_ref-34">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGehani1985" class="citation journal cs1">Gehani, N. (June 1985). "Ada's derived types and units of measure". Software: Practice and Experience. 15 (6): 555–569. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fspe.4380150604">10.1002/spe.4380150604</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:40558757">40558757</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Software%3A+Practice+and+Experience&rft.atitle=Ada%27s+derived+types+and+units+of+measure&rft.volume=15&rft.issue=6&rft.pages=555-569&rft.date=1985-06&rft_id=info%3Adoi%2F10.1002%2Fspe.4380150604&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A40558757%23id-name%3DS2CID&rft.aulast=Gehani&rft.aufirst=N.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988"> </li> <li id="cite_note-35"><a href="#cite_ref-35">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCmelikGehani1988" class="citation journal cs1">Cmelik, R. F.; Gehani, N. H. (May 1988). "Dimensional analysis with C++". IEEE Software. 5 (3): 21–27. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2F52.2021">10.1109/52.2021</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:22450087">22450087</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=IEEE+Software&rft.atitle=Dimensional+analysis+with+C%2B%2B&rft.volume=5&rft.issue=3&rft.pages=21-27&rft.date=1988-05&rft_id=info%3Adoi%2F10.1109%2F52.2021&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A22450087%23id-name%3DS2CID&rft.aulast=Cmelik&rft.aufirst=R.+F.&rft.au=Gehani%2C+N.+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988"> </li> <li id="cite_note-36"><a href="#cite_ref-36">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKennedy1996" class="citation thesis cs1">Kennedy, Andrew J. (April 1996). <a rel="nofollow" class="external text" href="http://www.cl.cam.ac.uk/techreports/UCAM-CL-TR-391.html">Programming languages and dimensions</a> (Phd). Vol. 391. University of Cambridge. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1476-2986">1476-2986</a>. UCAM-CL-TR-391.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adissertation&rft.title=Programming+languages+and+dimensions&rft.inst=University+of+Cambridge&rft.date=1996-04&rft.issn=1476-2986&rft.aulast=Kennedy&rft.aufirst=Andrew+J.&rft_id=http%3A%2F%2Fwww.cl.cam.ac.uk%2Ftechreports%2FUCAM-CL-TR-391.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988"> </li> <li id="cite_note-37"><a href="#cite_ref-37">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKennedy2010" class="citation book cs1">Kennedy, A. (2010). "Types for Units-of-Measure: Theory and Practice". In Horváth, Z.; Plasmeijer, R.; Zsók, V. (eds.). Central European Functional Programming School. CEFP 2009. Lecture Notes in Computer Science. Vol. 6299. Springer. pp. 268–305. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.174.6901">10.1.1.174.6901</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-642-17685-2_8">10.1007/978-3-642-17685-2_8</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-642-17684-5" title="Special:BookSources/978-3-642-17684-5"><bdi>978-3-642-17684-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Types+for+Units-of-Measure%3A+Theory+and+Practice&rft.btitle=Central+European+Functional+Programming+School.+CEFP+2009&rft.series=Lecture+Notes+in+Computer+Science&rft.pages=268-305&rft.pub=Springer&rft.date=2010&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.174.6901%23id-name%3DCiteSeerX&rft_id=info%3Adoi%2F10.1007%2F978-3-642-17685-2_8&rft.isbn=978-3-642-17684-5&rft.aulast=Kennedy&rft.aufirst=A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988"> </li> <li id="cite_note-38"><a href="#cite_ref-38">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGundry2015" class="citation journal cs1">Gundry, Adam (December 2015). <a rel="nofollow" class="external text" href="http://adam.gundry.co.uk/pub/typechecker-plugins/typechecker-plugins-2015-07-17.pdf">"A typechecker plugin for units of measure: domain-specific constraint solving in GHC Haskell"</a> (PDF). SIGPLAN Notices. 50 (12): 11–22. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F2887747.2804305">10.1145/2887747.2804305</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170810000458/http://adam.gundry.co.uk/pub/typechecker-plugins/typechecker-plugins-2015-07-17.pdf">Archived</a> (PDF) from the original on 10 August 2017.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=SIGPLAN+Notices&rft.atitle=A+typechecker+plugin+for+units+of+measure%3A+domain-specific+constraint+solving+in+GHC+Haskell&rft.volume=50&rft.issue=12&rft.pages=11-22&rft.date=2015-12&rft_id=info%3Adoi%2F10.1145%2F2887747.2804305&rft.aulast=Gundry&rft.aufirst=Adam&rft_id=http%3A%2F%2Fadam.gundry.co.uk%2Fpub%2Ftypechecker-plugins%2Ftypechecker-plugins-2015-07-17.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988"> </li> <li id="cite_note-39"><a href="#cite_ref-39">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGarrigueLy2017" class="citation book cs1 cs1-prop-foreign-lang-source">Garrigue, J.; Ly, D. (2017). <a rel="nofollow" class="external text" href="https://www.math.nagoya-u.ac.jp/~garrigue/papers/ocamldim.pdf">"Des unités dans le typeur"</a> (PDF). 28ièmes Journées Francophones des Langaeges Applicatifs, Jan 2017, Gourette, France (in French). hal-01503084. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20201110105231/https://www.math.nagoya-u.ac.jp/~garrigue/papers/ocamldim.pdf">Archived</a> (PDF) from the original on 10 November 2020.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Des+unit%C3%A9s+dans+le+typeur&rft.btitle=28i%C3%A8mes+Journ%C3%A9es+Francophones+des+Langaeges+Applicatifs%2C+Jan+2017%2C+Gourette%2C+France&rft.date=2017&rft.aulast=Garrigue&rft.aufirst=J.&rft.au=Ly%2C+D.&rft_id=https%3A%2F%2Fwww.math.nagoya-u.ac.jp%2F~garrigue%2Fpapers%2Focamldim.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988"> </li> <li id="cite_note-40"><a href="#cite_ref-40">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTeller2020" class="citation web cs1">Teller, David (January 2020). <a rel="nofollow" class="external text" href="https://yoric.github.io/post/uom.rs/">"Units of Measure in Rust with Refinement Types"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Units+of+Measure+in+Rust+with+Refinement+Types&rft.date=2020-01&rft.aulast=Teller&rft.aufirst=David&rft_id=https%3A%2F%2Fyoric.github.io%2Fpost%2Fuom.rs%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988"> </li> <li id="cite_note-41"><a href="#cite_ref-41">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrecco2022" class="citation web cs1">Grecco, Hernan E. (2022). <a rel="nofollow" class="external text" href="https://pint.readthedocs.io/en/stable/">"Pint: makes units easy"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Pint%3A+makes+units+easy&rft.date=2022&rft.aulast=Grecco&rft.aufirst=Hernan+E.&rft_id=https%3A%2F%2Fpint.readthedocs.io%2Fen%2Fstable%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988"> </li> <li id="cite_note-42"><a href="#cite_ref-42">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://camfort.github.io/">"CamFort: Specify, verify, and refactor Fortran code"</a>. University of Cambridge; University of Kent. 2018.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=CamFort%3A+Specify%2C+verify%2C+and+refactor+Fortran+code&rft.pub=University+of+Cambridge%3B+University+of+Kent&rft.date=2018&rft_id=https%3A%2F%2Fcamfort.github.io%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988"> </li> <li id="cite_note-43"><a href="#cite_ref-43">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBennich-BjörkmanMcKeever2018" class="citation book cs1">Bennich-Björkman, O.; McKeever, S. (2018). "The next 700 unit of measurement checkers". Proceedings of the 11th ACM SIGPLAN International Conference on Software Language Engineering. pp. 121–132. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F3276604.3276613">10.1145/3276604.3276613</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4503-6029-6" title="Special:BookSources/978-1-4503-6029-6"><bdi>978-1-4503-6029-6</bdi></a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:53089559">53089559</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+next+700+unit+of+measurement+checkers&rft.btitle=Proceedings+of+the+11th+ACM+SIGPLAN+International+Conference+on+Software+Language+Engineering&rft.pages=121-132&rft.date=2018&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A53089559%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1145%2F3276604.3276613&rft.isbn=978-1-4503-6029-6&rft.aulast=Bennich-Bj%C3%B6rkman&rft.aufirst=O.&rft.au=McKeever%2C+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988"> </li> <li id="cite_note-44"><a href="#cite_ref-44">^</a> <a href="#CITEREFHart1995">Hart 1995</a> </li> <li id="cite_note-45"><a href="#cite_ref-45">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGriffioen2019" class="citation thesis cs1">Griffioen, P. (2019). <a rel="nofollow" class="external text" href="https://pure.uva.nl/ws/files/42206706/Thesis.pdf">A unit-aware matrix language and its application in control and auditing</a> (PDF) (Thesis). University of Amsterdam. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/11245.1%2Ffd7be191-700f-4468-a329-4c8ecd9007ba">11245.1/fd7be191-700f-4468-a329-4c8ecd9007ba</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20200221072448/https://pure.uva.nl/ws/files/42206706/Thesis.pdf">Archived</a> (PDF) from the original on 21 February 2020.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adissertation&rft.title=A+unit-aware+matrix+language+and+its+application+in+control+and+auditing&rft.inst=University+of+Amsterdam&rft.date=2019&rft_id=info%3Ahdl%2F11245.1%2Ffd7be191-700f-4468-a329-4c8ecd9007ba&rft.aulast=Griffioen&rft.aufirst=P.&rft_id=https%3A%2F%2Fpure.uva.nl%2Fws%2Ffiles%2F42206706%2FThesis.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988"> </li> <li id="cite_note-46"><a href="#cite_ref-46">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMcBrideNordvall-Forsberg2022" class="citation book cs1"><a href="/wiki/Conor_McBride" title="Conor McBride">McBride, Conor</a>; Nordvall-Forsberg, Fredrik (2022). <a rel="nofollow" class="external text" href="https://strathprints.strath.ac.uk/76626/1/McBride_etal_amctmtxii2021_type_systems_for_programs_respecting_dimensions.pdf">"Type systems for programs respecting dimensions"</a> (PDF). Advanced Mathematical and Computational Tools in Metrology and Testing XII. Advances in Mathematics for Applied Sciences. World Scientific. pp. 331–345. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1142%2F9789811242380_0020">10.1142/9789811242380_0020</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9789811242380" title="Special:BookSources/9789811242380"><bdi>9789811242380</bdi></a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:243831207">243831207</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20220517115417/https://strathprints.strath.ac.uk/76626/1/McBride_etal_amctmtxii2021_type_systems_for_programs_respecting_dimensions.pdf">Archived</a> (PDF) from the original on 17 May 2022.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Type+systems+for+programs+respecting+dimensions&rft.btitle=Advanced+Mathematical+and+Computational+Tools+in+Metrology+and+Testing+XII&rft.series=Advances+in+Mathematics+for+Applied+Sciences&rft.pages=331-345&rft.pub=World+Scientific&rft.date=2022&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A243831207%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1142%2F9789811242380_0020&rft.isbn=9789811242380&rft.aulast=McBride&rft.aufirst=Conor&rft.au=Nordvall-Forsberg%2C+Fredrik&rft_id=https%3A%2F%2Fstrathprints.strath.ac.uk%2F76626%2F1%2FMcBride_etal_amctmtxii2021_type_systems_for_programs_respecting_dimensions.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988"> </li> <li id="cite_note-47"><a href="#cite_ref-47">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://reference.wolfram.com/language/ref/NondimensionalizationTransform.html.en">"NondimensionalizationTransform—Wolfram Language Documentation"</a>. reference.wolfram.com. Retrieved 19 April 2023.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=reference.wolfram.com&rft.atitle=NondimensionalizationTransform%E2%80%94Wolfram+Language+Documentation&rft_id=https%3A%2F%2Freference.wolfram.com%2Flanguage%2Fref%2FNondimensionalizationTransform.html.en&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988"> </li> <li id="cite_note-48"><a href="#cite_ref-48">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://reference.wolfram.com/language/ref/UnitDimensions.html.en">"UnitDimensions—Wolfram Language Documentation"</a>. reference.wolfram.com. 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Retrieved 19 April 2023.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=reference.wolfram.com&rft.atitle=DimensionalCombinations%E2%80%94Wolfram+Language+Documentation&rft_id=https%3A%2F%2Freference.wolfram.com%2Flanguage%2Fref%2FDimensionalCombinations.html.en&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988"> </li> <li id="cite_note-:0-50">^ <a href="#cite_ref-:0_50-0">a</a> <a href="#cite_ref-:0_50-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://reference.wolfram.com/language/ref/UnityDimensions.html.en">"UnityDimensions—Wolfram Language Documentation"</a>. reference.wolfram.com. Retrieved 19 April 2023.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=reference.wolfram.com&rft.atitle=UnityDimensions%E2%80%94Wolfram+Language+Documentation&rft_id=https%3A%2F%2Freference.wolfram.com%2Flanguage%2Fref%2FUnityDimensions.html.en&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988"> </li> <li id="cite_note-51"><a href="#cite_ref-51">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://reference.wolfram.com/language/ref/QuantityVariableDimensions.html.en">"QuantityVariableDimensions—Wolfram Language Documentation"</a>. reference.wolfram.com. Retrieved 19 April 2023.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=reference.wolfram.com&rft.atitle=QuantityVariableDimensions%E2%80%94Wolfram+Language+Documentation&rft_id=https%3A%2F%2Freference.wolfram.com%2Flanguage%2Fref%2FQuantityVariableDimensions.html.en&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988"> </li> <li id="cite_note-52"><a href="#cite_ref-52">^</a> (<a href="#CITEREFHuntley1967">Huntley 1967</a>) </li> <li id="cite_note-53"><a href="#cite_ref-53">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFQuincey2021" class="citation journal cs1">Quincey, Paul (2021). <a rel="nofollow" class="external text" href="https://dx.doi.org/10.1088/1681-7575/ac023f">"Angles in the SI: a detailed proposal for solving the problem"</a>. Metrologia. 58 (5): 053002. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/2108.05704">2108.05704</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F1681-7575%2Fac023f">10.1088/1681-7575/ac023f</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Metrologia&rft.atitle=Angles+in+the+SI%3A+a+detailed+proposal+for+solving+the+problem&rft.volume=58&rft.issue=5&rft.pages=053002&rft.date=2021&rft_id=info%3Aarxiv%2F2108.05704&rft_id=info%3Adoi%2F10.1088%2F1681-7575%2Fac023f&rft.aulast=Quincey&rft.aufirst=Paul&rft_id=http%3A%2F%2Fdx.doi.org%2F10.1088%2F1681-7575%2Fac023f&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988"> </li> <li id="cite_note-54"><a href="#cite_ref-54">^</a> Siano (<a href="#CITEREFSiano1985-I">1985-I</a>, <a href="#CITEREFSiano1985-II">1985-II</a>) </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=Dimensional_analysis&action=edit&section=40" title="Edit section: References">edit</a>]</div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBarenblatt1996" class="citation cs2"><a href="/wiki/Grigory_Barenblatt" title="Grigory Barenblatt">Barenblatt, G. I.</a> (1996), Scaling, Self-Similarity, and Intermediate Asymptotics, Cambridge, UK: Cambridge University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-43522-2" title="Special:BookSources/978-0-521-43522-2"><bdi>978-0-521-43522-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Scaling%2C+Self-Similarity%2C+and+Intermediate+Asymptotics&rft.place=Cambridge%2C+UK&rft.pub=Cambridge+University+Press&rft.date=1996&rft.isbn=978-0-521-43522-2&rft.aulast=Barenblatt&rft.aufirst=G.+I.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBhaskarNigam1990" class="citation cs2">Bhaskar, R.; Nigam, Anil (1990), "Qualitative Physics Using Dimensional Analysis", Artificial Intelligence, 45 (1–2): 73–111, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0004-3702%2890%2990038-2">10.1016/0004-3702(90)90038-2</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Artificial+Intelligence&rft.atitle=Qualitative+Physics+Using+Dimensional+Analysis&rft.volume=45&rft.issue=1%E2%80%932&rft.pages=73-111&rft.date=1990&rft_id=info%3Adoi%2F10.1016%2F0004-3702%2890%2990038-2&rft.aulast=Bhaskar&rft.aufirst=R.&rft.au=Nigam%2C+Anil&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBhaskarNigam1991" class="citation cs2">Bhaskar, R.; Nigam, Anil (1991), "Qualitative Explanations of Red Giant Formation", The Astrophysical Journal, 372: 592–6, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1991ApJ...372..592B">1991ApJ...372..592B</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1086%2F170003">10.1086/170003</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Astrophysical+Journal&rft.atitle=Qualitative+Explanations+of+Red+Giant+Formation&rft.volume=372&rft.pages=592-6&rft.date=1991&rft_id=info%3Adoi%2F10.1086%2F170003&rft_id=info%3Abibcode%2F1991ApJ...372..592B&rft.aulast=Bhaskar&rft.aufirst=R.&rft.au=Nigam%2C+Anil&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBoucherAlves1960" class="citation cs2">Boucher; Alves (1960), "Dimensionless Numbers", Chemical Engineering Progress, 55: 55–64</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Chemical+Engineering+Progress&rft.atitle=Dimensionless+Numbers&rft.volume=55&rft.pages=55-64&rft.date=1960&rft.au=Boucher&rft.au=Alves&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBridgman1922" class="citation cs2"><a href="/wiki/Percy_Williams_Bridgman" title="Percy Williams Bridgman">Bridgman, P. W.</a> (1922), Dimensional Analysis, Yale University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-548-91029-0" title="Special:BookSources/978-0-548-91029-0"><bdi>978-0-548-91029-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Dimensional+Analysis&rft.pub=Yale+University+Press&rft.date=1922&rft.isbn=978-0-548-91029-0&rft.aulast=Bridgman&rft.aufirst=P.+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBuckingham1914" class="citation cs2"><a href="/wiki/Edgar_Buckingham" title="Edgar Buckingham">Buckingham, Edgar</a> (1914), <a rel="nofollow" class="external text" href="https://babel.hathitrust.org/cgi/pt?id=uc1.31210014450082&view=1up&seq=905">"On Physically Similar Systems: Illustrations of the Use of Dimensional Analysis"</a>, Physical Review, 4 (4): 345–376, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1914PhRv....4..345B">1914PhRv....4..345B</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRev.4.345">10.1103/PhysRev.4.345</a>, <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/10338.dmlcz%2F101743">10338.dmlcz/101743</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review&rft.atitle=On+Physically+Similar+Systems%3A+Illustrations+of+the+Use+of+Dimensional+Analysis&rft.volume=4&rft.issue=4&rft.pages=345-376&rft.date=1914&rft_id=info%3Ahdl%2F10338.dmlcz%2F101743&rft_id=info%3Adoi%2F10.1103%2FPhysRev.4.345&rft_id=info%3Abibcode%2F1914PhRv....4..345B&rft.aulast=Buckingham&rft.aufirst=Edgar&rft_id=https%3A%2F%2Fbabel.hathitrust.org%2Fcgi%2Fpt%3Fid%3Duc1.31210014450082%26view%3D1up%26seq%3D905&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDrobot1953–1954" class="citation cs2">Drobot, S. (1953–1954), <a rel="nofollow" class="external text" href="http://matwbn.icm.edu.pl/ksiazki/sm/sm14/sm1418.pdf">"On the foundations of dimensional analysis"</a> (PDF), Studia Mathematica, 14: 84–99, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4064%2Fsm-14-1-84-99">10.4064/sm-14-1-84-99</a>, <a rel="nofollow" class="external text" href="https://web.archive.org/web/20040116160647/http://matwbn.icm.edu.pl/ksiazki/sm/sm14/sm1418.pdf">archived</a> (PDF) from the original on 16 January 2004</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Studia+Mathematica&rft.atitle=On+the+foundations+of+dimensional+analysis&rft.volume=14&rft.pages=84-99&rft.date=1953%2F1954&rft_id=info%3Adoi%2F10.4064%2Fsm-14-1-84-99&rft.aulast=Drobot&rft.aufirst=S.&rft_id=http%3A%2F%2Fmatwbn.icm.edu.pl%2Fksiazki%2Fsm%2Fsm14%2Fsm1418.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFourier1822" class="citation cs2 cs1-prop-foreign-lang-source">Fourier, Joseph (1822), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=TDQJAAAAIAAJ&pg=PR3">Theorie analytique de la chaleur</a> (in French), Paris: Firmin Didot</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theorie+analytique+de+la+chaleur&rft.place=Paris&rft.pub=Firmin+Didot&rft.date=1822&rft.aulast=Fourier&rft.aufirst=Joseph&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DTDQJAAAAIAAJ%26pg%3DPR3&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGibbings2011" class="citation cs2">Gibbings, J.C. (2011), Dimensional Analysis, Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-84996-316-9" title="Special:BookSources/978-1-84996-316-9"><bdi>978-1-84996-316-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Dimensional+Analysis&rft.pub=Springer&rft.date=2011&rft.isbn=978-1-84996-316-9&rft.aulast=Gibbings&rft.aufirst=J.C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHart1994" class="citation cs2"><a href="/wiki/George_W._Hart" title="George W. Hart">Hart, George W.</a> (1994), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=NeWVeylbeiQC&pg=PA186">"The theory of dimensioned matrices"</a>, in Lewis, John G. (ed.), Proceedings of the Fifth SIAM Conference on Applied Linear Algebra, SIAM, pp. 186–190, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-89871-336-7" title="Special:BookSources/978-0-89871-336-7"><bdi>978-0-89871-336-7</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+theory+of+dimensioned+matrices&rft.btitle=Proceedings+of+the+Fifth+SIAM+Conference+on+Applied+Linear+Algebra&rft.pages=186-190&rft.pub=SIAM&rft.date=1994&rft.isbn=978-0-89871-336-7&rft.aulast=Hart&rft.aufirst=George+W.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DNeWVeylbeiQC%26pg%3DPA186&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988"> As <a rel="nofollow" class="external text" href="http://www.georgehart.com/research/tdm.ps">postscript</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHart1995" class="citation cs2">Hart, George W. (1995), <a rel="nofollow" class="external text" href="http://www.georgehart.com/research/multanal.html">Multidimensional Analysis: Algebras and Systems for Science and Engineering</a>, Springer-Verlag, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-94417-3" title="Special:BookSources/978-0-387-94417-3"><bdi>978-0-387-94417-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Multidimensional+Analysis%3A+Algebras+and+Systems+for+Science+and+Engineering&rft.pub=Springer-Verlag&rft.date=1995&rft.isbn=978-0-387-94417-3&rft.aulast=Hart&rft.aufirst=George+W.&rft_id=http%3A%2F%2Fwww.georgehart.com%2Fresearch%2Fmultanal.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHuntley1967" class="citation cs2">Huntley, H. E. (1967), <a rel="nofollow" class="external text" href="https://openlibrary.org/books/OL6128830M">Dimensional Analysis</a>, Dover, <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/682090763">682090763</a>, <a href="/wiki/OL_(identifier)" class="mw-redirect" title="OL (identifier)">OL</a> <a rel="nofollow" class="external text" href="https://openlibrary.org/books/OL6128830M">6128830M</a>, LOC 67-17978</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Dimensional+Analysis&rft.pub=Dover&rft.date=1967&rft_id=info%3Aoclcnum%2F682090763&rft_id=https%3A%2F%2Fopenlibrary.org%2Fbooks%2FOL6128830M%23id-name%3DOL&rft.aulast=Huntley&rft.aufirst=H.+E.&rft_id=https%3A%2F%2Fopenlibrary.org%2Fbooks%2FOL6128830M&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKlinkenberg1955" class="citation cs2">Klinkenberg, A. (1955), "Dimensional systems and systems of units in physics with special reference to chemical engineering: Part I. The principles according to which dimensional systems and systems of units are constructed", Chemical Engineering Science, 4 (3): 130–140, 167–177, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1955ChEnS...4..130K">1955ChEnS...4..130K</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0009-2509%2855%2980004-8">10.1016/0009-2509(55)80004-8</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Chemical+Engineering+Science&rft.atitle=Dimensional+systems+and+systems+of+units+in+physics+with+special+reference+to+chemical+engineering%3A+Part+I.+The+principles+according+to+which+dimensional+systems+and+systems+of+units+are+constructed&rft.volume=4&rft.issue=3&rft.pages=130-140%2C+167-177&rft.date=1955&rft_id=info%3Adoi%2F10.1016%2F0009-2509%2855%2980004-8&rft_id=info%3Abibcode%2F1955ChEnS...4..130K&rft.aulast=Klinkenberg&rft.aufirst=A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLanghaar1951" class="citation cs2"><a href="/wiki/Henry_L._Langhaar" title="Henry L. Langhaar">Langhaar, Henry L.</a> (1951), Dimensional Analysis and Theory of Models, Wiley, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-88275-682-0" title="Special:BookSources/978-0-88275-682-0"><bdi>978-0-88275-682-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Dimensional+Analysis+and+Theory+of+Models&rft.pub=Wiley&rft.date=1951&rft.isbn=978-0-88275-682-0&rft.aulast=Langhaar&rft.aufirst=Henry+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMendezOrdóñez2005" class="citation cs2">Mendez, P.F.; Ordóñez, F. (September 2005), "Scaling Laws From Statistical Data and Dimensional Analysis", Journal of Applied Mechanics, 72 (5): 648–657, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2005JAM....72..648M">2005JAM....72..648M</a>, <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.422.610">10.1.1.422.610</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1115%2F1.1943434">10.1115/1.1943434</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Applied+Mechanics&rft.atitle=Scaling+Laws+From+Statistical+Data+and+Dimensional+Analysis&rft.volume=72&rft.issue=5&rft.pages=648-657&rft.date=2005-09&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.422.610%23id-name%3DCiteSeerX&rft_id=info%3Adoi%2F10.1115%2F1.1943434&rft_id=info%3Abibcode%2F2005JAM....72..648M&rft.aulast=Mendez&rft.aufirst=P.F.&rft.au=Ord%C3%B3%C3%B1ez%2C+F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMoody1944" class="citation cs2">Moody, L. F. (1944), "Friction Factors for Pipe Flow", Transactions of the American Society of Mechanical Engineers, 66 (671): 671–678, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1115%2F1.4018140">10.1115/1.4018140</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Transactions+of+the+American+Society+of+Mechanical+Engineers&rft.atitle=Friction+Factors+for+Pipe+Flow&rft.volume=66&rft.issue=671&rft.pages=671-678&rft.date=1944&rft_id=info%3Adoi%2F10.1115%2F1.4018140&rft.aulast=Moody&rft.aufirst=L.+F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMurphy1949" class="citation cs2">Murphy, N. F. (1949), "Dimensional Analysis", Bulletin of the Virginia Polytechnic Institute, 42 (6)</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bulletin+of+the+Virginia+Polytechnic+Institute&rft.atitle=Dimensional+Analysis&rft.volume=42&rft.issue=6&rft.date=1949&rft.aulast=Murphy&rft.aufirst=N.+F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPerry1944" class="citation cs2">Perry, J. H.; et al. (1944), "Standard System of Nomenclature for Chemical Engineering Unit Operations", Transactions of the American Institute of Chemical Engineers, 40 (251)</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Transactions+of+the+American+Institute+of+Chemical+Engineers&rft.atitle=Standard+System+of+Nomenclature+for+Chemical+Engineering+Unit+Operations&rft.volume=40&rft.issue=251&rft.date=1944&rft.aulast=Perry&rft.aufirst=J.+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPesic2005" class="citation cs2">Pesic, Peter (2005), <a rel="nofollow" class="external text" href="https://archive.org/details/skyinbottle00pesi/page/227">Sky in a Bottle</a>, MIT Press, pp. <a rel="nofollow" class="external text" href="https://archive.org/details/skyinbottle00pesi/page/227">227–8</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-262-16234-0" title="Special:BookSources/978-0-262-16234-0"><bdi>978-0-262-16234-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Sky+in+a+Bottle&rft.pages=227-8&rft.pub=MIT+Press&rft.date=2005&rft.isbn=978-0-262-16234-0&rft.aulast=Pesic&rft.aufirst=Peter&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fskyinbottle00pesi%2Fpage%2F227&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPetty2001" class="citation cs2">Petty, G. W. (2001), "Automated computation and consistency checking of physical dimensions and units in scientific programs", Software: Practice and Experience, 31 (11): 1067–76, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fspe.401">10.1002/spe.401</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:206506776">206506776</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Software%3A+Practice+and+Experience&rft.atitle=Automated+computation+and+consistency+checking+of+physical+dimensions+and+units+in+scientific+programs&rft.volume=31&rft.issue=11&rft.pages=1067-76&rft.date=2001&rft_id=info%3Adoi%2F10.1002%2Fspe.401&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A206506776%23id-name%3DS2CID&rft.aulast=Petty&rft.aufirst=G.+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPorter1933" class="citation cs2">Porter, Alfred W. (1933), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=SxguAQAAIAAJ">The Method of Dimensions</a> (3rd ed.), Methuen</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Method+of+Dimensions&rft.edition=3rd&rft.pub=Methuen&rft.date=1933&rft.aulast=Porter&rft.aufirst=Alfred+W.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DSxguAQAAIAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJ._W._Strutt_(3rd_Baron_Rayleigh)1915" class="citation cs2"><a href="/wiki/John_William_Strutt,_3rd_Baron_Rayleigh" title="John William Strutt, 3rd Baron Rayleigh">J. W. Strutt (3rd Baron Rayleigh)</a> (1915), "The Principle of Similitude", Nature, 95 (2368): 66–8, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1915Natur..95...66R">1915Natur..95...66R</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2F095066c0">10.1038/095066c0</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nature&rft.atitle=The+Principle+of+Similitude&rft.volume=95&rft.issue=2368&rft.pages=66-8&rft.date=1915&rft_id=info%3Adoi%2F10.1038%2F095066c0&rft_id=info%3Abibcode%2F1915Natur..95...66R&rft.au=J.+W.+Strutt+%283rd+Baron+Rayleigh%29&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSiano1985-I" class="citation cs2">Siano, Donald (1985), "Orientational Analysis – A Supplement to Dimensional Analysis – I", Journal of the Franklin Institute, 320 (6): 267–283, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0016-0032%2885%2990031-6">10.1016/0016-0032(85)90031-6</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+the+Franklin+Institute&rft.atitle=Orientational+Analysis+%E2%80%93+A+Supplement+to+Dimensional+Analysis+%E2%80%93+I&rft.volume=320&rft.issue=6&rft.pages=267-283&rft.date=1985&rft_id=info%3Adoi%2F10.1016%2F0016-0032%2885%2990031-6&rft.aulast=Siano&rft.aufirst=Donald&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSiano1985-II" class="citation cs2">Siano, Donald (1985), "Orientational Analysis, Tensor Analysis and The Group Properties of the SI Supplementary Units – II", Journal of the Franklin Institute, 320 (6): 285–302, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0016-0032%2885%2990032-8">10.1016/0016-0032(85)90032-8</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+the+Franklin+Institute&rft.atitle=Orientational+Analysis%2C+Tensor+Analysis+and+The+Group+Properties+of+the+SI+Supplementary+Units+%E2%80%93+II&rft.volume=320&rft.issue=6&rft.pages=285-302&rft.date=1985&rft_id=info%3Adoi%2F10.1016%2F0016-0032%2885%2990032-8&rft.aulast=Siano&rft.aufirst=Donald&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSilberbergMcKetta1953" class="citation cs2">Silberberg, I. H.; McKetta, J. J. Jr. (1953), "Learning How to Use Dimensional Analysis", Petroleum Refiner, 32 (4): 5</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Petroleum+Refiner&rft.atitle=Learning+How+to+Use+Dimensional+Analysis&rft.volume=32&rft.issue=4&rft.pages=5&rft.date=1953&rft.aulast=Silberberg&rft.aufirst=I.+H.&rft.au=McKetta%2C+J.+J.+Jr.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988">, (5): 147, (6): 101, (7): 129</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTao2012" class="citation web cs1"><a href="/wiki/Terence_Tao" title="Terence Tao">Tao, Terence</a> (2012). <a rel="nofollow" class="external text" href="https://terrytao.wordpress.com/2012/12/29/a-mathematical-formalisation-of-dimensional-analysis/">"A mathematical formalisation of dimensional analysis"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=A+mathematical+formalisation+of+dimensional+analysis&rft.date=2012&rft.aulast=Tao&rft.aufirst=Terence&rft_id=https%3A%2F%2Fterrytao.wordpress.com%2F2012%2F12%2F29%2Fa-mathematical-formalisation-of-dimensional-analysis%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVan_Driest1946" class="citation cs2">Van Driest, E. R. (March 1946), "On Dimensional Analysis and the Presentation of Data in Fluid Flow Problems", Journal of Applied Mechanics, 68 (A–34)</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Applied+Mechanics&rft.atitle=On+Dimensional+Analysis+and+the+Presentation+of+Data+in+Fluid+Flow+Problems&rft.volume=68&rft.issue=A%E2%80%9334&rft.date=1946-03&rft.aulast=Van+Driest&rft.aufirst=E.+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWhitney1968" class="citation cs2">Whitney, H. (1968), "The Mathematics of Physical Quantities, Parts I and II", American Mathematical Monthly, 75 (2): 115–138, 227–256, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2315883">10.2307/2315883</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2315883">2315883</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Mathematical+Monthly&rft.atitle=The+Mathematics+of+Physical+Quantities%2C+Parts+I+and+II&rft.volume=75&rft.issue=2&rft.pages=115-138%2C+227-256&rft.date=1968&rft_id=info%3Adoi%2F10.2307%2F2315883&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2315883%23id-name%3DJSTOR&rft.aulast=Whitney&rft.aufirst=H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988"></li> <li><a href="/wiki/Edwin_Bidwell_Wilson" title="Edwin Bidwell Wilson">Wilson, Edwin B.</a> (1920) <a rel="nofollow" class="external text" href="https://archive.org/details/aeronautics00wilsgoog/page/n197/mode/2up">"Theory of Dimensions"</a>, chapter XI of Aeronautics, via <a href="/wiki/Internet_Archive" title="Internet Archive">Internet Archive</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVignaux1992" class="citation cs2">Vignaux, GA (1992), "Dimensional Analysis in Data Modelling", in Erickson, Gary J.; Neudorfer, Paul O. (eds.), Maximum entropy and Bayesian methods: proceedings of the Eleventh International Workshop on Maximum Entropy and Bayesian Methods of Statistical Analysis, Seattle, 1991, Kluwer Academic, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7923-2031-9" title="Special:BookSources/978-0-7923-2031-9"><bdi>978-0-7923-2031-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Dimensional+Analysis+in+Data+Modelling&rft.btitle=Maximum+entropy+and+Bayesian+methods%3A+proceedings+of+the+Eleventh+International+Workshop+on+Maximum+Entropy+and+Bayesian+Methods+of+Statistical+Analysis%2C+Seattle%2C+1991&rft.pub=Kluwer+Academic&rft.date=1992&rft.isbn=978-0-7923-2031-9&rft.aulast=Vignaux&rft.aufirst=GA&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKasprzakLysikRybaczuk1990" class="citation cs2">Kasprzak, Wacław; Lysik, Bertold; Rybaczuk, Marek (1990), Dimensional Analysis in the Identification of Mathematical Models, World Scientific, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-02-0304-7" title="Special:BookSources/978-981-02-0304-7"><bdi>978-981-02-0304-7</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Dimensional+Analysis+in+the+Identification+of+Mathematical+Models&rft.pub=World+Scientific&rft.date=1990&rft.isbn=978-981-02-0304-7&rft.aulast=Kasprzak&rft.aufirst=Wac%C5%82aw&rft.au=Lysik%2C+Bertold&rft.au=Rybaczuk%2C+Marek&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988"></li></ul> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2>[<a href="/w/index.php?title=Dimensional_analysis&action=edit&section=41" title="Edit section: Further reading">edit</a>]</div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGiancoli2014" class="citation book cs1">Giancoli, Douglas C. (2014). "1. Introduction, Measurement, Estimating §1.8 Dimensions and Dimensional Analysis". Physics: Principles with Applications (7th ed.). Pearson. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-321-62592-2" title="Special:BookSources/978-0-321-62592-2"><bdi>978-0-321-62592-2</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/853154197">853154197</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=1.+Introduction%2C+Measurement%2C+Estimating+%C2%A71.8+Dimensions+and+Dimensional+Analysis&rft.btitle=Physics%3A+Principles+with+Applications&rft.edition=7th&rft.pub=Pearson&rft.date=2014&rft_id=info%3Aoclcnum%2F853154197&rft.isbn=978-0-321-62592-2&rft.aulast=Giancoli&rft.aufirst=Douglas+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988"></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2>[<a href="/w/index.php?title=Dimensional_analysis&action=edit&section=42" title="Edit section: External links">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style 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href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-flex"> <div class="side-box-image"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <a href="https://commons.wikimedia.org/wiki/Category:Dimensional_analysis" class="extiw" title="commons:Category:Dimensional analysis">Dimensional analysis</a>.</div></div> </div> <ul><li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20100410142839/http://www.roymech.co.uk/Related/Fluids/Dimension_Analysis.html">List of dimensions for variety of physical quantities</a></li> <li><a rel="nofollow" class="external text" href="http://www.calchemy.com/uclive.htm">Unicalc Live web calculator doing units conversion by dimensional analysis</a></li> <li><a rel="nofollow" class="external text" href="http://www.boost.org/doc/libs/1_66_0/doc/html/boost_units.html">A C++ implementation of compile-time dimensional analysis in the Boost open-source libraries</a></li> <li><a rel="nofollow" class="external text" href="http://www.math.ntnu.no/~hanche/notes/buckingham/buckingham-a4.pdf">Buckingham's pi-theorem</a></li> <li><a rel="nofollow" class="external text" href="http://QuantitySystem.CodePlex.com">Quantity System calculator for units conversion based on dimensional approach</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20171224025732/http://quantitysystem.codeplex.com/">Archived</a> 24 December 2017 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li> <li><a rel="nofollow" class="external text" href="http://www.outlawmapofphysics.com">Units, quantities, and fundamental constants project dimensional analysis maps</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBowley2009" class="citation web cs1">Bowley, Roger (2009). <a rel="nofollow" class="external text" href="http://www.sixtysymbols.com/videos/dimensional.htm">"Dimensional Analysis"</a>. Sixty Symbols. <a href="/wiki/Brady_Haran" title="Brady Haran">Brady Haran</a> for the <a href="/wiki/University_of_Nottingham" title="University of Nottingham">University of Nottingham</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Sixty+Symbols&rft.atitle=Dimensional+Analysis&rft.date=2009&rft.aulast=Bowley&rft.aufirst=Roger&rft_id=http%3A%2F%2Fwww.sixtysymbols.com%2Fvideos%2Fdimensional.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDureisseix2019" class="citation thesis cs1">Dureisseix, David (2019). <a rel="nofollow" class="external text" href="https://cel.archives-ouvertes.fr/cel-01380149">An introduction to dimensional analysis</a> (lecture). INSA Lyon.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adissertation&rft.title=An+introduction+to+dimensional+analysis&rft.inst=INSA+Lyon&rft.date=2019&rft.aulast=Dureisseix&rft.aufirst=David&rft_id=https%3A%2F%2Fcel.archives-ouvertes.fr%2Fcel-01380149&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADimensional+analysis" class="Z3988"></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol 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href="/wiki/Imperial_units" title="Imperial units">UK imperial system</a></li> <li><a href="/wiki/United_States_customary_units" title="United States customary units">US customary units (USCS/USC)</a></li> <li><a href="/wiki/Chinese_units_of_measurement" title="Chinese units of measurement">Chinese</a> <ul><li><a href="/wiki/Hong_Kong_units_of_measurement" title="Hong Kong units of measurement">Hong Kong</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:5em">Specific</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Apothecaries%27_system" title="Apothecaries' system">Apothecaries'</a></li> <li><a href="/wiki/Avoirdupois_system" class="mw-redirect" title="Avoirdupois system">Avoirdupois</a></li> <li><a href="/wiki/Troy_weight" title="Troy weight">Troy</a></li> <li><a href="/wiki/Astronomical_system_of_units" title="Astronomical system of units">Astronomical</a></li> <li><a href="/wiki/Conventional_electrical_unit" title="Conventional electrical unit">Electrical</a></li> <li><a href="/wiki/English_Engineering_Units" title="English Engineering Units">English Engineering Units</a> (US)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:5em"><a href="/wiki/Natural_units" title="Natural units">Natural</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Atomic_units" title="Atomic units">Atomic</a></li> <li><a href="/wiki/Geometrized_unit_system" title="Geometrized unit system">Geometrised</a></li> <li><a href="/wiki/Heaviside%E2%80%93Lorentz_units" title="Heaviside–Lorentz units">Heaviside–Lorentz</a></li> <li><a href="/wiki/Planck_units" title="Planck units">Planck</a></li> <li><a href="/wiki/Natural_units#Quantum_chromodynamics_units" title="Natural units">Quantum chromodynamical</a></li> <li><a href="/wiki/Stoney_units" title="Stoney units">Stoney</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Background</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:5em">Metric</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Metric_system" title="Metric system">Overview</a></li> <li><a href="/wiki/Outline_of_the_metric_system" title="Outline of the metric system">Outline</a></li> <li><a href="/wiki/History_of_the_metric_system" title="History of the metric system">History</a></li> <li><a href="/wiki/Metrication" title="Metrication">Metrication</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:5em">UK/US</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Imperial_and_US_customary_measurement_systems" title="Imperial and US customary measurement systems">Overview</a></li> <li><a href="/wiki/Comparison_of_the_imperial_and_US_customary_measurement_systems" title="Comparison of the imperial and US customary measurement systems">Comparison</a></li> <li><a href="/wiki/Foot%E2%80%93pound%E2%80%93second_system" class="mw-redirect" title="Foot–pound–second system">Foot–pound–second</a> (FPS)</li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Historic</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:5em">Metric</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/MKS_system_of_units" class="mw-redirect" title="MKS system of units">metre–kilogram–second</a> (MKS)</li> <li><a href="/wiki/Metre%E2%80%93tonne%E2%80%93second_system_of_units" title="Metre–tonne–second system of units">metre–tonne–second</a> (MTS)</li> <li><a href="/wiki/Centimetre%E2%80%93gram%E2%80%93second_system_of_units" title="Centimetre–gram–second system of units">centimetre–gram–second</a> (CGS)</li> <li><a href="/wiki/Gravitational_metric_system" title="Gravitational metric system">gravitational</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:5em">Europe</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Old_Cornish_units_of_measurement" title="Old Cornish units of measurement">Cornish</a></li> <li><a href="/wiki/Cypriot_units_of_measurement" title="Cypriot units of measurement">Cypriot</a></li> <li><a href="/wiki/Czech_units_of_measurement" title="Czech units of measurement">Czech</a></li> <li><a href="/wiki/Danish_units_of_measurement" title="Danish units of measurement">Danish</a></li> <li><a href="/wiki/Dutch_units_of_measurement" title="Dutch units of measurement">Dutch</a></li> <li><a href="/wiki/English_units" title="English units">English</a> <ul><li><a href="/wiki/Winchester_measure" title="Winchester measure">Winchester</a></li> <li><a href="/wiki/Exchequer_Standards" title="Exchequer Standards">Exchequer</a></li></ul></li> <li><a href="/wiki/Estonian_units_of_measurement" title="Estonian units of measurement">Estonian</a></li> <li><a href="/wiki/Obsolete_Finnish_units_of_measurement" title="Obsolete Finnish units of measurement">Finnish</a></li> <li><a href="/wiki/French_units_of_measurement" title="French units of measurement">French</a> <ul><li><a href="/wiki/Traditional_French_units_of_measurement" title="Traditional French units of measurement">Traditional</a></li> <li><a href="/wiki/Mesures_usuelles" title="Mesures usuelles">Mesures usuelles</a></li></ul></li> <li><a href="/wiki/Obsolete_German_units_of_measurement" class="mw-redirect" title="Obsolete German units of measurement">German</a></li> <li><a href="/wiki/Greek_units_of_measurement" title="Greek units of measurement">Greek</a> <ul><li><a href="/wiki/Byzantine_units_of_measurement" title="Byzantine units of measurement">Byzantine</a></li></ul></li> <li><a href="/wiki/Hungarian_units_of_measurement" title="Hungarian units of measurement">Hungarian</a></li> <li><a href="/wiki/Icelandic_units_of_measurement" title="Icelandic units of measurement">Icelandic</a></li> <li><a href="/wiki/Old_Irish_units_of_measurement" class="mw-redirect" title="Old Irish units of measurement">Irish</a></li> <li><a href="/wiki/Italian_units_of_measurement" title="Italian units of measurement">Italian</a></li> <li><a href="/wiki/Latvian_units_of_measurement" title="Latvian units of measurement">Latvian</a></li> <li><a href="/wiki/Luxembourgian_units_of_measurement" class="mw-redirect" title="Luxembourgian units of measurement">Luxembourgian</a></li> <li><a href="/wiki/Maltese_units_of_measurement" title="Maltese units of measurement">Maltese</a></li> <li><a href="/wiki/Norwegian_units_of_measurement" title="Norwegian units of measurement">Norwegian</a></li> <li><a href="/wiki/Ottoman_units_of_measurement" title="Ottoman units of measurement">Ottoman</a></li> <li><a href="/wiki/Polish_units_of_measurement" title="Polish units of measurement">Polish</a></li> <li><a href="/wiki/Portuguese_units_of_measurement" title="Portuguese units of measurement">Portuguese</a></li> <li><a href="/wiki/Romanian_units_of_measurement" title="Romanian units of measurement">Romanian</a></li> <li><a href="/wiki/Obsolete_Russian_units_of_measurement" class="mw-redirect" title="Obsolete Russian units of measurement">Russian</a></li> <li><a href="/wiki/Scottish_units" title="Scottish units">Scottish</a></li> <li><a href="/wiki/Obsolete_Serbian_units_of_measurement" title="Obsolete Serbian units of measurement">Serbian</a></li> <li><a href="/wiki/Slovak_units_of_measurement" title="Slovak units of measurement">Slovak</a></li> <li><a href="/wiki/Spanish_units_of_measurement" title="Spanish units of measurement">Spanish</a></li> <li><a href="/wiki/Swedish_units_of_measurement" title="Swedish units of measurement">Swedish</a></li> <li><a href="/wiki/Swiss_units_of_measurement" title="Swiss units of measurement">Swiss</a></li> <li><a href="/wiki/Welsh_units" title="Welsh units">Welsh</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:5em">Asia</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Afghan_units_of_measurement" title="Afghan units of measurement">Afghan</a></li> <li><a href="/wiki/Cambodian_units_of_measurement" title="Cambodian units of measurement">Cambodian</a></li> <li><a href="/wiki/Indian_units_of_measurement" title="Indian units of measurement">Indian</a></li> <li><a href="/wiki/Indonesian_units_of_measurement" title="Indonesian units of measurement">Indonesian</a></li> <li><a href="/wiki/Japanese_units_of_measurement" title="Japanese units of measurement">Japanese</a></li> <li><a href="/wiki/Korean_units_of_measurement" title="Korean units of measurement">Korean</a></li> <li><a href="/wiki/Mongolian_units" title="Mongolian units">Mongolian</a></li> <li><a href="/wiki/Myanmar_units_of_measurement" title="Myanmar units of measurement">Myanmar</a></li> <li><a href="/wiki/Nepalese_customary_units_of_measurement" title="Nepalese customary units of measurement">Nepalese</a></li> <li><a href="/wiki/Omani_units_of_measurement" title="Omani units of measurement">Omani</a></li> <li><a href="/wiki/History_of_measurement_systems_in_Pakistan" title="History of measurement systems in Pakistan">Pakistani</a></li> <li><a href="/wiki/Philippine_units_of_measurement" title="Philippine units of measurement">Philippine</a></li> <li><a href="/wiki/Abucco" title="Abucco">Pegu</a></li> <li><a href="/wiki/Singaporean_units_of_measurement" title="Singaporean units of measurement">Singaporean</a></li> <li><a href="/wiki/Sri_Lankan_units_of_measurement" title="Sri Lankan units of measurement">Sri Lankan</a></li> <li><a href="/wiki/Syrian_units_of_measurement" title="Syrian units of measurement">Syrian</a></li> <li><a href="/wiki/Taiwanese_units_of_measurement" title="Taiwanese units of measurement">Taiwanese</a></li> <li><a href="/wiki/Obsolete_Tatar_units_of_measurement" class="mw-redirect" title="Obsolete Tatar units of measurement">Tatar</a></li> <li><a href="/wiki/Thai_units_of_measurement" title="Thai units of measurement">Thai</a></li> <li><a href="/wiki/Vietnamese_units_of_measurement" title="Vietnamese units of measurement">Vietnamese</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:5em">Africa</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Algerian_units_of_measurement" title="Algerian units of measurement">Algerian</a></li> <li><a href="/wiki/Ethiopian_units_of_measurement" title="Ethiopian units of measurement">Ethiopian</a></li> <li><a href="/wiki/Egyptian_units_of_measurement" title="Egyptian units of measurement">Egyptian</a></li> <li><a href="/wiki/Eritrean_units_of_measurement" title="Eritrean units of measurement">Eritrean</a></li> <li><a href="/wiki/Guinean_units_of_measurement" title="Guinean units of measurement">Guinean</a></li> <li><a href="/wiki/Libyan_units_of_measurement" title="Libyan units of measurement">Libyan</a></li> <li><a href="/wiki/Malagasy_units_of_measurement" title="Malagasy units of measurement">Malagasy</a></li> <li><a href="/wiki/Mauritian_units_of_measurement" title="Mauritian units of measurement">Mauritian</a></li> <li><a href="/wiki/Moroccan_units_of_measurement" title="Moroccan units of measurement">Moroccan</a></li> <li><a href="/wiki/Seychellois_units_of_measurement" title="Seychellois units of measurement">Seychellois</a></li> <li><a href="/wiki/Somali_units_of_measurement" title="Somali units of measurement">Somali</a></li> <li><a href="/wiki/South_African_units_of_measurement" title="South African units of measurement">South African</a></li> <li><a href="/wiki/Tunisian_units_of_measurement" title="Tunisian units of measurement">Tunisian</a></li> <li><a href="/wiki/Tanzanian_units_of_measurement" title="Tanzanian units of measurement">Tanzanian</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:5em">North America</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Costa_Rican_units_of_measurement" title="Costa Rican units of measurement">Costa Rican</a></li> <li><a href="/wiki/Cuban_units_of_measurement" title="Cuban units of measurement">Cuban</a></li> <li><a href="/wiki/Haitian_units_of_measurement" title="Haitian units of measurement">Haitian</a></li> <li><a href="/wiki/Honduran_units_of_measurement" title="Honduran units of measurement">Honduran</a></li> <li><a href="/wiki/Mexican_units_of_measurement" title="Mexican units of measurement">Mexican</a></li> <li><a href="/wiki/Nicaraguan_units_of_measurement" title="Nicaraguan units of measurement">Nicaraguan</a></li> <li><a href="/wiki/Puerto_Rican_units_of_measurement" title="Puerto Rican units of measurement">Puerto Rican</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:5em">South America</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Argentine_units_of_measurement" title="Argentine units of measurement">Argentine</a></li> <li><a href="/wiki/Bolivian_units_of_measurement" title="Bolivian units of measurement">Bolivian</a></li> <li><a href="/wiki/Brazilian_units_of_measurement" title="Brazilian units of measurement">Brazilian</a></li> <li><a href="/wiki/Chilean_units_of_measurement" title="Chilean units of measurement">Chilean</a></li> <li><a href="/wiki/Colombian_units_of_measurement" title="Colombian units of measurement">Colombian</a></li> <li><a href="/wiki/Paraguayan_units_of_measurement" title="Paraguayan units of measurement">Paraguayan</a></li> <li><a href="/wiki/Peruvian_units_of_measurement" title="Peruvian units of measurement">Peruvian</a></li> <li><a href="/wiki/Uruguayan_units_of_measurement" title="Uruguayan units of measurement">Uruguayan</a></li> <li><a href="/wiki/Venezuelan_units_of_measurement" title="Venezuelan units of measurement">Venezuelan</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Ancient</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Ancient_Arabic_units_of_measurement" title="Ancient Arabic units of measurement">Arabic</a></li> <li><a href="/wiki/Biblical_and_Talmudic_units_of_measurement" title="Biblical and Talmudic units of measurement">Biblical and Talmudic</a></li> <li><a href="/wiki/Ancient_Egyptian_units_of_measurement" title="Ancient Egyptian units of measurement">Egyptian</a></li> <li><a href="/wiki/Ancient_Greek_units_of_measurement" title="Ancient Greek units of measurement">Greek</a></li> <li><a href="/wiki/Hindu_units_of_time" title="Hindu units of time">Hindu</a></li> <li><a href="/wiki/History_of_measurement_systems_in_India" title="History of measurement systems in India">Indian</a></li> <li><a href="/wiki/Ancient_Mesopotamian_units_of_measurement" title="Ancient Mesopotamian units of measurement">Mesopotamian</a></li> <li><a href="/wiki/Persian_units_of_measurement" title="Persian units of measurement">Persian</a></li> <li><a href="/wiki/Ancient_Roman_units_of_measurement" title="Ancient Roman units of measurement">Roman</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">List articles</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/List_of_humorous_units_of_measurement" title="List of humorous units of measurement">Humorous</a></li> <li><a href="/wiki/List_of_obsolete_units_of_measurement" title="List of obsolete units of measurement">Obsolete</a></li> <li><a href="/wiki/List_of_unusual_units_of_measurement" title="List of unusual units of measurement">Unusual</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Absolute_scale" title="Absolute scale">Absolute scale</a></li> <li><a href="/wiki/N-body_units" title="N-body units">N-body</a></li> <li><a href="/wiki/Modulor" title="Modulor">Modulor</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="SI_base_quantities" style="padding:3px"><table class="nowraplinks hlist mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="3"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:SI_base_quantities" title="Template:SI base quantities"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:SI_base_quantities" title="Template talk:SI base quantities"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:SI_base_quantities" title="Special:EditPage/Template:SI base quantities"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="SI_base_quantities" style="font-size:114%;margin:0 4em"><a href="/wiki/SI_base_unit" title="SI base unit">SI base quantities</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Base quantities</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <table class="wikitable sortable" style="text-align:center; margin:0;"> <tbody><tr> <th colspan="3">Quantity </th> <th> </th> <th colspan="2"><a href="/wiki/SI_unit" class="mw-redirect" title="SI unit">SI unit</a> </th></tr> <tr> <th style="text-align:center;">Name </th> <th style="text-align:center;">Symbol </th> <th style="text-align:center;"><a class="mw-selflink selflink">Dimension symbol</a> </th> <th class="unsortable"> </th> <th style="text-align:center;">Unit name </th> <th style="text-align:center;">Unit symbol </th></tr> <tr> <td style="text-align:left;"><a href="/wiki/Time" title="Time">time, duration</a> </td> <td>t </td> <td style="font-family:sans-serif;font-style:normal;">T </td> <td> </td> <td style="text-align:left;"><a href="/wiki/Second" title="Second">second</a> </td> <td style="text-align:left;">s </td></tr> <tr> <td style="text-align:left;"><a href="/wiki/Length" title="Length">length</a> </td> <td>l, x, r, etc. </td> <td style="font-family:sans-serif;font-style:normal;">L </td> <td> </td> <td style="text-align:left;"><a href="/wiki/Metre" title="Metre">metre</a> </td> <td style="text-align:left;">m </td></tr> <tr> <td style="text-align:left;"><a href="/wiki/Mass" title="Mass">mass</a> </td> <td>m </td> <td style="font-family:sans-serif;font-style:normal;">M </td> <td> </td> <td style="text-align:left;"><a href="/wiki/Kilogram" title="Kilogram">kilogram</a> </td> <td style="text-align:left;">kg </td></tr> <tr> <td style="text-align:left;"><a href="/wiki/Electric_current" title="Electric current">electric current</a> </td> <td><abbr title="Uppercase italic letter 'i'"> I </abbr>, i </td> <td style="font-family:sans-serif;font-style:normal;"><abbr title="Uppercase sans-serif roman letter 'i'"> I </abbr> </td> <td> </td> <td style="text-align:left;"><a href="/wiki/Ampere" title="Ampere">ampere</a> </td> <td style="text-align:left;">A </td></tr> <tr> <td style="text-align:left;"><a href="/wiki/Thermodynamic_temperature" title="Thermodynamic temperature">thermodynamic temperature</a> </td> <td>T </td> <td style="font-family:sans-serif;font-style:normal;"><abbr title="Greek capital letter theta">Θ</abbr> </td> <td> </td> <td style="text-align:left;"><a href="/wiki/Kelvin" title="Kelvin">kelvin</a> </td> <td style="text-align:left;">K </td></tr> <tr> <td style="text-align:left;"><a href="/wiki/Amount_of_substance" title="Amount of substance">amount of substance</a> </td> <td>n </td> <td style="font-family:sans-serif;font-style:normal;">N </td> <td> </td> <td style="text-align:left;"><a href="/wiki/Mole_(unit)" title="Mole (unit)">mole</a> </td> <td style="text-align:left;">mol </td></tr> <tr> <td style="text-align:left;"><a href="/wiki/Luminous_intensity" title="Luminous intensity">luminous intensity</a> </td> <td>Iv </td> <td style="font-family:sans-serif;font-style:normal;">J </td> <td> </td> <td style="text-align:left;"><a href="/wiki/Candela" title="Candela">candela</a> </td> <td style="text-align:left;">cd </td></tr></tbody></table> </div></td><td class="noviewer navbox-image" rowspan="2" style="width:1px;padding:0 0 0 2px"><div><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ab/Unit_relations_in_the_new_SI.svg/200px-Unit_relations_in_the_new_SI.svg.png" decoding="async" width="200" height="192" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ab/Unit_relations_in_the_new_SI.svg/300px-Unit_relations_in_the_new_SI.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ab/Unit_relations_in_the_new_SI.svg/400px-Unit_relations_in_the_new_SI.svg.png 2x" data-file-width="625" data-file-height="600" /></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">See also</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/History_of_the_metric_system" title="History of the metric system">History of the metric system</a></li> <li><a href="/wiki/International_System_of_Quantities" title="International System of Quantities">International System of Quantities</a></li> <li><a href="/wiki/2019_revision_of_the_SI" title="2019 revision of the SI">2019 revision</a></li> <li><a href="/wiki/System_of_measurement" class="mw-redirect" title="System of measurement">Systems of measurement</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="3"><div> <ul><li><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /> <a href="/wiki/Category:SI_base_units" title="Category:SI base units">Category</a></li> <li><a href="/wiki/Outline_of_the_metric_system" title="Outline of the metric system">Outline</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"><style data-mw-deduplicate="TemplateStyles:r1038841319">.mw-parser-output .tooltip-dotted{border-bottom:1px dotted;cursor:help}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1038841319"></div><div role="navigation" class="navbox authority-control" aria-labelledby="Authority_control_databases_frameless&#124;text-top&#124;10px&#124;alt=Edit_this_at_Wikidata&#124;link=https&#58;//www.wikidata.org/wiki/Q217113#identifiers&#124;class=noprint&#124;Edit_this_at_Wikidata" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Authority_control_databases_frameless&#124;text-top&#124;10px&#124;alt=Edit_this_at_Wikidata&#124;link=https&#58;//www.wikidata.org/wiki/Q217113#identifiers&#124;class=noprint&#124;Edit_this_at_Wikidata" style="font-size:114%;margin:0 4em"><a href="/wiki/Help:Authority_control" title="Help:Authority control">Authority control databases</a> <a href="https://www.wikidata.org/wiki/Q217113#identifiers" title="Edit this at Wikidata"><img alt="Edit this at Wikidata" src="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/10px-OOjs_UI_icon_edit-ltr-progressive.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/15px-OOjs_UI_icon_edit-ltr-progressive.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/20px-OOjs_UI_icon_edit-ltr-progressive.svg.png 2x" data-file-width="20" data-file-height="20" /></a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">International</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><a rel="nofollow" class="external text" href="http://id.worldcat.org/fast/893849/">FAST</a></li></ul></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">National</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4133116-3">Germany</a></li><li><a rel="nofollow" class="external text" href="https://id.loc.gov/authorities/sh85038036">United States</a></li><li><a rel="nofollow" class="external text" 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Template:Cite_journal"," 6.47% 84.376 1 Template:Systems_of_measurement"," 5.94% 77.482 1 Template:Short_description"," 4.51% 58.790 1 Template:Sfnp"," 4.40% 57.365 2 Template:Citation_needed"]},"scribunto":{"limitreport-timeusage":{"value":"0.751","limit":"10.000"},"limitreport-memusage":{"value":11612141,"limit":52428800},"limitreport-logs":"anchor_id_list = table#1 {\n [\"CITEREFBIPM2019\"] = 1,\n [\"CITEREFBarenblatt1996\"] = 1,\n [\"CITEREFBennich-BjörkmanMcKeever2018\"] = 1,\n [\"CITEREFBerberan-SantosPogliani1999\"] = 1,\n [\"CITEREFBhaskarNigam1990\"] = 1,\n [\"CITEREFBhaskarNigam1991\"] = 1,\n [\"CITEREFBolsterHershbergerDonnelly2011\"] = 1,\n [\"CITEREFBoucherAlves1960\"] = 1,\n [\"CITEREFBowley2009\"] = 1,\n [\"CITEREFBridgman1922\"] = 1,\n [\"CITEREFBuckingham1914\"] = 1,\n [\"CITEREFCimbalaÇengel2006\"] = 1,\n [\"CITEREFCmelikGehani1988\"] = 1,\n [\"CITEREFDrobot1953–1954\"] = 1,\n [\"CITEREFDuff2004\"] = 1,\n [\"CITEREFDuffOkunVeneziano2002\"] = 1,\n [\"CITEREFDureisseix2019\"] = 1,\n [\"CITEREFFourier1822\"] = 1,\n [\"CITEREFGarrigueLy2017\"] = 1,\n [\"CITEREFGehani1977\"] = 1,\n [\"CITEREFGehani1985\"] = 1,\n [\"CITEREFGiancoli2014\"] = 1,\n [\"CITEREFGibbings2011\"] = 1,\n [\"CITEREFGrecco2022\"] = 1,\n [\"CITEREFGriffioen2019\"] = 1,\n [\"CITEREFGundry2015\"] = 1,\n [\"CITEREFHart1994\"] = 1,\n [\"CITEREFHart1995\"] = 1,\n [\"CITEREFHuntley1967\"] = 1,\n [\"CITEREFJ._W._Strutt_(3rd_Baron_Rayleigh)1915\"] = 1,\n [\"CITEREFJCGM2012\"] = 1,\n [\"CITEREFKasprzakLysikRybaczuk1990\"] = 1,\n [\"CITEREFKennedy1996\"] = 1,\n [\"CITEREFKennedy2010\"] = 1,\n [\"CITEREFKlinkenberg1955\"] = 1,\n [\"CITEREFLanghaar1951\"] = 1,\n [\"CITEREFMacagno1971\"] = 1,\n [\"CITEREFMartinShawManchester_Physics2008\"] = 1,\n [\"CITEREFMartins1981\"] = 1,\n [\"CITEREFMason1962\"] = 1,\n [\"CITEREFMaxwell1873\"] = 3,\n [\"CITEREFMcBrideNordvall-Forsberg2022\"] = 1,\n [\"CITEREFMendezOrdóñez2005\"] = 1,\n [\"CITEREFMoody1944\"] = 1,\n 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{\n}\n"},"cachereport":{"origin":"mw-web.codfw.main-6b7f745dd4-2x7bg","timestamp":"20241125133902","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Dimensional analysis","url":"https:\/\/en.wikipedia.org\/wiki\/Dimensional_analysis","sameAs":"http:\/\/www.wikidata.org\/entity\/Q217113","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q217113","author":{"@type":"Organization","name":"Contributors to Wikimedia projects"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2001-11-04T03:00:52Z","dateModified":"2024-10-29T16:04:58Z","headline":"analysis of the relationships between different physical quantities by identifying their base quantities and units of measure"}</script> </body> </html>