Levi-Civita symbol

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id="toc-Generalization_to_n_dimensions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Generalization_to_n_dimensions"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.4</span> <span>Generalization to <i>n</i> dimensions</span> </div> </a> <ul id="toc-Generalization_to_n_dimensions-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Properties" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Properties"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Properties</span> </div> </a> <button aria-controls="toc-Properties-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Properties subsection</span> </button> <ul id="toc-Properties-sublist" class="vector-toc-list"> <li id="toc-Two_dimensions_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Two_dimensions_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Two dimensions</span> </div> </a> <ul id="toc-Two_dimensions_2-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Three_dimensions_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Three_dimensions_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Three dimensions</span> </div> </a> <ul id="toc-Three_dimensions_2-sublist" class="vector-toc-list"> <li id="toc-Index_and_symbol_values" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Index_and_symbol_values"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.1</span> <span>Index and symbol values</span> </div> </a> <ul id="toc-Index_and_symbol_values-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Product" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Product"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.2</span> <span>Product</span> </div> </a> <ul id="toc-Product-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-n_dimensions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#n_dimensions"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span><i>n</i> dimensions</span> </div> </a> <ul id="toc-n_dimensions-sublist" class="vector-toc-list"> <li id="toc-Index_and_symbol_values_2" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Index_and_symbol_values_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3.1</span> <span>Index and symbol values</span> </div> </a> <ul id="toc-Index_and_symbol_values_2-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Product_2" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Product_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3.2</span> <span>Product</span> </div> </a> <ul id="toc-Product_2-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Proofs" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Proofs"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Proofs</span> </div> </a> <ul id="toc-Proofs-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Applications_and_examples" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Applications_and_examples"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Applications and examples</span> </div> </a> <button aria-controls="toc-Applications_and_examples-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Applications and examples subsection</span> </button> <ul id="toc-Applications_and_examples-sublist" class="vector-toc-list"> <li id="toc-Determinants" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Determinants"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Determinants</span> </div> </a> <ul id="toc-Determinants-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Vector_cross_product" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Vector_cross_product"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Vector cross product</span> </div> </a> <ul id="toc-Vector_cross_product-sublist" class="vector-toc-list"> <li id="toc-Cross_product_(two_vectors)" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Cross_product_(two_vectors)"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.1</span> <span>Cross product (two vectors)</span> </div> </a> <ul id="toc-Cross_product_(two_vectors)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Triple_scalar_product_(three_vectors)" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Triple_scalar_product_(three_vectors)"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.2</span> <span>Triple scalar product (three vectors)</span> </div> </a> <ul id="toc-Triple_scalar_product_(three_vectors)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Curl_(one_vector_field)" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Curl_(one_vector_field)"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.3</span> <span>Curl (one vector field)</span> </div> </a> <ul id="toc-Curl_(one_vector_field)-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Tensor_density" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Tensor_density"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Tensor density</span> </div> </a> <ul id="toc-Tensor_density-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Levi-Civita_tensors" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Levi-Civita_tensors"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Levi-Civita tensors</span> </div> </a> <button aria-controls="toc-Levi-Civita_tensors-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Levi-Civita tensors subsection</span> </button> <ul id="toc-Levi-Civita_tensors-sublist" class="vector-toc-list"> <li id="toc-Example:_Minkowski_space" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Example:_Minkowski_space"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Example: Minkowski space</span> </div> </a> <ul id="toc-Example:_Minkowski_space-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Notes</span> </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 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>External links</span> </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" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </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"><span class="mw-page-title-main">Levi-Civita symbol</span></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 28 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-28" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">28 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/S%C3%ADmbol_de_Levi-Civita" title="Símbol de Levi-Civita – Catalan" lang="ca" hreflang="ca" data-title="Símbol de Levi-Civita" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Levi-Civit%C5%AFv_symbol" title="Levi-Civitův symbol – Czech" lang="cs" hreflang="cs" data-title="Levi-Civitův symbol" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Levi-Civita-Symbol" title="Levi-Civita-Symbol – German" lang="de" hreflang="de" data-title="Levi-Civita-Symbol" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Levi-Civita_s%C3%BCmbol" title="Levi-Civita sümbol – Estonian" lang="et" hreflang="et" data-title="Levi-Civita sümbol" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A3%CF%8D%CE%BC%CE%B2%CE%BF%CE%BB%CE%BF_%CE%BC%CE%B5%CF%84%CE%AC%CE%B8%CE%B5%CF%83%CE%B7%CF%82" title="Σύμβολο μετάθεσης – Greek" lang="el" hreflang="el" data-title="Σύμβολο μετάθεσης" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/S%C3%ADmbolo_de_Levi-Civita" title="Símbolo de Levi-Civita – Spanish" lang="es" hreflang="es" data-title="Símbolo de Levi-Civita" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Simbolo_de_Levi-Civita" title="Simbolo de Levi-Civita – Esperanto" lang="eo" hreflang="eo" data-title="Simbolo de Levi-Civita" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%86%D9%85%D8%A7%D8%AF_%D9%84%D9%88%DB%8C-%DA%86%DB%8C%D9%88%DB%8C%D8%AA%D8%A7" title="نماد لوی-چیویتا – Persian" lang="fa" hreflang="fa" data-title="نماد لوی-چیویتا" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Symbole_de_Levi-Civita" title="Symbole de Levi-Civita – French" lang="fr" hreflang="fr" data-title="Symbole de Levi-Civita" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%A0%88%EB%B9%84%EC%B9%98%EB%B9%84%ED%83%80_%EA%B8%B0%ED%98%B8" title="레비치비타 기호 – Korean" lang="ko" hreflang="ko" data-title="레비치비타 기호" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Simbolo_di_Levi-Civita" title="Simbolo di Levi-Civita – Italian" lang="it" hreflang="it" data-title="Simbolo di Levi-Civita" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A1%D7%99%D7%9E%D7%9F_%D7%9C%D7%95%D7%99-%D7%A6%27%D7%99%D7%95%D7%95%D7%99%D7%98%D7%94" title="סימן לוי-צ'יוויטה – Hebrew" lang="he" hreflang="he" data-title="סימן לוי-צ'יוויטה" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%9A%E1%83%94%E1%83%95%E1%83%98-%E1%83%A9%E1%83%98%E1%83%95%E1%83%98%E1%83%A2%E1%83%90%E1%83%A1_%E1%83%A1%E1%83%98%E1%83%9B%E1%83%91%E1%83%9D%E1%83%9A%E1%83%9D" title="ლევი-ჩივიტას სიმბოლო – Georgian" lang="ka" hreflang="ka" data-title="ლევი-ჩივიტას სიმბოლო" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/S%C3%ADmbul_da_Levi-Civita" title="Símbul da Levi-Civita – Lombard" lang="lmo" hreflang="lmo" data-title="Símbul da Levi-Civita" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Levi-Civita-szimb%C3%B3lum" title="Levi-Civita-szimbólum – Hungarian" lang="hu" hreflang="hu" data-title="Levi-Civita-szimbólum" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%B2%E0%B5%86%E0%B4%B5%E0%B4%BF-%E0%B4%B8%E0%B4%BF%E0%B4%B5%E0%B4%BF%E0%B4%B1%E0%B5%8D%E0%B4%B1_%E0%B4%9A%E0%B4%BF%E0%B4%B9%E0%B5%8D%E0%B4%A8%E0%B4%82" title="ലെവി-സിവിറ്റ ചിഹ്നം – Malayalam" lang="ml" hreflang="ml" data-title="ലെവി-സിവിറ്റ ചിഹ്നം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Levi-civita-symbool" title="Levi-civita-symbool – Dutch" lang="nl" hreflang="nl" data-title="Levi-civita-symbool" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%82%A8%E3%83%87%E3%82%A3%E3%83%B3%E3%83%88%E3%83%B3%E3%81%AE%E3%82%A4%E3%83%97%E3%82%B7%E3%83%AD%E3%83%B3" title="エディントンのイプシロン – Japanese" lang="ja" hreflang="ja" data-title="エディントンのイプシロン" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Levi-Civita-symbol" title="Levi-Civita-symbol – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Levi-Civita-symbol" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Symbol_Leviego-Civity" title="Symbol Leviego-Civity – Polish" lang="pl" hreflang="pl" data-title="Symbol Leviego-Civity" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/S%C3%ADmbolo_de_Levi-Civita" title="Símbolo de Levi-Civita – Portuguese" lang="pt" hreflang="pt" data-title="Símbolo de Levi-Civita" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A1%D0%B8%D0%BC%D0%B2%D0%BE%D0%BB_%D0%9B%D0%B5%D0%B2%D0%B8-%D0%A7%D0%B8%D0%B2%D0%B8%D1%82%D1%8B" title="Символ Леви-Чивиты – Russian" lang="ru" hreflang="ru" data-title="Символ Леви-Чивиты" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Levi-Civitajev_simbol" title="Levi-Civitajev simbol – Slovenian" lang="sl" hreflang="sl" data-title="Levi-Civitajev simbol" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9B%D0%B5%D0%B2%D0%B8-%D0%A7%D0%B8%D0%B2%D0%B8%D1%82%D0%B0_%D1%81%D0%B8%D0%BC%D0%B1%D0%BE%D0%BB" title="Леви-Чивита симбол – Serbian" lang="sr" hreflang="sr" data-title="Леви-Чивита симбол" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Levi-Civita-symboli" title="Levi-Civita-symboli – Finnish" lang="fi" hreflang="fi" data-title="Levi-Civita-symboli" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Permutationssymbol" title="Permutationssymbol – Swedish" lang="sv" hreflang="sv" data-title="Permutationssymbol" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A1%D0%B8%D0%BC%D0%B2%D0%BE%D0%BB_%D0%9B%D0%B5%D0%B2%D1%96-%D0%A7%D0%B8%D0%B2%D1%96%D1%82%D0%B8" title="Символ Леві-Чивіти – Ukrainian" lang="uk" hreflang="uk" data-title="Символ Леві-Чивіти" data-language-autonym="Українська" data-language-local-name="Ukrainian" 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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">Antisymmetric permutation object acting on tensors</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">Not to be confused with <a href="/wiki/Levi-Civita_connection" title="Levi-Civita connection">Levi-Civita connection</a>.</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See <a href="/wiki/Ricci_calculus" title="Ricci calculus">Ricci calculus</a>, <a href="/wiki/Einstein_notation" title="Einstein notation">Einstein notation</a>, and <a href="/wiki/Raising_and_lowering_indices" title="Raising and lowering indices">Raising and lowering indices</a> for the <a href="/wiki/Index_notation" title="Index notation">index notation</a> used in the article.</div> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, particularly in <a href="/wiki/Linear_algebra" title="Linear algebra">linear algebra</a>, <a href="/wiki/Tensor_analysis" class="mw-redirect" title="Tensor analysis">tensor analysis</a>, and <a href="/wiki/Differential_geometry" title="Differential geometry">differential geometry</a>, the <b>Levi-Civita symbol</b> or <b>Levi-Civita epsilon</b> represents a collection of numbers defined from the <a href="/wiki/Parity_of_a_permutation" title="Parity of a permutation">sign of a permutation</a> of the <a href="/wiki/Natural_number" title="Natural number">natural numbers</a> <span class="texhtml">1, 2, ..., <i>n</i></span>, for some positive integer <span class="texhtml mvar" style="font-style:italic;">n</span>. It is named after the Italian mathematician and physicist <a href="/wiki/Tullio_Levi-Civita" title="Tullio Levi-Civita">Tullio Levi-Civita</a>. Other names include the <b><a href="/wiki/Permutation" title="Permutation">permutation</a> symbol</b>, <b>antisymmetric symbol</b>, or <b>alternating symbol</b>, which refer to its <a href="/wiki/Antisymmetric_tensor" title="Antisymmetric tensor">antisymmetric</a> property and definition in terms of permutations. </p><p>The standard letters to denote the Levi-Civita symbol are the Greek lower case <a href="/wiki/Epsilon" title="Epsilon">epsilon</a> <span class="texhtml mvar" style="font-style:italic;">ε</span> or <span class="texhtml mvar" style="font-style:italic;">ϵ</span>, or less commonly the Latin lower case <span class="texhtml mvar" style="font-style:italic;">e</span>. Index notation allows one to display permutations in a way compatible with tensor analysis: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{i_{1}i_{2}\dots i_{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{i_{1}i_{2}\dots i_{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75be000de23194d18b8e0426ea0e43f592964fd8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.572ex; height:2.343ex;" alt="{\displaystyle \varepsilon _{i_{1}i_{2}\dots i_{n}}}"></span> where <i>each</i> index <span class="texhtml"><i>i</i><sub>1</sub>, <i>i</i><sub>2</sub>, ..., <i>i</i><sub><i>n</i></sub></span> takes values <span class="texhtml">1, 2, ..., <i>n</i></span>. There are <span class="texhtml"><i>n</i><sup><i>n</i></sup></span> indexed values of <span class="texhtml"><i>ε</i><sub><i>i</i><sub>1</sub><i>i</i><sub>2</sub>...<i>i<sub>n</sub></i></sub></span>, which can be arranged into an <span class="texhtml mvar" style="font-style:italic;">n</span>-dimensional array. The key defining property of the symbol is <i>total antisymmetry</i> in the indices. When any two indices are interchanged, equal or not, the symbol is negated: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{\dots i_{p}\dots i_{q}\dots }=-\varepsilon _{\dots i_{q}\dots i_{p}\dots }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>…</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>…</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> <mo>…</mo> </mrow> </msub> <mo>=</mo> <mo>−</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>…</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> <mo>…</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>…</mo> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{\dots i_{p}\dots i_{q}\dots }=-\varepsilon _{\dots i_{q}\dots i_{p}\dots }.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35d766798eabbe2dee596bc417931838a4ac3842" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:25.236ex; height:2.843ex;" alt="{\displaystyle \varepsilon _{\dots i_{p}\dots i_{q}\dots }=-\varepsilon _{\dots i_{q}\dots i_{p}\dots }.}"></span> </p><p>If any two indices are equal, the symbol is zero. When all indices are unequal, we have: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{i_{1}i_{2}\dots i_{n}}=(-1)^{p}\varepsilon _{1\,2\,\dots n},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mspace width="thinmathspace" /> <mn>2</mn> <mspace width="thinmathspace" /> <mo>…</mo> <mi>n</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{i_{1}i_{2}\dots i_{n}}=(-1)^{p}\varepsilon _{1\,2\,\dots n},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac125c56b2a3ac7919961e27096a4b2569f186d2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.802ex; height:3.009ex;" alt="{\displaystyle \varepsilon _{i_{1}i_{2}\dots i_{n}}=(-1)^{p}\varepsilon _{1\,2\,\dots n},}"></span> where <span class="texhtml mvar" style="font-style:italic;">p</span> (called the parity of the permutation) is the number of pairwise interchanges of indices necessary to unscramble <span class="texhtml"><i>i</i><sub>1</sub>, <i>i</i><sub>2</sub>, ..., <i>i</i><sub><i>n</i></sub></span> into the order <span class="texhtml">1, 2, ..., <i>n</i></span>, and the factor <span class="texhtml">(−1)<sup><i>p</i></sup></span> is called the <a href="/wiki/Parity_of_a_permutation" title="Parity of a permutation">sign, or signature</a> of the permutation. The value <span class="texhtml"><i>ε</i><sub>1 2 ... <i>n</i></sub></span> must be defined, else the particular values of the symbol for all permutations are indeterminate. Most authors choose <span class="texhtml"><i>ε</i><sub>1 2 ... <i>n</i></sub> = +1</span>, which means the Levi-Civita symbol equals the sign of a permutation when the indices are all unequal. This choice is used throughout this article. </p><p>The term "<span class="texhtml mvar" style="font-style:italic;">n</span>-dimensional Levi-Civita symbol" refers to the fact that the number of indices on the symbol <span class="texhtml mvar" style="font-style:italic;">n</span> matches the <a href="/wiki/Dimension" title="Dimension">dimensionality</a> of the <a href="/wiki/Vector_space" title="Vector space">vector space</a> in question, which may be <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean</a> or <a href="/wiki/Non-Euclidean_space" class="mw-redirect" title="Non-Euclidean space">non-Euclidean</a>, for example, <span class="texhtml"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}"></span></span> or <a href="/wiki/Minkowski_space" title="Minkowski space">Minkowski space</a>. The values of the Levi-Civita symbol are independent of any <a href="/wiki/Metric_tensor" title="Metric tensor">metric tensor</a> and <a href="/wiki/Coordinate_system" title="Coordinate system">coordinate system</a>. Also, the specific term "symbol" emphasizes that it is not a <a href="/wiki/Tensor" title="Tensor">tensor</a> because of how it transforms between coordinate systems; however it can be interpreted as a <a href="/wiki/Tensor_density" title="Tensor density">tensor density</a>. </p><p>The Levi-Civita symbol allows the <a href="/wiki/Determinant" title="Determinant">determinant</a> of a square matrix, and the <a href="/wiki/Cross_product" title="Cross product">cross product</a> of two vectors in three-dimensional Euclidean space, to be expressed in <a href="/wiki/Einstein_notation" title="Einstein notation">Einstein index notation</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Levi-Civita_symbol&action=edit&section=1" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Levi-Civita symbol is most often used in three and four dimensions, and to some extent in two dimensions, so these are given here before defining the general case. </p> <div class="mw-heading mw-heading3"><h3 id="Two_dimensions">Two dimensions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Levi-Civita_symbol&action=edit&section=2" title="Edit section: Two dimensions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Two_dimensions" class="mw-redirect" title="Two dimensions">two dimensions</a>, the Levi-Civita symbol is defined by: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{ij}={\begin{cases}+1&{\text{if }}(i,j)=(1,2)\\-1&{\text{if }}(i,j)=(2,1)\\\;\;\,0&{\text{if }}i=j\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mo>+</mo> <mn>1</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mo stretchy="false">(</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mo stretchy="false">(</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thinmathspace" /> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>i</mi> <mo>=</mo> <mi>j</mi> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{ij}={\begin{cases}+1&{\text{if }}(i,j)=(1,2)\\-1&{\text{if }}(i,j)=(2,1)\\\;\;\,0&{\text{if }}i=j\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8515954e478faa56dc63174923dada31cfd90244" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:28.579ex; height:8.509ex;" alt="{\displaystyle \varepsilon _{ij}={\begin{cases}+1&{\text{if }}(i,j)=(1,2)\\-1&{\text{if }}(i,j)=(2,1)\\\;\;\,0&{\text{if }}i=j\end{cases}}}"></span> The values can be arranged into a 2 × 2 <a href="/wiki/Antisymmetric_matrix" class="mw-redirect" title="Antisymmetric matrix">antisymmetric matrix</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}\varepsilon _{11}&\varepsilon _{12}\\\varepsilon _{21}&\varepsilon _{22}\end{pmatrix}}={\begin{pmatrix}0&1\\-1&0\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}\varepsilon _{11}&\varepsilon _{12}\\\varepsilon _{21}&\varepsilon _{22}\end{pmatrix}}={\begin{pmatrix}0&1\\-1&0\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d71a29ba9609b6b42b24748f3c19c9a7df9a884" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.141ex; height:6.176ex;" alt="{\displaystyle {\begin{pmatrix}\varepsilon _{11}&\varepsilon _{12}\\\varepsilon _{21}&\varepsilon _{22}\end{pmatrix}}={\begin{pmatrix}0&1\\-1&0\end{pmatrix}}}"></span> </p><p>Use of the two-dimensional symbol is common in condensed matter, and in certain specialized high-energy topics like <a href="/wiki/Supersymmetry" title="Supersymmetry">supersymmetry</a><sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/Twistor_theory" title="Twistor theory">twistor theory</a>,<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> where it appears in the context of 2-<a href="/wiki/Spinors" class="mw-redirect" title="Spinors">spinors</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Three_dimensions">Three dimensions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Levi-Civita_symbol&action=edit&section=3" title="Edit section: Three dimensions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Permutation_indices_3d_numerical.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/50/Permutation_indices_3d_numerical.svg/210px-Permutation_indices_3d_numerical.svg.png" decoding="async" width="210" height="113" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/50/Permutation_indices_3d_numerical.svg/315px-Permutation_indices_3d_numerical.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/50/Permutation_indices_3d_numerical.svg/420px-Permutation_indices_3d_numerical.svg.png 2x" data-file-width="388" data-file-height="208" /></a><figcaption>For the indices <span class="texhtml">(<i>i</i>, <i>j</i>, <i>k</i>)</span> in <span class="texhtml"><i>ε</i><sub><i>ijk</i></sub></span>, the values <span class="texhtml">1, 2, 3</span> occurring in the <style data-mw-deduplicate="TemplateStyles:r981673959">.mw-parser-output .legend{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .legend-color{display:inline-block;min-width:1.25em;height:1.25em;line-height:1.25;margin:1px 0;text-align:center;border:1px solid black;background-color:transparent;color:black}.mw-parser-output .legend-text{}</style><span class="legend-color mw-no-invert" style="background-color:orange; color:black;"> </span> cyclic order <span class="texhtml">(1, 2, 3)</span> correspond to <span class="texhtml"><i>ε</i> = +1</span>, while occurring in the <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r981673959"><span class="legend-color mw-no-invert" style="background-color:red; color:black;"> </span> reverse cyclic order correspond to <span class="texhtml"><i>ε</i> = −1</span>, otherwise <span class="texhtml"><i>ε</i> = 0</span>.</figcaption></figure> <p>In <a href="/wiki/Three_dimensions" class="mw-redirect" title="Three dimensions">three dimensions</a>, the Levi-Civita symbol is defined by:<sup id="cite_ref-Tyldesley_3-0" class="reference"><a href="#cite_note-Tyldesley-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{ijk}={\begin{cases}+1&{\text{if }}(i,j,k){\text{ is }}(1,2,3),(2,3,1),{\text{ or }}(3,1,2),\\-1&{\text{if }}(i,j,k){\text{ is }}(3,2,1),(1,3,2),{\text{ or }}(2,1,3),\\\;\;\,0&{\text{if }}i=j,{\text{ or }}j=k,{\text{ or }}k=i\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mo>+</mo> <mn>1</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mo stretchy="false">(</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> is </mtext> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> or </mtext> </mrow> <mo stretchy="false">(</mo> <mn>3</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mo stretchy="false">(</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> is </mtext> </mrow> <mo stretchy="false">(</mo> <mn>3</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> or </mtext> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thinmathspace" /> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>i</mi> <mo>=</mo> <mi>j</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> or </mtext> </mrow> <mi>j</mi> <mo>=</mo> <mi>k</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> or </mtext> </mrow> <mi>k</mi> <mo>=</mo> <mi>i</mi> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{ijk}={\begin{cases}+1&{\text{if }}(i,j,k){\text{ is }}(1,2,3),(2,3,1),{\text{ or }}(3,1,2),\\-1&{\text{if }}(i,j,k){\text{ is }}(3,2,1),(1,3,2),{\text{ or }}(2,1,3),\\\;\;\,0&{\text{if }}i=j,{\text{ or }}j=k,{\text{ or }}k=i\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/903ee1efe5f3e041da93aac5f55c82679c3329ad" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:54.183ex; height:8.509ex;" alt="{\displaystyle \varepsilon _{ijk}={\begin{cases}+1&{\text{if }}(i,j,k){\text{ is }}(1,2,3),(2,3,1),{\text{ or }}(3,1,2),\\-1&{\text{if }}(i,j,k){\text{ is }}(3,2,1),(1,3,2),{\text{ or }}(2,1,3),\\\;\;\,0&{\text{if }}i=j,{\text{ or }}j=k,{\text{ or }}k=i\end{cases}}}"></span> </p><p>That is, <span class="texhtml"><i>ε</i><sub><i>ijk</i></sub></span> is <span class="texhtml">1</span> if <span class="texhtml">(<i>i</i>, <i>j</i>, <i>k</i>)</span> is an <a href="/wiki/Even_and_odd_permutations" class="mw-redirect" title="Even and odd permutations">even permutation</a> of <span class="texhtml">(1, 2, 3)</span>, <span class="texhtml">−1</span> if it is an <a href="/wiki/Odd_permutation" class="mw-redirect" title="Odd permutation">odd permutation</a>, and 0 if any index is repeated. In three dimensions only, the <a href="/wiki/Cyclic_permutation" title="Cyclic permutation">cyclic permutations</a> of <span class="texhtml">(1, 2, 3)</span> are all even permutations, similarly the <a href="/wiki/Anticyclic_permutation" class="mw-redirect" title="Anticyclic permutation">anticyclic permutations</a> are all odd permutations. This means in 3d it is sufficient to take cyclic or anticyclic permutations of <span class="texhtml">(1, 2, 3)</span> and easily obtain all the even or odd permutations. </p><p>Analogous to 2-dimensional matrices, the values of the 3-dimensional Levi-Civita symbol can be arranged into a <span class="nowrap">3 × 3 × 3</span> array: </p> <dl><dd><span typeof="mw:File"><a href="/wiki/File:Epsilontensor.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/71/Epsilontensor.svg/200px-Epsilontensor.svg.png" decoding="async" width="200" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/71/Epsilontensor.svg/300px-Epsilontensor.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/71/Epsilontensor.svg/400px-Epsilontensor.svg.png 2x" data-file-width="500" data-file-height="250" /></a></span></dd></dl> <p>where <span class="texhtml mvar" style="font-style:italic;">i</span> is the depth (<style data-mw-deduplicate="TemplateStyles:r1239334494">@media screen{html.skin-theme-clientpref-night .mw-parser-output div:not(.notheme)>.tmp-color,html.skin-theme-clientpref-night .mw-parser-output p>.tmp-color,html.skin-theme-clientpref-night .mw-parser-output table:not(.notheme) .tmp-color{color:inherit!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output div:not(.notheme)>.tmp-color,html.skin-theme-clientpref-os .mw-parser-output p>.tmp-color,html.skin-theme-clientpref-os .mw-parser-output table:not(.notheme) .tmp-color{color:inherit!important}}</style><span class="tmp-color" style="color:blue">blue</span>: <span class="texhtml"><i>i</i> = 1</span>; <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:red">red</span>: <span class="texhtml"><i>i</i> = 2</span>; <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239334494"><span class="tmp-color" style="color:green">green</span>: <span class="texhtml"><i>i</i> = 3</span>), <span class="texhtml mvar" style="font-style:italic;">j</span> is the row and <span class="texhtml mvar" style="font-style:italic;">k</span> is the column. </p><p>Some examples: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\varepsilon _{\color {BrickRed}{1}\color {Violet}{3}\color {Orange}{2}}=-\varepsilon _{\color {BrickRed}{1}\color {Orange}{2}\color {Violet}{3}}&=-1\\\varepsilon _{\color {Violet}{3}\color {BrickRed}{1}\color {Orange}{2}}=-\varepsilon _{\color {Orange}{2}\color {BrickRed}{1}\color {Violet}{3}}&=-(-\varepsilon _{\color {BrickRed}{1}\color {Orange}{2}\color {Violet}{3}})=1\\\varepsilon _{\color {Orange}{2}\color {Violet}{3}\color {BrickRed}{1}}=-\varepsilon _{\color {BrickRed}{1}\color {Violet}{3}\color {Orange}{2}}&=-(-\varepsilon _{\color {BrickRed}{1}\color {Orange}{2}\color {Violet}{3}})=1\\\varepsilon _{\color {Orange}{2}\color {Violet}{3}\color {Orange}{2}}=-\varepsilon _{\color {Orange}{2}\color {Violet}{3}\color {Orange}{2}}&=0\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>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#B6321C"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mstyle mathcolor="#58429B"> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mstyle mathcolor="#F58137"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mstyle> </mstyle> </mstyle> </mrow> </msub> <mo>=</mo> <mo>−</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#B6321C"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mstyle mathcolor="#F58137"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mstyle mathcolor="#58429B"> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mstyle> </mstyle> </mstyle> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#58429B"> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mstyle mathcolor="#B6321C"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mstyle mathcolor="#F58137"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mstyle> </mstyle> </mstyle> </mrow> </msub> <mo>=</mo> <mo>−</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#F58137"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mstyle mathcolor="#B6321C"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mstyle mathcolor="#58429B"> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mstyle> </mstyle> </mstyle> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mo>−</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#B6321C"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mstyle mathcolor="#F58137"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mstyle mathcolor="#58429B"> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mstyle> </mstyle> </mstyle> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#F58137"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mstyle mathcolor="#58429B"> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mstyle mathcolor="#B6321C"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mstyle> </mstyle> </mstyle> </mrow> </msub> <mo>=</mo> <mo>−</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#B6321C"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mstyle mathcolor="#58429B"> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mstyle mathcolor="#F58137"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mstyle> </mstyle> </mstyle> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mo>−</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#B6321C"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mstyle mathcolor="#F58137"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mstyle mathcolor="#58429B"> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mstyle> </mstyle> </mstyle> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#F58137"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mstyle mathcolor="#58429B"> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mstyle mathcolor="#F58137"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mstyle> </mstyle> </mstyle> </mrow> </msub> <mo>=</mo> <mo>−</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#F58137"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mstyle mathcolor="#58429B"> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mstyle mathcolor="#F58137"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mstyle> </mstyle> </mstyle> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\varepsilon _{\color {BrickRed}{1}\color {Violet}{3}\color {Orange}{2}}=-\varepsilon _{\color {BrickRed}{1}\color {Orange}{2}\color {Violet}{3}}&=-1\\\varepsilon _{\color {Violet}{3}\color {BrickRed}{1}\color {Orange}{2}}=-\varepsilon _{\color {Orange}{2}\color {BrickRed}{1}\color {Violet}{3}}&=-(-\varepsilon _{\color {BrickRed}{1}\color {Orange}{2}\color {Violet}{3}})=1\\\varepsilon _{\color {Orange}{2}\color {Violet}{3}\color {BrickRed}{1}}=-\varepsilon _{\color {BrickRed}{1}\color {Violet}{3}\color {Orange}{2}}&=-(-\varepsilon _{\color {BrickRed}{1}\color {Orange}{2}\color {Violet}{3}})=1\\\varepsilon _{\color {Orange}{2}\color {Violet}{3}\color {Orange}{2}}=-\varepsilon _{\color {Orange}{2}\color {Violet}{3}\color {Orange}{2}}&=0\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b24d2da21b82ebed1afec775b13bb8f998d2fa0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:29.788ex; height:12.176ex;" alt="{\displaystyle {\begin{aligned}\varepsilon _{\color {BrickRed}{1}\color {Violet}{3}\color {Orange}{2}}=-\varepsilon _{\color {BrickRed}{1}\color {Orange}{2}\color {Violet}{3}}&=-1\\\varepsilon _{\color {Violet}{3}\color {BrickRed}{1}\color {Orange}{2}}=-\varepsilon _{\color {Orange}{2}\color {BrickRed}{1}\color {Violet}{3}}&=-(-\varepsilon _{\color {BrickRed}{1}\color {Orange}{2}\color {Violet}{3}})=1\\\varepsilon _{\color {Orange}{2}\color {Violet}{3}\color {BrickRed}{1}}=-\varepsilon _{\color {BrickRed}{1}\color {Violet}{3}\color {Orange}{2}}&=-(-\varepsilon _{\color {BrickRed}{1}\color {Orange}{2}\color {Violet}{3}})=1\\\varepsilon _{\color {Orange}{2}\color {Violet}{3}\color {Orange}{2}}=-\varepsilon _{\color {Orange}{2}\color {Violet}{3}\color {Orange}{2}}&=0\end{aligned}}}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Four_dimensions">Four dimensions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Levi-Civita_symbol&action=edit&section=4" title="Edit section: Four dimensions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Four-dimensional_space" title="Four-dimensional space">four dimensions</a>, the Levi-Civita symbol is defined by: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{ijkl}={\begin{cases}+1&{\text{if }}(i,j,k,l){\text{ is an even permutation of }}(1,2,3,4)\\-1&{\text{if }}(i,j,k,l){\text{ is an odd permutation of }}(1,2,3,4)\\\;\;\,0&{\text{otherwise}}\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> <mi>k</mi> <mi>l</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mo>+</mo> <mn>1</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mo stretchy="false">(</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>,</mo> <mi>k</mi> <mo>,</mo> <mi>l</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> is an even permutation of </mtext> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mo stretchy="false">(</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>,</mo> <mi>k</mi> <mo>,</mo> <mi>l</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> is an odd permutation of </mtext> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thinmathspace" /> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>otherwise</mtext> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{ijkl}={\begin{cases}+1&{\text{if }}(i,j,k,l){\text{ is an even permutation of }}(1,2,3,4)\\-1&{\text{if }}(i,j,k,l){\text{ is an odd permutation of }}(1,2,3,4)\\\;\;\,0&{\text{otherwise}}\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88529e14cefd260f97a6165da0a58ef2c778c0fa" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:61.692ex; height:8.509ex;" alt="{\displaystyle \varepsilon _{ijkl}={\begin{cases}+1&{\text{if }}(i,j,k,l){\text{ is an even permutation of }}(1,2,3,4)\\-1&{\text{if }}(i,j,k,l){\text{ is an odd permutation of }}(1,2,3,4)\\\;\;\,0&{\text{otherwise}}\end{cases}}}"></span> </p><p>These values can be arranged into a <span class="nowrap">4 × 4 × 4 × 4</span> array, although in 4 dimensions and higher this is difficult to draw. </p><p>Some examples: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\varepsilon _{\color {BrickRed}{1}\color {RedViolet}{4}\color {Violet}{3}\color {Orange}{\color {Orange}{2}}}=-\varepsilon _{\color {BrickRed}{1}\color {Orange}{\color {Orange}{2}}\color {Violet}{3}\color {RedViolet}{4}}&=-1\\\varepsilon _{\color {Orange}{\color {Orange}{2}}\color {BrickRed}{1}\color {Violet}{3}\color {RedViolet}{4}}=-\varepsilon _{\color {BrickRed}{1}\color {Orange}{\color {Orange}{2}}\color {Violet}{3}\color {RedViolet}{4}}&=-1\\\varepsilon _{\color {RedViolet}{4}\color {Violet}{3}\color {Orange}{\color {Orange}{2}}\color {BrickRed}{1}}=-\varepsilon _{\color {BrickRed}{1}\color {Violet}{3}\color {Orange}{\color {Orange}{2}}\color {RedViolet}{4}}&=-(-\varepsilon _{\color {BrickRed}{1}\color {Orange}{\color {Orange}{2}}\color {Violet}{3}\color {RedViolet}{4}})=1\\\varepsilon _{\color {Violet}{3}\color {Orange}{\color {Orange}{2}}\color {RedViolet}{4}\color {Violet}{3}}=-\varepsilon _{\color {Violet}{3}\color {Orange}{\color {Orange}{2}}\color {RedViolet}{4}\color {Violet}{3}}&=0\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>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#B6321C"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mstyle mathcolor="#A1246B"> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> <mstyle mathcolor="#58429B"> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mstyle mathcolor="#F58137"> <mrow class="MJX-TeXAtom-ORD"> <mstyle> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mstyle> </mrow> </mstyle> </mstyle> </mstyle> </mstyle> </mrow> </msub> <mo>=</mo> <mo>−</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#B6321C"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mstyle mathcolor="#F58137"> <mrow class="MJX-TeXAtom-ORD"> <mstyle> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mstyle> </mrow> <mstyle mathcolor="#58429B"> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mstyle mathcolor="#A1246B"> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </mstyle> </mstyle> </mstyle> </mstyle> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#F58137"> <mrow class="MJX-TeXAtom-ORD"> <mstyle> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mstyle> </mrow> <mstyle mathcolor="#B6321C"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mstyle mathcolor="#58429B"> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mstyle mathcolor="#A1246B"> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </mstyle> </mstyle> </mstyle> </mstyle> </mrow> </msub> <mo>=</mo> <mo>−</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#B6321C"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mstyle mathcolor="#F58137"> <mrow class="MJX-TeXAtom-ORD"> <mstyle> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mstyle> </mrow> <mstyle mathcolor="#58429B"> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mstyle mathcolor="#A1246B"> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </mstyle> </mstyle> </mstyle> </mstyle> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#A1246B"> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> <mstyle mathcolor="#58429B"> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mstyle mathcolor="#F58137"> <mrow class="MJX-TeXAtom-ORD"> <mstyle> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mstyle> </mrow> <mstyle mathcolor="#B6321C"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mstyle> </mstyle> </mstyle> </mstyle> </mrow> </msub> <mo>=</mo> <mo>−</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#B6321C"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mstyle mathcolor="#58429B"> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mstyle mathcolor="#F58137"> <mrow class="MJX-TeXAtom-ORD"> <mstyle> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mstyle> </mrow> <mstyle mathcolor="#A1246B"> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </mstyle> </mstyle> </mstyle> </mstyle> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mo>−</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#B6321C"> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mstyle mathcolor="#F58137"> <mrow class="MJX-TeXAtom-ORD"> <mstyle> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mstyle> </mrow> <mstyle mathcolor="#58429B"> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mstyle mathcolor="#A1246B"> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </mstyle> </mstyle> </mstyle> </mstyle> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#58429B"> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mstyle mathcolor="#F58137"> <mrow class="MJX-TeXAtom-ORD"> <mstyle> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mstyle> </mrow> <mstyle mathcolor="#A1246B"> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> <mstyle mathcolor="#58429B"> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mstyle> </mstyle> </mstyle> </mstyle> </mrow> </msub> <mo>=</mo> <mo>−</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#58429B"> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mstyle mathcolor="#F58137"> <mrow class="MJX-TeXAtom-ORD"> <mstyle> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mstyle> </mrow> <mstyle mathcolor="#A1246B"> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> <mstyle mathcolor="#58429B"> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mstyle> </mstyle> </mstyle> </mstyle> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\varepsilon _{\color {BrickRed}{1}\color {RedViolet}{4}\color {Violet}{3}\color {Orange}{\color {Orange}{2}}}=-\varepsilon _{\color {BrickRed}{1}\color {Orange}{\color {Orange}{2}}\color {Violet}{3}\color {RedViolet}{4}}&=-1\\\varepsilon _{\color {Orange}{\color {Orange}{2}}\color {BrickRed}{1}\color {Violet}{3}\color {RedViolet}{4}}=-\varepsilon _{\color {BrickRed}{1}\color {Orange}{\color {Orange}{2}}\color {Violet}{3}\color {RedViolet}{4}}&=-1\\\varepsilon _{\color {RedViolet}{4}\color {Violet}{3}\color {Orange}{\color {Orange}{2}}\color {BrickRed}{1}}=-\varepsilon _{\color {BrickRed}{1}\color {Violet}{3}\color {Orange}{\color {Orange}{2}}\color {RedViolet}{4}}&=-(-\varepsilon _{\color {BrickRed}{1}\color {Orange}{\color {Orange}{2}}\color {Violet}{3}\color {RedViolet}{4}})=1\\\varepsilon _{\color {Violet}{3}\color {Orange}{\color {Orange}{2}}\color {RedViolet}{4}\color {Violet}{3}}=-\varepsilon _{\color {Violet}{3}\color {Orange}{\color {Orange}{2}}\color {RedViolet}{4}\color {Violet}{3}}&=0\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c0730691c94c2a66989818fa988e33393d81f69" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:32.254ex; height:12.009ex;" alt="{\displaystyle {\begin{aligned}\varepsilon _{\color {BrickRed}{1}\color {RedViolet}{4}\color {Violet}{3}\color {Orange}{\color {Orange}{2}}}=-\varepsilon _{\color {BrickRed}{1}\color {Orange}{\color {Orange}{2}}\color {Violet}{3}\color {RedViolet}{4}}&=-1\\\varepsilon _{\color {Orange}{\color {Orange}{2}}\color {BrickRed}{1}\color {Violet}{3}\color {RedViolet}{4}}=-\varepsilon _{\color {BrickRed}{1}\color {Orange}{\color {Orange}{2}}\color {Violet}{3}\color {RedViolet}{4}}&=-1\\\varepsilon _{\color {RedViolet}{4}\color {Violet}{3}\color {Orange}{\color {Orange}{2}}\color {BrickRed}{1}}=-\varepsilon _{\color {BrickRed}{1}\color {Violet}{3}\color {Orange}{\color {Orange}{2}}\color {RedViolet}{4}}&=-(-\varepsilon _{\color {BrickRed}{1}\color {Orange}{\color {Orange}{2}}\color {Violet}{3}\color {RedViolet}{4}})=1\\\varepsilon _{\color {Violet}{3}\color {Orange}{\color {Orange}{2}}\color {RedViolet}{4}\color {Violet}{3}}=-\varepsilon _{\color {Violet}{3}\color {Orange}{\color {Orange}{2}}\color {RedViolet}{4}\color {Violet}{3}}&=0\end{aligned}}}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Generalization_to_n_dimensions">Generalization to <i>n</i> dimensions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Levi-Civita_symbol&action=edit&section=5" title="Edit section: Generalization to n dimensions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>More generally, in <a href="/wiki/N-dimensional_space" class="mw-redirect" title="N-dimensional space"><span class="texhtml"><i>n</i></span> dimensions</a>, the Levi-Civita symbol is defined by:<sup id="cite_ref-Kay_4-0" class="reference"><a href="#cite_note-Kay-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{a_{1}a_{2}a_{3}\ldots a_{n}}={\begin{cases}+1&{\text{if }}(a_{1},a_{2},a_{3},\ldots ,a_{n}){\text{ is an even permutation of }}(1,2,3,\dots ,n)\\-1&{\text{if }}(a_{1},a_{2},a_{3},\ldots ,a_{n}){\text{ is an odd permutation of }}(1,2,3,\dots ,n)\\\;\;\,0&{\text{otherwise}}\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mo>+</mo> <mn>1</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> is an even permutation of </mtext> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> is an odd permutation of </mtext> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thinmathspace" /> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>otherwise</mtext> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{a_{1}a_{2}a_{3}\ldots a_{n}}={\begin{cases}+1&{\text{if }}(a_{1},a_{2},a_{3},\ldots ,a_{n}){\text{ is an even permutation of }}(1,2,3,\dots ,n)\\-1&{\text{if }}(a_{1},a_{2},a_{3},\ldots ,a_{n}){\text{ is an odd permutation of }}(1,2,3,\dots ,n)\\\;\;\,0&{\text{otherwise}}\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23c0782db4dd1de2fb3c92a86bc0275bc5dae0cf" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:82.121ex; height:8.509ex;" alt="{\displaystyle \varepsilon _{a_{1}a_{2}a_{3}\ldots a_{n}}={\begin{cases}+1&{\text{if }}(a_{1},a_{2},a_{3},\ldots ,a_{n}){\text{ is an even permutation of }}(1,2,3,\dots ,n)\\-1&{\text{if }}(a_{1},a_{2},a_{3},\ldots ,a_{n}){\text{ is an odd permutation of }}(1,2,3,\dots ,n)\\\;\;\,0&{\text{otherwise}}\end{cases}}}"></span> </p><p>Thus, it is the <a href="/wiki/Even_and_odd_permutations" class="mw-redirect" title="Even and odd permutations">sign of the permutation</a> in the case of a permutation, and zero otherwise. </p><p>Using the <a href="/wiki/Multiplication#Capital_pi_notation" title="Multiplication">capital pi notation</a> <span class="texhtml">Π</span> for ordinary multiplication of numbers, an explicit expression for the symbol is:<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (July 2021)">citation needed</span></a></i>]</sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\varepsilon _{a_{1}a_{2}a_{3}\ldots a_{n}}&=\prod _{1\leq i<j\leq n}\operatorname {sgn}(a_{j}-a_{i})\\&=\operatorname {sgn}(a_{2}-a_{1})\operatorname {sgn}(a_{3}-a_{1})\dotsm \operatorname {sgn}(a_{n}-a_{1})\operatorname {sgn}(a_{3}-a_{2})\operatorname {sgn}(a_{4}-a_{2})\dotsm \operatorname {sgn}(a_{n}-a_{2})\dotsm \operatorname {sgn}(a_{n}-a_{n-1})\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>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>≤</mo> <mi>i</mi> <mo><</mo> <mi>j</mi> <mo>≤</mo> <mi>n</mi> </mrow> </munder> <mi>sgn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>sgn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mi>sgn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⋯</mo> <mi>sgn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mi>sgn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mi>sgn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⋯</mo> <mi>sgn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⋯</mo> <mi>sgn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\varepsilon _{a_{1}a_{2}a_{3}\ldots a_{n}}&=\prod _{1\leq i<j\leq n}\operatorname {sgn}(a_{j}-a_{i})\\&=\operatorname {sgn}(a_{2}-a_{1})\operatorname {sgn}(a_{3}-a_{1})\dotsm \operatorname {sgn}(a_{n}-a_{1})\operatorname {sgn}(a_{3}-a_{2})\operatorname {sgn}(a_{4}-a_{2})\dotsm \operatorname {sgn}(a_{n}-a_{2})\dotsm \operatorname {sgn}(a_{n}-a_{n-1})\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6c1b64bf16e4deedbda479359870756c7b313ee" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:116.564ex; height:8.843ex;" alt="{\displaystyle {\begin{aligned}\varepsilon _{a_{1}a_{2}a_{3}\ldots a_{n}}&=\prod _{1\leq i<j\leq n}\operatorname {sgn}(a_{j}-a_{i})\\&=\operatorname {sgn}(a_{2}-a_{1})\operatorname {sgn}(a_{3}-a_{1})\dotsm \operatorname {sgn}(a_{n}-a_{1})\operatorname {sgn}(a_{3}-a_{2})\operatorname {sgn}(a_{4}-a_{2})\dotsm \operatorname {sgn}(a_{n}-a_{2})\dotsm \operatorname {sgn}(a_{n}-a_{n-1})\end{aligned}}}"></span> where the <a href="/wiki/Signum_function" class="mw-redirect" title="Signum function">signum function</a> (denoted <span class="texhtml">sgn</span>) returns the sign of its argument while discarding the <a href="/wiki/Absolute_value" title="Absolute value">absolute value</a> if nonzero. The formula is valid for all index values, and for any <span class="texhtml mvar" style="font-style:italic;">n</span> (when <span class="texhtml"><i>n</i> = 0</span> or <span class="texhtml"><i>n</i> = 1</span>, this is the <a href="/wiki/Empty_product" title="Empty product">empty product</a>). However, computing the formula above naively has a <a href="/wiki/Time_complexity" title="Time complexity">time complexity</a> of <span class="texhtml">O(<i>n</i><sup>2</sup>)</span>, whereas the sign can be computed from the parity of the permutation from its <a href="/wiki/Permutation#Cycle_notation" title="Permutation">disjoint cycles</a> in only <span class="texhtml">O(<i>n</i> log(<i>n</i>))</span> cost. </p> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Levi-Civita_symbol&action=edit&section=6" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A tensor whose components in an <a href="/wiki/Orthonormal_basis" title="Orthonormal basis">orthonormal basis</a> are given by the Levi-Civita symbol (a tensor of <a href="/wiki/Covariance_and_contravariance_of_vectors" title="Covariance and contravariance of vectors">covariant</a> rank <span class="texhtml mvar" style="font-style:italic;">n</span>) is sometimes called a <b>permutation tensor</b>. </p><p>Under the ordinary transformation rules for tensors the Levi-Civita symbol is unchanged under pure rotations, consistent with that it is (by definition) the same in all coordinate systems related by orthogonal transformations. However, the Levi-Civita symbol is a <a href="/wiki/Pseudotensor" title="Pseudotensor">pseudotensor</a> because under an <a href="/wiki/Orthogonal_matrix" title="Orthogonal matrix">orthogonal transformation</a> of <a href="/wiki/Jacobian_matrix_and_determinant" title="Jacobian matrix and determinant">Jacobian determinant</a> −1, for example, a <a href="/wiki/Reflection_(mathematics)" title="Reflection (mathematics)">reflection</a> in an odd number of dimensions, it <i>should</i> acquire a minus sign if it were a tensor. As it does not change at all, the Levi-Civita symbol is, by definition, a pseudotensor. </p><p>As the Levi-Civita symbol is a pseudotensor, the result of taking a cross product is a <a href="/wiki/Pseudovector" title="Pseudovector">pseudovector</a>, not a vector.<sup id="cite_ref-Riley_et_al_5-0" class="reference"><a href="#cite_note-Riley_et_al-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>Under a general <a href="/wiki/Coordinate_change" class="mw-redirect" title="Coordinate change">coordinate change</a>, the components of the permutation tensor are multiplied by the <a href="/wiki/Jacobian_matrix_and_determinant" title="Jacobian matrix and determinant">Jacobian</a> of the <a href="/wiki/Transformation_matrix" title="Transformation matrix">transformation matrix</a>. This implies that in coordinate frames different from the one in which the tensor was defined, its components can differ from those of the Levi-Civita symbol by an overall factor. If the frame is orthonormal, the factor will be ±1 depending on whether the orientation of the frame is the same or not.<sup id="cite_ref-Riley_et_al_5-1" class="reference"><a href="#cite_note-Riley_et_al-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>In index-free tensor notation, the Levi-Civita symbol is replaced by the concept of the <a href="/wiki/Hodge_dual" class="mw-redirect" title="Hodge dual">Hodge dual</a>.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (March 2024)">citation needed</span></a></i>]</sup> </p><p>Summation symbols can be eliminated by using <a href="/wiki/Einstein_notation" title="Einstein notation">Einstein notation</a>, where an index repeated between two or more terms indicates summation over that index. For example, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{ijk}\varepsilon ^{imn}\equiv \sum _{i=1,2,3}\varepsilon _{ijk}\varepsilon ^{imn}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> <mi>k</mi> </mrow> </msub> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>m</mi> <mi>n</mi> </mrow> </msup> <mo>≡</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> </mrow> </munder> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> <mi>k</mi> </mrow> </msub> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>m</mi> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{ijk}\varepsilon ^{imn}\equiv \sum _{i=1,2,3}\varepsilon _{ijk}\varepsilon ^{imn}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbc09a3f0070b1a956cb975e30d8e6a3f7f1d856" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:24.171ex; height:5.843ex;" alt="{\displaystyle \varepsilon _{ijk}\varepsilon ^{imn}\equiv \sum _{i=1,2,3}\varepsilon _{ijk}\varepsilon ^{imn}}"></span>.</dd></dl> <p>In the following examples, Einstein notation is used. </p> <div class="mw-heading mw-heading3"><h3 id="Two_dimensions_2">Two dimensions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Levi-Civita_symbol&action=edit&section=7" title="Edit section: Two dimensions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In two dimensions, when all <span class="texhtml"><i>i</i>, <i>j</i>, <i>m</i>, <i>n</i></span> each take the values 1 and 2:<sup id="cite_ref-Tyldesley_3-1" class="reference"><a href="#cite_note-Tyldesley-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{ij}\varepsilon ^{mn}={\delta _{i}}^{m}{\delta _{j}}^{n}-{\delta _{i}}^{n}{\delta _{j}}^{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mi>n</mi> </mrow> </msup> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{ij}\varepsilon ^{mn}={\delta _{i}}^{m}{\delta _{j}}^{n}-{\delta _{i}}^{n}{\delta _{j}}^{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c7f25b08600de78499393e979de85a962e77d06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.58ex; height:3.009ex;" alt="{\displaystyle \varepsilon _{ij}\varepsilon ^{mn}={\delta _{i}}^{m}{\delta _{j}}^{n}-{\delta _{i}}^{n}{\delta _{j}}^{m}}"></span> </td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_1" class="reference nourlexpansion" style="font-weight:bold;">1</span>)</b></td></tr></tbody></table> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{ij}\varepsilon ^{in}={\delta _{j}}^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>n</mi> </mrow> </msup> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{ij}\varepsilon ^{in}={\delta _{j}}^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aef7668dc81ade8f19a46edabd78f95e96aae87e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.689ex; height:3.343ex;" alt="{\displaystyle \varepsilon _{ij}\varepsilon ^{in}={\delta _{j}}^{n}}"></span> </td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_2" class="reference nourlexpansion" style="font-weight:bold;">2</span>)</b></td></tr></tbody></table> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{ij}\varepsilon ^{ij}=2.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msup> <mo>=</mo> <mn>2.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{ij}\varepsilon ^{ij}=2.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/664a30dec7e95704f014d093e13972de5309dff5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.029ex; height:3.343ex;" alt="{\displaystyle \varepsilon _{ij}\varepsilon ^{ij}=2.}"></span> </td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_3" class="reference nourlexpansion" style="font-weight:bold;">3</span>)</b></td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Three_dimensions_2">Three dimensions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Levi-Civita_symbol&action=edit&section=8" title="Edit section: Three dimensions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Index_and_symbol_values">Index and symbol values</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Levi-Civita_symbol&action=edit&section=9" title="Edit section: Index and symbol values"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In three dimensions, when all <span class="texhtml"><i>i</i>, <i>j</i>, <i>k</i>, <i>m</i>, <i>n</i></span> each take values 1, 2, and 3:<sup id="cite_ref-Tyldesley_3-2" class="reference"><a href="#cite_note-Tyldesley-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{ijk}\varepsilon ^{pqk}=\delta _{i}{}^{p}\delta _{j}{}^{q}-\delta _{i}{}^{q}\delta _{j}{}^{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> <mi>k</mi> </mrow> </msub> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mi>q</mi> <mi>k</mi> </mrow> </msup> <mo>=</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msup> <mo>−</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msup> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{ijk}\varepsilon ^{pqk}=\delta _{i}{}^{p}\delta _{j}{}^{q}-\delta _{i}{}^{q}\delta _{j}{}^{p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fb1d15fc60b7b1dd2ee81077525c9dfee02df9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.755ex; height:3.343ex;" alt="{\displaystyle \varepsilon _{ijk}\varepsilon ^{pqk}=\delta _{i}{}^{p}\delta _{j}{}^{q}-\delta _{i}{}^{q}\delta _{j}{}^{p}}"></span> </td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_4" class="reference nourlexpansion" style="font-weight:bold;">4</span>)</b></td></tr></tbody></table> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{jmn}\varepsilon ^{imn}=2{\delta _{j}}^{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>m</mi> <mi>n</mi> </mrow> </msub> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>m</mi> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mn>2</mn> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{jmn}\varepsilon ^{imn}=2{\delta _{j}}^{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a455c2935ac3a24f5d869a0fa9031b866a3485df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.737ex; height:3.343ex;" alt="{\displaystyle \varepsilon _{jmn}\varepsilon ^{imn}=2{\delta _{j}}^{i}}"></span> </td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_5" class="reference nourlexpansion" style="font-weight:bold;">5</span>)</b></td></tr></tbody></table> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{ijk}\varepsilon ^{ijk}=6.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> <mi>k</mi> </mrow> </msub> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> <mi>k</mi> </mrow> </msup> <mo>=</mo> <mn>6.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{ijk}\varepsilon ^{ijk}=6.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0369815b9ebaaf33c91e733f806cf9c92564280e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.742ex; height:3.343ex;" alt="{\displaystyle \varepsilon _{ijk}\varepsilon ^{ijk}=6.}"></span> </td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_6" class="reference nourlexpansion" style="font-weight:bold;">6</span>)</b></td></tr></tbody></table> <div class="mw-heading mw-heading4"><h4 id="Product">Product</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Levi-Civita_symbol&action=edit&section=10" title="Edit section: Product"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Levi-Civita symbol is related to the <a href="/wiki/Kronecker_delta" title="Kronecker delta">Kronecker delta</a>. In three dimensions, the relationship is given by the following equations (vertical lines denote the determinant):<sup id="cite_ref-Kay_4-1" class="reference"><a href="#cite_note-Kay-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\varepsilon _{ijk}\varepsilon _{lmn}&={\begin{vmatrix}\delta _{il}&\delta _{im}&\delta _{in}\\\delta _{jl}&\delta _{jm}&\delta _{jn}\\\delta _{kl}&\delta _{km}&\delta _{kn}\\\end{vmatrix}}\\[6pt]&=\delta _{il}\left(\delta _{jm}\delta _{kn}-\delta _{jn}\delta _{km}\right)-\delta _{im}\left(\delta _{jl}\delta _{kn}-\delta _{jn}\delta _{kl}\right)+\delta _{in}\left(\delta _{jl}\delta _{km}-\delta _{jm}\delta _{kl}\right).\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="0.9em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> <mi>k</mi> </mrow> </msub> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mi>m</mi> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>l</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>m</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>l</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>m</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>l</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>m</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>l</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>m</mi> </mrow> </msub> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>n</mi> </mrow> </msub> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>m</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>m</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>l</mi> </mrow> </msub> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>n</mi> </mrow> </msub> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>l</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>n</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>l</mi> </mrow> </msub> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>m</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>m</mi> </mrow> </msub> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>l</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\varepsilon _{ijk}\varepsilon _{lmn}&={\begin{vmatrix}\delta _{il}&\delta _{im}&\delta _{in}\\\delta _{jl}&\delta _{jm}&\delta _{jn}\\\delta _{kl}&\delta _{km}&\delta _{kn}\\\end{vmatrix}}\\[6pt]&=\delta _{il}\left(\delta _{jm}\delta _{kn}-\delta _{jn}\delta _{km}\right)-\delta _{im}\left(\delta _{jl}\delta _{kn}-\delta _{jn}\delta _{kl}\right)+\delta _{in}\left(\delta _{jl}\delta _{km}-\delta _{jm}\delta _{kl}\right).\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/649e6209e5af520ca1a5ea07c33b58591565ab3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:77.799ex; height:14.176ex;" alt="{\displaystyle {\begin{aligned}\varepsilon _{ijk}\varepsilon _{lmn}&={\begin{vmatrix}\delta _{il}&\delta _{im}&\delta _{in}\\\delta _{jl}&\delta _{jm}&\delta _{jn}\\\delta _{kl}&\delta _{km}&\delta _{kn}\\\end{vmatrix}}\\[6pt]&=\delta _{il}\left(\delta _{jm}\delta _{kn}-\delta _{jn}\delta _{km}\right)-\delta _{im}\left(\delta _{jl}\delta _{kn}-\delta _{jn}\delta _{kl}\right)+\delta _{in}\left(\delta _{jl}\delta _{km}-\delta _{jm}\delta _{kl}\right).\end{aligned}}}"></span></dd></dl> <p>A special case of this result occurs when one of the indices is repeated and summed over: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=1}^{3}\varepsilon _{ijk}\varepsilon _{imn}=\delta _{jm}\delta _{kn}-\delta _{jn}\delta _{km}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </munderover> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> <mi>k</mi> </mrow> </msub> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>m</mi> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>m</mi> </mrow> </msub> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>n</mi> </mrow> </msub> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{3}\varepsilon _{ijk}\varepsilon _{imn}=\delta _{jm}\delta _{kn}-\delta _{jn}\delta _{km}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e19d7782236c042cf847d11be2e7d1142738671" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:30.395ex; height:7.176ex;" alt="{\displaystyle \sum _{i=1}^{3}\varepsilon _{ijk}\varepsilon _{imn}=\delta _{jm}\delta _{kn}-\delta _{jn}\delta _{km}}"></span></dd></dl> <p>In Einstein notation, the duplication of the <span class="texhtml mvar" style="font-style:italic;">i</span> index implies the sum on <span class="texhtml mvar" style="font-style:italic;">i</span>. The previous is then denoted <span class="texhtml"><i>ε<sub>ijk</sub>ε<sub>imn</sub></i> = <i>δ<sub>jm</sub>δ<sub>kn</sub></i> − <i>δ<sub>jn</sub>δ<sub>km</sub></i></span>. </p><p>If two indices are repeated (and summed over), this further reduces to: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=1}^{3}\sum _{j=1}^{3}\varepsilon _{ijk}\varepsilon _{ijn}=2\delta _{kn}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </munderover> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </munderover> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> <mi>k</mi> </mrow> </msub> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mn>2</mn> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{3}\sum _{j=1}^{3}\varepsilon _{ijk}\varepsilon _{ijn}=2\delta _{kn}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1a3304b4629a17252d777c79f1a7a9f82534899" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:21.816ex; height:7.509ex;" alt="{\displaystyle \sum _{i=1}^{3}\sum _{j=1}^{3}\varepsilon _{ijk}\varepsilon _{ijn}=2\delta _{kn}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="n_dimensions"><i>n</i> dimensions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Levi-Civita_symbol&action=edit&section=11" title="Edit section: n dimensions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Index_and_symbol_values_2">Index and symbol values</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Levi-Civita_symbol&action=edit&section=12" title="Edit section: Index and symbol values"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <span class="texhtml mvar" style="font-style:italic;">n</span> dimensions, when all <span class="texhtml"><i>i</i><sub>1</sub>, ...,<i>i<sub>n</sub></i>, <i>j</i><sub>1</sub>, ..., <i>j</i><sub><i>n</i></sub></span> take values <span class="texhtml">1, 2, ..., <i>n</i></span>:<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (March 2024)">citation needed</span></a></i>]</sup> </p> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{i_{1}\dots i_{n}}\varepsilon ^{j_{1}\dots j_{n}}=\delta _{i_{1}\dots i_{n}}^{j_{1}\dots j_{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msub> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msup> <mo>=</mo> <msubsup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{i_{1}\dots i_{n}}\varepsilon ^{j_{1}\dots j_{n}}=\delta _{i_{1}\dots i_{n}}^{j_{1}\dots j_{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cbdd6de2cbbe92fff61efce87b8a535aaed6f87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:22.027ex; height:3.843ex;" alt="{\displaystyle \varepsilon _{i_{1}\dots i_{n}}\varepsilon ^{j_{1}\dots j_{n}}=\delta _{i_{1}\dots i_{n}}^{j_{1}\dots j_{n}}}"></span> </td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_7" class="reference nourlexpansion" style="font-weight:bold;">7</span>)</b></td></tr></tbody></table> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{i_{1}\dots i_{k}~i_{k+1}\dots i_{n}}\varepsilon ^{i_{1}\dots i_{k}~j_{k+1}\dots j_{n}}=\delta _{i_{1}\ldots i_{k}~i_{k+1}\ldots i_{n}}^{i_{1}\dots i_{k}~j_{k+1}\ldots j_{n}}=k!~\delta _{i_{k+1}\dots i_{n}}^{j_{k+1}\dots j_{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mtext> </mtext> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msub> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mtext> </mtext> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msup> <mo>=</mo> <msubsup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mtext> </mtext> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mtext> </mtext> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msubsup> <mo>=</mo> <mi>k</mi> <mo>!</mo> <mtext> </mtext> <msubsup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{i_{1}\dots i_{k}~i_{k+1}\dots i_{n}}\varepsilon ^{i_{1}\dots i_{k}~j_{k+1}\dots j_{n}}=\delta _{i_{1}\ldots i_{k}~i_{k+1}\ldots i_{n}}^{i_{1}\dots i_{k}~j_{k+1}\ldots j_{n}}=k!~\delta _{i_{k+1}\dots i_{n}}^{j_{k+1}\dots j_{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d75551e5dcf32fda99e8713e4a39c25f3e54fdd4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:56.857ex; height:4.176ex;" alt="{\displaystyle \varepsilon _{i_{1}\dots i_{k}~i_{k+1}\dots i_{n}}\varepsilon ^{i_{1}\dots i_{k}~j_{k+1}\dots j_{n}}=\delta _{i_{1}\ldots i_{k}~i_{k+1}\ldots i_{n}}^{i_{1}\dots i_{k}~j_{k+1}\ldots j_{n}}=k!~\delta _{i_{k+1}\dots i_{n}}^{j_{k+1}\dots j_{n}}}"></span> </td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_8" class="reference nourlexpansion" style="font-weight:bold;">8</span>)</b></td></tr></tbody></table> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{i_{1}\dots i_{n}}\varepsilon ^{i_{1}\dots i_{n}}=n!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msub> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msup> <mo>=</mo> <mi>n</mi> <mo>!</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{i_{1}\dots i_{n}}\varepsilon ^{i_{1}\dots i_{n}}=n!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63666d4a8d5b216bfaf7eb50426f6dc81397ed91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.485ex; height:3.343ex;" alt="{\displaystyle \varepsilon _{i_{1}\dots i_{n}}\varepsilon ^{i_{1}\dots i_{n}}=n!}"></span> </td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><b>(<span id="math_9" class="reference nourlexpansion" style="font-weight:bold;">9</span>)</b></td></tr></tbody></table> <p>where the exclamation mark (<span class="texhtml">!</span>) denotes the <a href="/wiki/Factorial" title="Factorial">factorial</a>, and <span class="texhtml"><i>δ</i><span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.2em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline"><i>α</i>...</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline"><i>β</i>...</sub></span></span></span> is the <a href="/wiki/Generalized_Kronecker_delta" class="mw-redirect" title="Generalized Kronecker delta">generalized Kronecker delta</a>. For any <span class="texhtml mvar" style="font-style:italic;">n</span>, the property </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i,j,k,\dots =1}^{n}\varepsilon _{ijk\dots }\varepsilon _{ijk\dots }=n!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>,</mo> <mi>k</mi> <mo>,</mo> <mo>⋯</mo> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> <mi>k</mi> <mo>…</mo> </mrow> </msub> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> <mi>k</mi> <mo>…</mo> </mrow> </msub> <mo>=</mo> <mi>n</mi> <mo>!</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i,j,k,\dots =1}^{n}\varepsilon _{ijk\dots }\varepsilon _{ijk\dots }=n!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04755c7661b3e7aa2931e71e2e57663103204add" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:23.712ex; height:7.176ex;" alt="{\displaystyle \sum _{i,j,k,\dots =1}^{n}\varepsilon _{ijk\dots }\varepsilon _{ijk\dots }=n!}"></span></dd></dl> <p>follows from the facts that </p> <ul><li>every permutation is either even or odd,</li> <li><span class="texhtml">(+1)<sup>2</sup> = (−1)<sup>2</sup> = 1</span>, and</li> <li>the number of permutations of any <span class="texhtml mvar" style="font-style:italic;">n</span>-element set number is exactly <span class="texhtml"><i>n</i>!</span>.</li></ul> <p>The particular case of (<b><a href="#math_8">8</a></b>) with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle k=n-2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle k=n-2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/195d89b84a63d835982daf57112845858d3052ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.707ex; height:2.343ex;" alt="{\textstyle k=n-2}"></span> is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{i_{1}\dots i_{n-2}jk}\varepsilon ^{i_{1}\dots i_{n-2}lm}=(n-2)!(\delta _{j}^{l}\delta _{k}^{m}-\delta _{j}^{m}\delta _{k}^{l})\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mi>j</mi> <mi>k</mi> </mrow> </msub> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mi>l</mi> <mi>m</mi> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>!</mo> <mo stretchy="false">(</mo> <msubsup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msubsup> <msubsup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msubsup> <msubsup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msubsup> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{i_{1}\dots i_{n-2}jk}\varepsilon ^{i_{1}\dots i_{n-2}lm}=(n-2)!(\delta _{j}^{l}\delta _{k}^{m}-\delta _{j}^{m}\delta _{k}^{l})\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30f4f8c7e6191a625a49982cb6c2efee393a7073" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:45.379ex; height:3.676ex;" alt="{\displaystyle \varepsilon _{i_{1}\dots i_{n-2}jk}\varepsilon ^{i_{1}\dots i_{n-2}lm}=(n-2)!(\delta _{j}^{l}\delta _{k}^{m}-\delta _{j}^{m}\delta _{k}^{l})\,.}"></span> </p> <div class="mw-heading mw-heading4"><h4 id="Product_2">Product</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Levi-Civita_symbol&action=edit&section=13" title="Edit section: Product"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In general, for <span class="texhtml mvar" style="font-style:italic;">n</span> dimensions, one can write the product of two Levi-Civita symbols as: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{i_{1}i_{2}\dots i_{n}}\varepsilon _{j_{1}j_{2}\dots j_{n}}={\begin{vmatrix}\delta _{i_{1}j_{1}}&\delta _{i_{1}j_{2}}&\dots &\delta _{i_{1}j_{n}}\\\delta _{i_{2}j_{1}}&\delta _{i_{2}j_{2}}&\dots &\delta _{i_{2}j_{n}}\\\vdots &\vdots &\ddots &\vdots \\\delta _{i_{n}j_{1}}&\delta _{i_{n}j_{2}}&\dots &\delta _{i_{n}j_{n}}\\\end{vmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msub> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> </mtd> <mtd> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msub> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> </mtd> <mtd> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msub> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> </mtd> <mtd> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msub> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msub> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{i_{1}i_{2}\dots i_{n}}\varepsilon _{j_{1}j_{2}\dots j_{n}}={\begin{vmatrix}\delta _{i_{1}j_{1}}&\delta _{i_{1}j_{2}}&\dots &\delta _{i_{1}j_{n}}\\\delta _{i_{2}j_{1}}&\delta _{i_{2}j_{2}}&\dots &\delta _{i_{2}j_{n}}\\\vdots &\vdots &\ddots &\vdots \\\delta _{i_{n}j_{1}}&\delta _{i_{n}j_{2}}&\dots &\delta _{i_{n}j_{n}}\\\end{vmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5fffeeee806317c30f5edfa3d7b4afc1866d425c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.005ex; width:44.262ex; height:15.176ex;" alt="{\displaystyle \varepsilon _{i_{1}i_{2}\dots i_{n}}\varepsilon _{j_{1}j_{2}\dots j_{n}}={\begin{vmatrix}\delta _{i_{1}j_{1}}&\delta _{i_{1}j_{2}}&\dots &\delta _{i_{1}j_{n}}\\\delta _{i_{2}j_{1}}&\delta _{i_{2}j_{2}}&\dots &\delta _{i_{2}j_{n}}\\\vdots &\vdots &\ddots &\vdots \\\delta _{i_{n}j_{1}}&\delta _{i_{n}j_{2}}&\dots &\delta _{i_{n}j_{n}}\\\end{vmatrix}}.}"></span><b>Proof:</b> Both sides change signs upon switching two indices, so without loss of generality assume <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i_{1}\leq \cdots \leq i_{n},j_{1}\leq \cdots \leq j_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>≤</mo> <mo>⋯</mo> <mo>≤</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>≤</mo> <mo>⋯</mo> <mo>≤</mo> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i_{1}\leq \cdots \leq i_{n},j_{1}\leq \cdots \leq j_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8a009b405b197a067543c4852910f70b46a6c9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:26.941ex; height:2.509ex;" alt="{\displaystyle i_{1}\leq \cdots \leq i_{n},j_{1}\leq \cdots \leq j_{n}}"></span>. If some <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i_{c}=i_{c+1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i_{c}=i_{c+1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/265754ca3646fccef947be80ce01149797c7e6b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.692ex; height:2.509ex;" alt="{\displaystyle i_{c}=i_{c+1}}"></span> then left side is zero, and right side is also zero since two of its rows are equal. Similarly for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j_{c}=j_{c+1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j_{c}=j_{c+1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09ba271618360683345c0896c7dd65e430eb3ebb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:9.03ex; height:2.509ex;" alt="{\displaystyle j_{c}=j_{c+1}}"></span>. Finally, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i_{1}<\cdots <i_{n},j_{1}<\cdots <j_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo><</mo> <mo>⋯</mo> <mo><</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo><</mo> <mo>⋯</mo> <mo><</mo> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i_{1}<\cdots <i_{n},j_{1}<\cdots <j_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5250d19e08fe4527f60f9966951d7b4dc0102182" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:26.941ex; height:2.509ex;" alt="{\displaystyle i_{1}<\cdots <i_{n},j_{1}<\cdots <j_{n}}"></span>, then both sides are 1. </p> <div class="mw-heading mw-heading3"><h3 id="Proofs">Proofs</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Levi-Civita_symbol&action=edit&section=14" title="Edit section: Proofs"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For (<b><a href="#math_1">1</a></b>), both sides are antisymmetric with respect of <span class="texhtml mvar" style="font-style:italic;">ij</span> and <span class="texhtml mvar" style="font-style:italic;">mn</span>. We therefore only need to consider the case <span class="texhtml"><i>i</i> ≠ <i>j</i></span> and <span class="texhtml"><i>m</i> ≠ <i>n</i></span>. By substitution, we see that the equation holds for <span class="texhtml"><i>ε</i><sub>12</sub><i>ε</i><sup>12</sup></span>, that is, for <span class="texhtml"><i>i</i> = <i>m</i> = 1</span> and <span class="texhtml"><i>j</i> = <i>n</i> = 2</span>. (Both sides are then one). Since the equation is antisymmetric in <span class="texhtml mvar" style="font-style:italic;">ij</span> and <span class="texhtml mvar" style="font-style:italic;">mn</span>, any set of values for these can be reduced to the above case (which holds). The equation thus holds for all values of <span class="texhtml mvar" style="font-style:italic;">ij</span> and <span class="texhtml mvar" style="font-style:italic;">mn</span>. </p><p>Using (<b><a href="#math_1">1</a></b>), we have for (<b><a href="#math_2">2</a></b>) </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{ij}\varepsilon ^{in}=\delta _{i}{}^{i}\delta _{j}{}^{n}-\delta _{i}{}^{n}\delta _{j}{}^{i}=2\delta _{j}{}^{n}-\delta _{j}{}^{n}=\delta _{j}{}^{n}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>n</mi> </mrow> </msup> <mo>=</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>=</mo> <mn>2</mn> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{ij}\varepsilon ^{in}=\delta _{i}{}^{i}\delta _{j}{}^{n}-\delta _{i}{}^{n}\delta _{j}{}^{i}=2\delta _{j}{}^{n}-\delta _{j}{}^{n}=\delta _{j}{}^{n}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30310b0d588a96f8a8efc22cf0bc78a864eca56a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:43.669ex; height:3.343ex;" alt="{\displaystyle \varepsilon _{ij}\varepsilon ^{in}=\delta _{i}{}^{i}\delta _{j}{}^{n}-\delta _{i}{}^{n}\delta _{j}{}^{i}=2\delta _{j}{}^{n}-\delta _{j}{}^{n}=\delta _{j}{}^{n}\,.}"></span></dd></dl> <p>Here we used the <a href="/wiki/Einstein_summation_convention" class="mw-redirect" title="Einstein summation convention">Einstein summation convention</a> with <span class="texhtml mvar" style="font-style:italic;">i</span> going from 1 to 2. Next, (<b><a href="#math_3">3</a></b>) follows similarly from (<b><a href="#math_2">2</a></b>). </p><p>To establish (<b><a href="#math_5">5</a></b>), notice that both sides vanish when <span class="texhtml"><i>i</i> ≠ <i>j</i></span>. Indeed, if <span class="texhtml"><i>i</i> ≠ <i>j</i></span>, then one can not choose <span class="texhtml mvar" style="font-style:italic;">m</span> and <span class="texhtml mvar" style="font-style:italic;">n</span> such that both permutation symbols on the left are nonzero. Then, with <span class="texhtml"><i>i</i> = <i>j</i></span> fixed, there are only two ways to choose <span class="texhtml mvar" style="font-style:italic;">m</span> and <span class="texhtml mvar" style="font-style:italic;">n</span> from the remaining two indices. For any such indices, we have </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{jmn}\varepsilon ^{imn}=\left(\varepsilon ^{imn}\right)^{2}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>m</mi> <mi>n</mi> </mrow> </msub> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>m</mi> <mi>n</mi> </mrow> </msup> <mo>=</mo> <msup> <mrow> <mo>(</mo> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>m</mi> <mi>n</mi> </mrow> </msup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{jmn}\varepsilon ^{imn}=\left(\varepsilon ^{imn}\right)^{2}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ebcbc59cb4dc2d3da520bf461e617b9f8d91021" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.59ex; height:3.843ex;" alt="{\displaystyle \varepsilon _{jmn}\varepsilon ^{imn}=\left(\varepsilon ^{imn}\right)^{2}=1}"></span></dd></dl> <p>(no summation), and the result follows. </p><p>Then (<b><a href="#math_6">6</a></b>) follows since <span class="nowrap">3! = 6</span> and for any distinct indices <span class="texhtml"><i>i</i>, <i>j</i>, <i>k</i></span> taking values <span class="nowrap">1, 2, 3</span>, we have </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{ijk}\varepsilon ^{ijk}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> <mi>k</mi> </mrow> </msub> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> <mi>k</mi> </mrow> </msup> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{ijk}\varepsilon ^{ijk}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ba3b3478ae1012e5485c43d07cf7767c6aa5a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.095ex; height:3.343ex;" alt="{\displaystyle \varepsilon _{ijk}\varepsilon ^{ijk}=1}"></span><span style="padding-left:3em;"> </span>(no summation, distinct <span class="texhtml"><i>i</i>, <i>j</i>, <i>k</i></span>)</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Applications_and_examples">Applications and examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Levi-Civita_symbol&action=edit&section=15" title="Edit section: Applications and examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Determinants">Determinants</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Levi-Civita_symbol&action=edit&section=16" title="Edit section: Determinants"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In linear algebra, the <a href="/wiki/Determinant" title="Determinant">determinant</a> of a <span class="nowrap">3 × 3</span> <a href="/wiki/Square_matrix" title="Square matrix">square matrix</a> <span class="texhtml"><b>A</b> = [<i>a<sub>ij</sub></i>]</span> can be written<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det(\mathbf {A} )=\sum _{i=1}^{3}\sum _{j=1}^{3}\sum _{k=1}^{3}\varepsilon _{ijk}a_{1i}a_{2j}a_{3k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </munderover> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </munderover> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </munderover> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> <mi>k</mi> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>i</mi> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>j</mi> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det(\mathbf {A} )=\sum _{i=1}^{3}\sum _{j=1}^{3}\sum _{k=1}^{3}\varepsilon _{ijk}a_{1i}a_{2j}a_{3k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/beb98573a2c43f7234607ca48ac0605e2bef26a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:33.754ex; height:7.509ex;" alt="{\displaystyle \det(\mathbf {A} )=\sum _{i=1}^{3}\sum _{j=1}^{3}\sum _{k=1}^{3}\varepsilon _{ijk}a_{1i}a_{2j}a_{3k}}"></span></dd></dl> <p>Similarly the determinant of an <span class="texhtml"><i>n</i> × <i>n</i></span> matrix <span class="texhtml"><b>A</b> = [<i>a<sub>ij</sub></i>]</span> can be written as<sup id="cite_ref-Riley_et_al_5-2" class="reference"><a href="#cite_note-Riley_et_al-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det(\mathbf {A} )=\varepsilon _{i_{1}\dots i_{n}}a_{1i_{1}}\dots a_{ni_{n}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mo>…</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det(\mathbf {A} )=\varepsilon _{i_{1}\dots i_{n}}a_{1i_{1}}\dots a_{ni_{n}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/624037b5e0d67e88f75559b3d8b8eef62708d378" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:28.137ex; height:3.009ex;" alt="{\displaystyle \det(\mathbf {A} )=\varepsilon _{i_{1}\dots i_{n}}a_{1i_{1}}\dots a_{ni_{n}},}"></span></dd></dl> <p>where each <span class="texhtml mvar" style="font-style:italic;">i<sub>r</sub></span> should be summed over <span class="texhtml">1, ..., <i>n</i></span>, or equivalently: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det(\mathbf {A} )={\frac {1}{n!}}\varepsilon _{i_{1}\dots i_{n}}\varepsilon _{j_{1}\dots j_{n}}a_{i_{1}j_{1}}\dots a_{i_{n}j_{n}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>n</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msub> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mo>…</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det(\mathbf {A} )={\frac {1}{n!}}\varepsilon _{i_{1}\dots i_{n}}\varepsilon _{j_{1}\dots j_{n}}a_{i_{1}j_{1}}\dots a_{i_{n}j_{n}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddb53f31327ebfd673ba1d3ebc7e85c15a1a9226" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:38.751ex; height:5.343ex;" alt="{\displaystyle \det(\mathbf {A} )={\frac {1}{n!}}\varepsilon _{i_{1}\dots i_{n}}\varepsilon _{j_{1}\dots j_{n}}a_{i_{1}j_{1}}\dots a_{i_{n}j_{n}},}"></span></dd></dl> <p>where now each <span class="texhtml mvar" style="font-style:italic;">i<sub>r</sub></span> and each <span class="texhtml mvar" style="font-style:italic;">j<sub>r</sub></span> should be summed over <span class="texhtml">1, ..., <i>n</i></span>. More generally, we have the identity<sup id="cite_ref-Riley_et_al_5-3" class="reference"><a href="#cite_note-Riley_et_al-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i_{1},i_{2},\dots }\varepsilon _{i_{1}\dots i_{n}}a_{i_{1}\,j_{1}}\dots a_{i_{n}\,j_{n}}=\det(\mathbf {A} )\varepsilon _{j_{1}\dots j_{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> </mrow> </munder> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mspace width="thinmathspace" /> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mo>…</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mspace width="thinmathspace" /> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msub> <mo>=</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">)</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i_{1},i_{2},\dots }\varepsilon _{i_{1}\dots i_{n}}a_{i_{1}\,j_{1}}\dots a_{i_{n}\,j_{n}}=\det(\mathbf {A} )\varepsilon _{j_{1}\dots j_{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ca34c714ff51a545258d30ba70f5d7e90597dc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:42.026ex; height:5.843ex;" alt="{\displaystyle \sum _{i_{1},i_{2},\dots }\varepsilon _{i_{1}\dots i_{n}}a_{i_{1}\,j_{1}}\dots a_{i_{n}\,j_{n}}=\det(\mathbf {A} )\varepsilon _{j_{1}\dots j_{n}}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Vector_cross_product">Vector cross product</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Levi-Civita_symbol&action=edit&section=17" title="Edit section: Vector cross product"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Cross_product" title="Cross product">cross product</a></div> <div class="mw-heading mw-heading4"><h4 id="Cross_product_(two_vectors)"><span id="Cross_product_.28two_vectors.29"></span>Cross product (two vectors)</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Levi-Civita_symbol&action=edit&section=18" title="Edit section: Cross product (two vectors)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbf {e_{1}} ,\mathbf {e_{2}} ,\mathbf {e_{3}} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </msub> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">2</mn> </mrow> </msub> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">3</mn> </mrow> </msub> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbf {e_{1}} ,\mathbf {e_{2}} ,\mathbf {e_{3}} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d45b9a817716f203f184d855ed4c51207ec6e068" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.085ex; height:2.843ex;" alt="{\displaystyle (\mathbf {e_{1}} ,\mathbf {e_{2}} ,\mathbf {e_{3}} )}"></span> a <a href="/wiki/Orientation_(vector_space)" title="Orientation (vector space)">positively oriented</a> orthonormal basis of a vector space. If <span class="texhtml">(<i>a</i><sup>1</sup>, <i>a</i><sup>2</sup>, <i>a</i><sup>3</sup>)</span> and <span class="texhtml">(<i>b</i><sup>1</sup>, <i>b</i><sup>2</sup>, <i>b</i><sup>3</sup>)</span> are the coordinates of the <a href="/wiki/Vector_(geometry)" class="mw-redirect" title="Vector (geometry)">vectors</a> <span class="texhtml"><b>a</b></span> and <span class="texhtml"><b>b</b></span> in this basis, then their cross product can be written as a determinant:<sup id="cite_ref-Riley_et_al_5-4" class="reference"><a href="#cite_note-Riley_et_al-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a\times b} ={\begin{vmatrix}\mathbf {e_{1}} &\mathbf {e_{2}} &\mathbf {e_{3}} \\a^{1}&a^{2}&a^{3}\\b^{1}&b^{2}&b^{3}\\\end{vmatrix}}=\sum _{i=1}^{3}\sum _{j=1}^{3}\sum _{k=1}^{3}\varepsilon _{ijk}\mathbf {e} _{i}a^{j}b^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> <mo>×</mo> <mi mathvariant="bold">b</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </msub> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">2</mn> </mrow> </msub> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">3</mn> </mrow> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mtd> <mtd> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mtd> <mtd> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </munderover> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </munderover> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </munderover> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> <mi>k</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a\times b} ={\begin{vmatrix}\mathbf {e_{1}} &\mathbf {e_{2}} &\mathbf {e_{3}} \\a^{1}&a^{2}&a^{3}\\b^{1}&b^{2}&b^{3}\\\end{vmatrix}}=\sum _{i=1}^{3}\sum _{j=1}^{3}\sum _{k=1}^{3}\varepsilon _{ijk}\mathbf {e} _{i}a^{j}b^{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/889cde8a79a4213e00c27d6ae60d404f2de87f1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:46.614ex; height:9.509ex;" alt="{\displaystyle \mathbf {a\times b} ={\begin{vmatrix}\mathbf {e_{1}} &\mathbf {e_{2}} &\mathbf {e_{3}} \\a^{1}&a^{2}&a^{3}\\b^{1}&b^{2}&b^{3}\\\end{vmatrix}}=\sum _{i=1}^{3}\sum _{j=1}^{3}\sum _{k=1}^{3}\varepsilon _{ijk}\mathbf {e} _{i}a^{j}b^{k}}"></span></dd></dl> <p>hence also using the Levi-Civita symbol, and more simply: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbf {a\times b} )^{i}=\sum _{j=1}^{3}\sum _{k=1}^{3}\varepsilon _{ijk}a^{j}b^{k}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> <mo>×</mo> <mi mathvariant="bold">b</mi> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </munderover> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </munderover> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> <mi>k</mi> </mrow> </msub> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbf {a\times b} )^{i}=\sum _{j=1}^{3}\sum _{k=1}^{3}\varepsilon _{ijk}a^{j}b^{k}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b798fa46e21c7f9d393a74a7c8abaf1de3d85f35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:27.107ex; height:7.509ex;" alt="{\displaystyle (\mathbf {a\times b} )^{i}=\sum _{j=1}^{3}\sum _{k=1}^{3}\varepsilon _{ijk}a^{j}b^{k}.}"></span></dd></dl> <p>In Einstein notation, the summation symbols may be omitted, and the <span class="texhtml mvar" style="font-style:italic;">i</span>th component of their cross product equals<sup id="cite_ref-Kay_4-2" class="reference"><a href="#cite_note-Kay-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbf {a\times b} )^{i}=\varepsilon _{ijk}a^{j}b^{k}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> <mo>×</mo> <mi mathvariant="bold">b</mi> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>=</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> <mi>k</mi> </mrow> </msub> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbf {a\times b} )^{i}=\varepsilon _{ijk}a^{j}b^{k}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d753218e19a64d5d50e10ee54c8b0e4d970fb842" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.622ex; height:3.343ex;" alt="{\displaystyle (\mathbf {a\times b} )^{i}=\varepsilon _{ijk}a^{j}b^{k}.}"></span></dd></dl> <p>The first component is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbf {a\times b} )^{1}=a^{2}b^{3}-a^{3}b^{2}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> <mo>×</mo> <mi mathvariant="bold">b</mi> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbf {a\times b} )^{1}=a^{2}b^{3}-a^{3}b^{2}\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0aa629f188edc5fc1a4dda7c8b7da5883e9a07eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.133ex; height:3.176ex;" alt="{\displaystyle (\mathbf {a\times b} )^{1}=a^{2}b^{3}-a^{3}b^{2}\,,}"></span></dd></dl> <p>then by cyclic permutations of <span class="nowrap">1, 2, 3</span> the others can be derived immediately, without explicitly calculating them from the above formulae: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}(\mathbf {a\times b} )^{2}&=a^{3}b^{1}-a^{1}b^{3}\,,\\(\mathbf {a\times b} )^{3}&=a^{1}b^{2}-a^{2}b^{1}\,.\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> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> <mo>×</mo> <mi mathvariant="bold">b</mi> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> <mo>×</mo> <mi mathvariant="bold">b</mi> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}(\mathbf {a\times b} )^{2}&=a^{3}b^{1}-a^{1}b^{3}\,,\\(\mathbf {a\times b} )^{3}&=a^{1}b^{2}-a^{2}b^{1}\,.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/030ec58e48a15541825567135d08a17784bc404d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:24.885ex; height:6.509ex;" alt="{\displaystyle {\begin{aligned}(\mathbf {a\times b} )^{2}&=a^{3}b^{1}-a^{1}b^{3}\,,\\(\mathbf {a\times b} )^{3}&=a^{1}b^{2}-a^{2}b^{1}\,.\end{aligned}}}"></span></dd></dl> <div class="mw-heading mw-heading4"><h4 id="Triple_scalar_product_(three_vectors)"><span id="Triple_scalar_product_.28three_vectors.29"></span>Triple scalar product (three vectors)</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Levi-Civita_symbol&action=edit&section=19" title="Edit section: Triple scalar product (three vectors)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>From the above expression for the cross product, we have: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a\times b} =-\mathbf {b\times a} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> <mo>×</mo> <mi mathvariant="bold">b</mi> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> <mo>×</mo> <mi mathvariant="bold">a</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a\times b} =-\mathbf {b\times a} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d9fd296c37cd456c261c10a6981ee312c7b96d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.157ex; height:2.343ex;" alt="{\displaystyle \mathbf {a\times b} =-\mathbf {b\times a} }"></span>.</dd></dl> <p>If <span class="texhtml"><b>c</b> = (<i>c</i><sup>1</sup>, <i>c</i><sup>2</sup>, <i>c</i><sup>3</sup>)</span> is a third vector, then the <a href="/wiki/Triple_scalar_product" class="mw-redirect" title="Triple scalar product">triple scalar product</a> equals </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} \cdot (\mathbf {b\times c} )=\varepsilon _{ijk}a^{i}b^{j}c^{k}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> <mo>×</mo> <mi mathvariant="bold">c</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> <mi>k</mi> </mrow> </msub> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \cdot (\mathbf {b\times c} )=\varepsilon _{ijk}a^{i}b^{j}c^{k}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d059442d5d989e316bd62c383bd567249e8c1f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.496ex; height:3.343ex;" alt="{\displaystyle \mathbf {a} \cdot (\mathbf {b\times c} )=\varepsilon _{ijk}a^{i}b^{j}c^{k}.}"></span></dd></dl> <p>From this expression, it can be seen that the triple scalar product is antisymmetric when exchanging any pair of arguments. For example, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} \cdot (\mathbf {b\times c} )=-\mathbf {b} \cdot (\mathbf {a\times c} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> <mo>×</mo> <mi mathvariant="bold">c</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> <mo>×</mo> <mi mathvariant="bold">c</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} \cdot (\mathbf {b\times c} )=-\mathbf {b} \cdot (\mathbf {a\times c} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c51a7d073c282de6715a15845434e9212fcc8b5d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.51ex; height:2.843ex;" alt="{\displaystyle \mathbf {a} \cdot (\mathbf {b\times c} )=-\mathbf {b} \cdot (\mathbf {a\times c} )}"></span>.</dd></dl> <div class="mw-heading mw-heading4"><h4 id="Curl_(one_vector_field)"><span id="Curl_.28one_vector_field.29"></span>Curl (one vector field)</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Levi-Civita_symbol&action=edit&section=20" title="Edit section: Curl (one vector field)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If <span class="texhtml"><b>F</b> = (<i>F</i><sup>1</sup>, <i>F</i><sup>2</sup>, <i>F</i><sup>3</sup>)</span> is a vector field defined on some <a href="/wiki/Open_set" title="Open set">open set</a> of <span class="texhtml"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}"></span></span> as a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> of <a href="/wiki/Position_vector" class="mw-redirect" title="Position vector">position</a> <span class="texhtml"><b>x</b> = (<i>x</i><sup>1</sup>, <i>x</i><sup>2</sup>, <i>x</i><sup>3</sup>)</span> (using <a href="/wiki/Cartesian_coordinates" class="mw-redirect" title="Cartesian coordinates">Cartesian coordinates</a>). Then the <span class="texhtml mvar" style="font-style:italic;">i</span>th component of the <a href="/wiki/Curl_(mathematics)" title="Curl (mathematics)">curl</a> of <span class="texhtml"><b>F</b></span> equals<sup id="cite_ref-Kay_4-3" class="reference"><a href="#cite_note-Kay-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\nabla \times \mathbf {F} )^{i}(\mathbf {x} )=\varepsilon _{ijk}{\frac {\partial }{\partial x^{j}}}F^{k}(\mathbf {x} ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi mathvariant="normal">∇</mi> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> <mi>k</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\nabla \times \mathbf {F} )^{i}(\mathbf {x} )=\varepsilon _{ijk}{\frac {\partial }{\partial x^{j}}}F^{k}(\mathbf {x} ),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91b4a87f13cc5c475f19792c5daa72103a43b5e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:29.968ex; height:5.676ex;" alt="{\displaystyle (\nabla \times \mathbf {F} )^{i}(\mathbf {x} )=\varepsilon _{ijk}{\frac {\partial }{\partial x^{j}}}F^{k}(\mathbf {x} ),}"></span></dd></dl> <p>which follows from the cross product expression above, substituting components of the <a href="/wiki/Gradient" title="Gradient">gradient</a> vector <a href="/wiki/Linear_operator" class="mw-redirect" title="Linear operator">operator</a> (nabla). </p> <div class="mw-heading mw-heading2"><h2 id="Tensor_density">Tensor density</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Levi-Civita_symbol&action=edit&section=21" title="Edit section: Tensor density"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In any arbitrary <a href="/wiki/Curvilinear_coordinate_system" class="mw-redirect" title="Curvilinear coordinate system">curvilinear coordinate system</a> and even in the absence of a <a href="/wiki/Metric_tensor" title="Metric tensor">metric</a> on the <a href="/wiki/Manifold" title="Manifold">manifold</a>, the Levi-Civita symbol as defined above may be considered to be a <a href="/wiki/Tensor_density" title="Tensor density">tensor density</a> field in two different ways. It may be regarded as a <a href="/wiki/Covariance_and_contravariance_of_vectors" title="Covariance and contravariance of vectors">contravariant</a> tensor density of weight +1 or as a covariant tensor density of weight −1. In <i>n</i> dimensions using the generalized Kronecker delta,<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\varepsilon ^{\mu _{1}\dots \mu _{n}}&=\delta _{\,1\,\dots \,n}^{\mu _{1}\dots \mu _{n}}\,\\\varepsilon _{\nu _{1}\dots \nu _{n}}&=\delta _{\nu _{1}\dots \nu _{n}}^{\,1\,\dots \,n}\,.\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> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mspace width="thinmathspace" /> <mn>1</mn> <mspace width="thinmathspace" /> <mo>…</mo> <mspace width="thinmathspace" /> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msubsup> <mspace width="thinmathspace" /> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mspace width="thinmathspace" /> <mn>1</mn> <mspace width="thinmathspace" /> <mo>…</mo> <mspace width="thinmathspace" /> <mi>n</mi> </mrow> </msubsup> <mspace width="thinmathspace" /> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\varepsilon ^{\mu _{1}\dots \mu _{n}}&=\delta _{\,1\,\dots \,n}^{\mu _{1}\dots \mu _{n}}\,\\\varepsilon _{\nu _{1}\dots \nu _{n}}&=\delta _{\nu _{1}\dots \nu _{n}}^{\,1\,\dots \,n}\,.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31def93af1b3981aef59ed1713797d2e3b5f92b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:18.515ex; height:6.509ex;" alt="{\displaystyle {\begin{aligned}\varepsilon ^{\mu _{1}\dots \mu _{n}}&=\delta _{\,1\,\dots \,n}^{\mu _{1}\dots \mu _{n}}\,\\\varepsilon _{\nu _{1}\dots \nu _{n}}&=\delta _{\nu _{1}\dots \nu _{n}}^{\,1\,\dots \,n}\,.\end{aligned}}}"></span></dd></dl> <p>Notice that these are numerically identical. In particular, the sign is the same. </p> <div class="mw-heading mw-heading2"><h2 id="Levi-Civita_tensors">Levi-Civita tensors</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Levi-Civita_symbol&action=edit&section=22" title="Edit section: Levi-Civita tensors"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>On a <a href="/wiki/Pseudo-Riemannian_manifold" title="Pseudo-Riemannian manifold">pseudo-Riemannian manifold</a>, one may define a coordinate-invariant covariant tensor field whose coordinate representation agrees with the Levi-Civita symbol wherever the coordinate system is such that the basis of the tangent space is orthonormal with respect to the metric and matches a selected orientation. This tensor should not be confused with the tensor density field mentioned above. The presentation in this section closely follows <a href="#CITEREFCarroll2004">Carroll 2004</a>. </p><p>The covariant Levi-Civita tensor (also known as the <a href="/wiki/Riemannian_volume_form" class="mw-redirect" title="Riemannian volume form">Riemannian volume form</a>) in any coordinate system that matches the selected orientation is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{a_{1}\dots a_{n}}={\sqrt {\left|\det[g_{ab}]\right|}}\,\varepsilon _{a_{1}\dots a_{n}}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow> <mo>|</mo> <mrow> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">[</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mrow> <mo>|</mo> </mrow> </msqrt> </mrow> <mspace width="thinmathspace" /> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msub> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{a_{1}\dots a_{n}}={\sqrt {\left|\det[g_{ab}]\right|}}\,\varepsilon _{a_{1}\dots a_{n}}\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37e9a92e440539e0513ae569542bc390a4f154c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:29.762ex; height:4.843ex;" alt="{\displaystyle E_{a_{1}\dots a_{n}}={\sqrt {\left|\det[g_{ab}]\right|}}\,\varepsilon _{a_{1}\dots a_{n}}\,,}"></span></dd></dl> <p>where <span class="texhtml mvar" style="font-style:italic;">g<sub>ab</sub></span> is the representation of the metric in that coordinate system. We can similarly consider a contravariant Levi-Civita tensor by raising the indices with the metric as usual, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E^{a_{1}\dots a_{n}}=E_{b_{1}\dots b_{n}}\prod _{i=1}^{n}g^{a_{i}b_{i}}={\sqrt {\left|\det[g_{ab}]\right|}}\,\varepsilon ^{a_{1}\dots a_{n}}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msup> <mo>=</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msub> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow> <mo>|</mo> <mrow> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">[</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mrow> <mo>|</mo> </mrow> </msqrt> </mrow> <mspace width="thinmathspace" /> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msup> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E^{a_{1}\dots a_{n}}=E_{b_{1}\dots b_{n}}\prod _{i=1}^{n}g^{a_{i}b_{i}}={\sqrt {\left|\det[g_{ab}]\right|}}\,\varepsilon ^{a_{1}\dots a_{n}}\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/440658cfde9b69c8e20623e60d83ffeffe9ccdd1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:47.939ex; height:6.843ex;" alt="{\displaystyle E^{a_{1}\dots a_{n}}=E_{b_{1}\dots b_{n}}\prod _{i=1}^{n}g^{a_{i}b_{i}}={\sqrt {\left|\det[g_{ab}]\right|}}\,\varepsilon ^{a_{1}\dots a_{n}}\,,}"></span></dd></dl> <p>but notice that if the <a href="/wiki/Metric_signature" title="Metric signature">metric signature</a> contains an odd number of negative eigenvalues <span class="texhtml"><i>q</i></span>, then the sign of the components of this tensor differ from the standard Levi-Civita symbol:<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E^{a_{1}\dots a_{n}}={\frac {\operatorname {sgn} \left(\det[g_{ab}]\right)}{\sqrt {\left|\det[g_{ab}]\right|}}}\,\varepsilon _{a_{1}\dots a_{n}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sgn</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">[</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mrow> <mo>)</mo> </mrow> </mrow> <msqrt> <mrow> <mo>|</mo> <mrow> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">[</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mrow> <mo>|</mo> </mrow> </msqrt> </mfrac> </mrow> <mspace width="thinmathspace" /> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E^{a_{1}\dots a_{n}}={\frac {\operatorname {sgn} \left(\det[g_{ab}]\right)}{\sqrt {\left|\det[g_{ab}]\right|}}}\,\varepsilon _{a_{1}\dots a_{n}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/844383e967cf0362f0e53708c63d60705dc4a0e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:31.853ex; height:7.009ex;" alt="{\displaystyle E^{a_{1}\dots a_{n}}={\frac {\operatorname {sgn} \left(\det[g_{ab}]\right)}{\sqrt {\left|\det[g_{ab}]\right|}}}\,\varepsilon _{a_{1}\dots a_{n}},}"></span></dd></dl> <p>where <span class="texhtml">sgn(det[g<sub><i>ab</i></sub>]) = (−1)<sup><i>q</i></sup></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{a_{1}\dots a_{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{a_{1}\dots a_{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/760a1af7c45078ae3218199d2f7abee8f7e63baf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.777ex; height:2.343ex;" alt="{\displaystyle \varepsilon _{a_{1}\dots a_{n}}}"></span> is the usual Levi-Civita symbol discussed in the rest of this article, and we used the definition of the metric <a href="/wiki/Determinant" title="Determinant">determinant</a> in the derivation. More explicitly, when the tensor and basis orientation are chosen such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle E_{01\dots n}=+{\sqrt {\left|\det[g_{ab}]\right|}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>01</mn> <mo>…</mo> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow> <mo>|</mo> <mrow> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">[</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mrow> <mo>|</mo> </mrow> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle E_{01\dots n}=+{\sqrt {\left|\det[g_{ab}]\right|}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17ace6282f43d49b5451560f7ac626e66a7bc64e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.467ex; height:3.343ex;" alt="{\textstyle E_{01\dots n}=+{\sqrt {\left|\det[g_{ab}]\right|}}}"></span>, we have that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E^{01\dots n}={\frac {\operatorname {sgn}(\det[g_{ab}])}{\sqrt {\left|\det[g_{ab}]\right|}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>01</mn> <mo>…</mo> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sgn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">[</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> </mrow> <msqrt> <mrow> <mo>|</mo> <mrow> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">[</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mrow> <mo>|</mo> </mrow> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E^{01\dots n}={\frac {\operatorname {sgn}(\det[g_{ab}])}{\sqrt {\left|\det[g_{ab}]\right|}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6b580c03f614708b26f6c3723071527fd82f130" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:23.137ex; height:7.009ex;" alt="{\displaystyle E^{01\dots n}={\frac {\operatorname {sgn}(\det[g_{ab}])}{\sqrt {\left|\det[g_{ab}]\right|}}}}"></span>. </p><p>From this we can infer the identity, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E^{\mu _{1}\dots \mu _{p}\alpha _{1}\dots \alpha _{n-p}}E_{\mu _{1}\dots \mu _{p}\beta _{1}\dots \beta _{n-p}}=(-1)^{q}p!\delta _{\beta _{1}\dots \beta _{n-p}}^{\alpha _{1}\dots \alpha _{n-p}}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>p</mi> </mrow> </msub> </mrow> </msup> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>p</mi> </mrow> </msub> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msup> <mi>p</mi> <mo>!</mo> <msubsup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>p</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>p</mi> </mrow> </msub> </mrow> </msubsup> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E^{\mu _{1}\dots \mu _{p}\alpha _{1}\dots \alpha _{n-p}}E_{\mu _{1}\dots \mu _{p}\beta _{1}\dots \beta _{n-p}}=(-1)^{q}p!\delta _{\beta _{1}\dots \beta _{n-p}}^{\alpha _{1}\dots \alpha _{n-p}}\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94b6c47b7e11fc7bf705827e38e10d07cf83380a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:50.489ex; height:4.009ex;" alt="{\displaystyle E^{\mu _{1}\dots \mu _{p}\alpha _{1}\dots \alpha _{n-p}}E_{\mu _{1}\dots \mu _{p}\beta _{1}\dots \beta _{n-p}}=(-1)^{q}p!\delta _{\beta _{1}\dots \beta _{n-p}}^{\alpha _{1}\dots \alpha _{n-p}}\,,}"></span></dd></dl> <p>where </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{\beta _{1}\dots \beta _{n-p}}^{\alpha _{1}\dots \alpha _{n-p}}=(n-p)!\delta _{\beta _{1}}^{\lbrack \alpha _{1}}\dots \delta _{\beta _{n-p}}^{\alpha _{n-p}\rbrack }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>p</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>p</mi> </mrow> </msub> </mrow> </msubsup> <mo>=</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>!</mo> <msubsup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">[</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msubsup> <mo>…</mo> <msubsup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>p</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>p</mi> </mrow> </msub> <mo fence="false" stretchy="false">]</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{\beta _{1}\dots \beta _{n-p}}^{\alpha _{1}\dots \alpha _{n-p}}=(n-p)!\delta _{\beta _{1}}^{\lbrack \alpha _{1}}\dots \delta _{\beta _{n-p}}^{\alpha _{n-p}\rbrack }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca9ee2d97ed7e446a894fedeeb47897413691cde" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:32.374ex; height:4.509ex;" alt="{\displaystyle \delta _{\beta _{1}\dots \beta _{n-p}}^{\alpha _{1}\dots \alpha _{n-p}}=(n-p)!\delta _{\beta _{1}}^{\lbrack \alpha _{1}}\dots \delta _{\beta _{n-p}}^{\alpha _{n-p}\rbrack }}"></span></dd></dl> <p>is the generalized Kronecker delta. </p> <div class="mw-heading mw-heading3"><h3 id="Example:_Minkowski_space">Example: Minkowski space</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Levi-Civita_symbol&action=edit&section=23" title="Edit section: Example: Minkowski space"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In Minkowski space (the four-dimensional <a href="/wiki/Spacetime" title="Spacetime">spacetime</a> of <a href="/wiki/Special_relativity" title="Special relativity">special relativity</a>), the covariant Levi-Civita tensor is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{\alpha \beta \gamma \delta }=\pm {\sqrt {\left|\det[g_{\mu \nu }]\right|}}\,\varepsilon _{\alpha \beta \gamma \delta }\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> <mi>γ</mi> <mi>δ</mi> </mrow> </msub> <mo>=</mo> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow> <mo>|</mo> <mrow> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">[</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mrow> <mo>|</mo> </mrow> </msqrt> </mrow> <mspace width="thinmathspace" /> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> <mi>γ</mi> <mi>δ</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{\alpha \beta \gamma \delta }=\pm {\sqrt {\left|\det[g_{\mu \nu }]\right|}}\,\varepsilon _{\alpha \beta \gamma \delta }\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69cea2c0f47be8b11eed0db78ca4d4cb4cf5230d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:28.191ex; height:4.843ex;" alt="{\displaystyle E_{\alpha \beta \gamma \delta }=\pm {\sqrt {\left|\det[g_{\mu \nu }]\right|}}\,\varepsilon _{\alpha \beta \gamma \delta }\,,}"></span></dd></dl> <p>where the sign depends on the orientation of the basis. The contravariant Levi-Civita tensor is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E^{\alpha \beta \gamma \delta }=g^{\alpha \zeta }g^{\beta \eta }g^{\gamma \theta }g^{\delta \iota }E_{\zeta \eta \theta \iota }\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> <mi>γ</mi> <mi>δ</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>ζ</mi> </mrow> </msup> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> <mi>η</mi> </mrow> </msup> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> <mi>θ</mi> </mrow> </msup> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> <mi>ι</mi> </mrow> </msup> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ζ</mi> <mi>η</mi> <mi>θ</mi> <mi>ι</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E^{\alpha \beta \gamma \delta }=g^{\alpha \zeta }g^{\beta \eta }g^{\gamma \theta }g^{\delta \iota }E_{\zeta \eta \theta \iota }\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b56d190e0afba6ce5313d59b47b0985e2cb3ba79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.673ex; height:3.343ex;" alt="{\displaystyle E^{\alpha \beta \gamma \delta }=g^{\alpha \zeta }g^{\beta \eta }g^{\gamma \theta }g^{\delta \iota }E_{\zeta \eta \theta \iota }\,.}"></span></dd></dl> <p>The following are examples of the general identity above specialized to Minkowski space (with the negative sign arising from the odd number of negatives in the signature of the metric tensor in either sign convention): </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}E_{\alpha \beta \gamma \delta }E_{\rho \sigma \mu \nu }&=-g_{\alpha \zeta }g_{\beta \eta }g_{\gamma \theta }g_{\delta \iota }\delta _{\rho \sigma \mu \nu }^{\zeta \eta \theta \iota }\\E^{\alpha \beta \gamma \delta }E^{\rho \sigma \mu \nu }&=-g^{\alpha \zeta }g^{\beta \eta }g^{\gamma \theta }g^{\delta \iota }\delta _{\zeta \eta \theta \iota }^{\rho \sigma \mu \nu }\\E^{\alpha \beta \gamma \delta }E_{\alpha \beta \gamma \delta }&=-24\\E^{\alpha \beta \gamma \delta }E_{\rho \beta \gamma \delta }&=-6\delta _{\rho }^{\alpha }\\E^{\alpha \beta \gamma \delta }E_{\rho \sigma \gamma \delta }&=-2\delta _{\rho \sigma }^{\alpha \beta }\\E^{\alpha \beta \gamma \delta }E_{\rho \sigma \theta \delta }&=-\delta _{\rho \sigma \theta }^{\alpha \beta \gamma }\,.\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>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> <mi>γ</mi> <mi>δ</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> <mi>σ</mi> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>ζ</mi> </mrow> </msub> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> <mi>η</mi> </mrow> </msub> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> <mi>θ</mi> </mrow> </msub> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> <mi>ι</mi> </mrow> </msub> <msubsup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> <mi>σ</mi> <mi>μ</mi> <mi>ν</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ζ</mi> <mi>η</mi> <mi>θ</mi> <mi>ι</mi> </mrow> </msubsup> </mtd> </mtr> <mtr> <mtd> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> <mi>γ</mi> <mi>δ</mi> </mrow> </msup> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> <mi>σ</mi> <mi>μ</mi> <mi>ν</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>ζ</mi> </mrow> </msup> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> <mi>η</mi> </mrow> </msup> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> <mi>θ</mi> </mrow> </msup> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>δ</mi> <mi>ι</mi> </mrow> </msup> <msubsup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ζ</mi> <mi>η</mi> <mi>θ</mi> <mi>ι</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> <mi>σ</mi> <mi>μ</mi> <mi>ν</mi> </mrow> </msubsup> </mtd> </mtr> <mtr> <mtd> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> <mi>γ</mi> <mi>δ</mi> </mrow> </msup> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> <mi>γ</mi> <mi>δ</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mn>24</mn> </mtd> </mtr> <mtr> <mtd> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> <mi>γ</mi> <mi>δ</mi> </mrow> </msup> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> <mi>β</mi> <mi>γ</mi> <mi>δ</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mn>6</mn> <msubsup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msubsup> </mtd> </mtr> <mtr> <mtd> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> <mi>γ</mi> <mi>δ</mi> </mrow> </msup> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> <mi>σ</mi> <mi>γ</mi> <mi>δ</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mn>2</mn> <msubsup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> <mi>σ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> </mrow> </msubsup> </mtd> </mtr> <mtr> <mtd> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> <mi>γ</mi> <mi>δ</mi> </mrow> </msup> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> <mi>σ</mi> <mi>θ</mi> <mi>δ</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <msubsup> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> <mi>σ</mi> <mi>θ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>β</mi> <mi>γ</mi> </mrow> </msubsup> <mspace width="thinmathspace" /> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}E_{\alpha \beta \gamma \delta }E_{\rho \sigma \mu \nu }&=-g_{\alpha \zeta }g_{\beta \eta }g_{\gamma \theta }g_{\delta \iota }\delta _{\rho \sigma \mu \nu }^{\zeta \eta \theta \iota }\\E^{\alpha \beta \gamma \delta }E^{\rho \sigma \mu \nu }&=-g^{\alpha \zeta }g^{\beta \eta }g^{\gamma \theta }g^{\delta \iota }\delta _{\zeta \eta \theta \iota }^{\rho \sigma \mu \nu }\\E^{\alpha \beta \gamma \delta }E_{\alpha \beta \gamma \delta }&=-24\\E^{\alpha \beta \gamma \delta }E_{\rho \beta \gamma \delta }&=-6\delta _{\rho }^{\alpha }\\E^{\alpha \beta \gamma \delta }E_{\rho \sigma \gamma \delta }&=-2\delta _{\rho \sigma }^{\alpha \beta }\\E^{\alpha \beta \gamma \delta }E_{\rho \sigma \theta \delta }&=-\delta _{\rho \sigma \theta }^{\alpha \beta \gamma }\,.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/007860ac0ae48711cb274321e1728c9b9ddee770" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.671ex; width:33.913ex; height:22.509ex;" alt="{\displaystyle {\begin{aligned}E_{\alpha \beta \gamma \delta }E_{\rho \sigma \mu \nu }&=-g_{\alpha \zeta }g_{\beta \eta }g_{\gamma \theta }g_{\delta \iota }\delta _{\rho \sigma \mu \nu }^{\zeta \eta \theta \iota }\\E^{\alpha \beta \gamma \delta }E^{\rho \sigma \mu \nu }&=-g^{\alpha \zeta }g^{\beta \eta }g^{\gamma \theta }g^{\delta \iota }\delta _{\zeta \eta \theta \iota }^{\rho \sigma \mu \nu }\\E^{\alpha \beta \gamma \delta }E_{\alpha \beta \gamma \delta }&=-24\\E^{\alpha \beta \gamma \delta }E_{\rho \beta \gamma \delta }&=-6\delta _{\rho }^{\alpha }\\E^{\alpha \beta \gamma \delta }E_{\rho \sigma \gamma \delta }&=-2\delta _{\rho \sigma }^{\alpha \beta }\\E^{\alpha \beta \gamma \delta }E_{\rho \sigma \theta \delta }&=-\delta _{\rho \sigma \theta }^{\alpha \beta \gamma }\,.\end{aligned}}}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Levi-Civita_symbol&action=edit&section=24" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/List_of_permutation_topics" title="List of permutation topics">List of permutation topics</a></li> <li><a href="/wiki/Symmetric_tensor" title="Symmetric tensor">Symmetric tensor</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Levi-Civita_symbol&action=edit&section=25" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><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 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(2010). <i>Supersymmetry</i>. Demystified. McGraw-Hill. pp. 57–58. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-163641-4" title="Special:BookSources/978-0-07-163641-4"><bdi>978-0-07-163641-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Supersymmetry&rft.series=Demystified&rft.pages=57-58&rft.pub=McGraw-Hill&rft.date=2010&rft.isbn=978-0-07-163641-4&rft.aulast=Labelle&rft.aufirst=P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALevi-Civita+symbol" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHadrovich" class="citation web cs1">Hadrovich, F. <a rel="nofollow" class="external text" href="http://users.ox.ac.uk/~tweb/00004/index.shtml">"Twistor Primer"</a><span class="reference-accessdate">. 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Monthly</i>, <b>32</b> (5): 233–241, <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%2F2299191">10.2307/2299191</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/2299191">2299191</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Amer.+Math.+Monthly&rft.atitle=The+generalized+Kronecker+symbol+and+its+application+to+the+theory+of+determinants&rft.volume=32&rft.issue=5&rft.pages=233-241&rft.date=1925&rft_id=info%3Adoi%2F10.2307%2F2299191&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2299191%23id-name%3DJSTOR&rft.aulast=Murnaghan&rft.aufirst=F.+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALevi-Civita+symbol" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLovelockRund1989" class="citation book cs1">Lovelock, David; Rund, Hanno (1989). <i>Tensors, Differential Forms, and Variational Principles</i>. Courier Dover Publications. p. 113. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-65840-6" title="Special:BookSources/0-486-65840-6"><bdi>0-486-65840-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Tensors%2C+Differential+Forms%2C+and+Variational+Principles&rft.pages=113&rft.pub=Courier+Dover+Publications&rft.date=1989&rft.isbn=0-486-65840-6&rft.aulast=Lovelock&rft.aufirst=David&rft.au=Rund%2C+Hanno&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALevi-Civita+symbol" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNakahara2017" class="citation book cs1">Nakahara, Mikio (2017-01-31). <a rel="nofollow" class="external text" href="https://www.taylorfrancis.com/books/mono/10.1201/9781315275826/geometry-topology-physics-mikio-nakahara"><i>Geometry, Topology and Physics</i></a> (2 ed.). Boca Raton: CRC Press. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1201%2F9781315275826">10.1201/9781315275826</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-315-27582-6" title="Special:BookSources/978-1-315-27582-6"><bdi>978-1-315-27582-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Geometry%2C+Topology+and+Physics&rft.place=Boca+Raton&rft.edition=2&rft.pub=CRC+Press&rft.date=2017-01-31&rft_id=info%3Adoi%2F10.1201%2F9781315275826&rft.isbn=978-1-315-27582-6&rft.aulast=Nakahara&rft.aufirst=Mikio&rft_id=https%3A%2F%2Fwww.taylorfrancis.com%2Fbooks%2Fmono%2F10.1201%2F9781315275826%2Fgeometry-topology-physics-mikio-nakahara&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALevi-Civita+symbol" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Levi-Civita_symbol&action=edit&section=26" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMisnerThorneWheeler1973" class="citation book cs1">Misner, C.; Thorne, K. S.; Wheeler, J. A. (1973). <i><a href="/wiki/Gravitation_(book)" title="Gravitation (book)">Gravitation</a></i>. W. H. Freeman & Co. pp. 85–86, §3.5. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7167-0344-0" title="Special:BookSources/0-7167-0344-0"><bdi>0-7167-0344-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Gravitation&rft.pages=85-86%2C+%C2%A73.5&rft.pub=W.+H.+Freeman+%26+Co&rft.date=1973&rft.isbn=0-7167-0344-0&rft.aulast=Misner&rft.aufirst=C.&rft.au=Thorne%2C+K.+S.&rft.au=Wheeler%2C+J.+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALevi-Civita+symbol" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNeuenschwander2015" class="citation book cs1">Neuenschwander, D. E. (2015). <i>Tensor Calculus for Physics</i>. Johns Hopkins University Press. pp. 11, 29, 95. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4214-1565-9" title="Special:BookSources/978-1-4214-1565-9"><bdi>978-1-4214-1565-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Tensor+Calculus+for+Physics&rft.pages=11%2C+29%2C+95&rft.pub=Johns+Hopkins+University+Press&rft.date=2015&rft.isbn=978-1-4214-1565-9&rft.aulast=Neuenschwander&rft.aufirst=D.+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALevi-Civita+symbol" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCarroll2004" class="citation cs2"><a href="/wiki/Sean_M._Carroll" title="Sean M. Carroll">Carroll, Sean M.</a> (2004), <a rel="nofollow" class="external text" href="https://www.preposterousuniverse.com/spacetimeandgeometry/"><i>Spacetime and Geometry</i></a>, Addison-Wesley, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8053-8732-3" title="Special:BookSources/0-8053-8732-3"><bdi>0-8053-8732-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Spacetime+and+Geometry&rft.pub=Addison-Wesley&rft.date=2004&rft.isbn=0-8053-8732-3&rft.aulast=Carroll&rft.aufirst=Sean+M.&rft_id=https%3A%2F%2Fwww.preposterousuniverse.com%2Fspacetimeandgeometry%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALevi-Civita+symbol" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Levi-Civita_symbol&action=edit&section=27" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><i>This article incorporates material from <a rel="nofollow" class="external text" href="https://planetmath.org/levicivitapermutationsymbol">Levi-Civita permutation symbol</a> on <a href="/wiki/PlanetMath" title="PlanetMath">PlanetMath</a>, which is licensed under the <a href="/wiki/Wikipedia:CC-BY-SA" class="mw-redirect" title="Wikipedia:CC-BY-SA">Creative Commons Attribution/Share-Alike License</a>.</i> </p> <ul><li><span class="citation mathworld" id="Reference-Mathworld-Permutation_Tensor"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/PermutationTensor.html">"Permutation Tensor"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Permutation+Tensor&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FPermutationTensor.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALevi-Civita+symbol" class="Z3988"></span></span></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 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scope="row" class="navbox-group" style="width:1%">Tensor<br />definitions</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/Tensor_(intrinsic_definition)" title="Tensor (intrinsic definition)">Tensor (intrinsic definition)</a></li> <li><a href="/wiki/Tensor_field" title="Tensor field">Tensor field</a></li> <li><a href="/wiki/Tensor_density" title="Tensor density">Tensor density</a></li> <li><a href="/wiki/Tensors_in_curvilinear_coordinates" title="Tensors in curvilinear coordinates">Tensors in curvilinear coordinates</a></li> <li><a href="/wiki/Mixed_tensor" title="Mixed tensor">Mixed tensor</a></li> <li><a href="/wiki/Antisymmetric_tensor" title="Antisymmetric tensor">Antisymmetric tensor</a></li> <li><a href="/wiki/Symmetric_tensor" title="Symmetric tensor">Symmetric tensor</a></li> <li><a href="/wiki/Tensor_operator" title="Tensor operator">Tensor operator</a></li> <li><a href="/wiki/Tensor_bundle" title="Tensor bundle">Tensor bundle</a></li> <li><a href="/wiki/Two-point_tensor" title="Two-point tensor">Two-point tensor</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Operation_(mathematics)" title="Operation (mathematics)">Operations</a></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 href="/wiki/Covariant_derivative" title="Covariant derivative">Covariant derivative</a></li> <li><a href="/wiki/Exterior_covariant_derivative" title="Exterior covariant derivative">Exterior covariant derivative</a></li> <li><a href="/wiki/Exterior_derivative" title="Exterior derivative">Exterior derivative</a></li> <li><a href="/wiki/Exterior_product" class="mw-redirect" title="Exterior product">Exterior product</a></li> <li><a href="/wiki/Hodge_star_operator" title="Hodge star operator">Hodge star operator</a></li> <li><a href="/wiki/Lie_derivative" title="Lie derivative">Lie derivative</a></li> <li><a href="/wiki/Raising_and_lowering_indices" title="Raising and lowering indices">Raising and lowering indices</a></li> <li><a href="/wiki/Symmetrization" title="Symmetrization">Symmetrization</a></li> <li><a href="/wiki/Tensor_contraction" title="Tensor contraction">Tensor contraction</a></li> <li><a href="/wiki/Tensor_product" title="Tensor product">Tensor product</a></li> <li><a href="/wiki/Transpose" title="Transpose">Transpose</a> (2nd-order tensors)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related<br />abstractions</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/Affine_connection" title="Affine connection">Affine connection</a></li> <li><a href="/wiki/Basis_(linear_algebra)" title="Basis (linear algebra)">Basis</a></li> <li><a href="/wiki/Cartan_formalism_(physics)" class="mw-redirect" title="Cartan formalism (physics)">Cartan formalism (physics)</a></li> <li><a href="/wiki/Connection_form" title="Connection form">Connection form</a></li> <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/Differential_form" title="Differential form">Differential form</a></li> <li><a href="/wiki/Dimension" title="Dimension">Dimension</a></li> <li><a href="/wiki/Exterior_form" class="mw-redirect" title="Exterior form">Exterior form</a></li> <li><a href="/wiki/Fiber_bundle" title="Fiber bundle">Fiber bundle</a></li> <li><a href="/wiki/Geodesic" title="Geodesic">Geodesic</a></li> <li><a href="/wiki/Levi-Civita_connection" title="Levi-Civita connection">Levi-Civita connection</a></li> <li><a href="/wiki/Linear_map" title="Linear map">Linear map</a></li> <li><a href="/wiki/Manifold" title="Manifold">Manifold</a></li> <li><a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">Matrix</a></li> <li><a href="/wiki/Multivector" title="Multivector">Multivector</a></li> <li><a href="/wiki/Pseudotensor" title="Pseudotensor">Pseudotensor</a></li> <li><a href="/wiki/Spinor" title="Spinor">Spinor</a></li> <li><a href="/wiki/Vector_(mathematics_and_physics)" title="Vector (mathematics and physics)">Vector</a></li> <li><a href="/wiki/Vector_space" title="Vector space">Vector space</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Notable tensors</th><td class="navbox-list-with-group navbox-list navbox-odd" 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:1%;font-weight:normal;">Mathematics</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 href="/wiki/Kronecker_delta" title="Kronecker delta">Kronecker delta</a></li> <li><a class="mw-selflink selflink">Levi-Civita symbol</a></li> <li><a href="/wiki/Metric_tensor" title="Metric tensor">Metric tensor</a></li> <li><a href="/wiki/Nonmetricity_tensor" title="Nonmetricity tensor">Nonmetricity tensor</a></li> <li><a href="/wiki/Ricci_curvature" title="Ricci curvature">Ricci curvature</a></li> <li><a href="/wiki/Riemann_curvature_tensor" title="Riemann curvature tensor">Riemann curvature tensor</a></li> <li><a href="/wiki/Torsion_tensor" title="Torsion tensor">Torsion tensor</a></li> <li><a href="/wiki/Weyl_tensor" title="Weyl tensor">Weyl tensor</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Physics</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/Moment_of_inertia#Inertia_tensor" title="Moment of inertia">Moment of inertia</a></li> <li><a href="/wiki/Angular_momentum#Angular_momentum_in_relativistic_mechanics" title="Angular momentum">Angular momentum tensor</a></li> <li><a href="/wiki/Spin_tensor" title="Spin tensor">Spin tensor</a></li> <li><a href="/wiki/Cauchy_stress_tensor" title="Cauchy stress tensor">Cauchy stress tensor</a></li> <li><a href="/wiki/Stress%E2%80%93energy_tensor" title="Stress–energy tensor">stress–energy tensor</a></li> <li><a href="/wiki/Einstein_tensor" title="Einstein tensor">Einstein tensor</a></li> <li><a href="/wiki/Electromagnetic_tensor" title="Electromagnetic tensor">EM tensor</a></li> <li><a href="/wiki/Gluon_field_strength_tensor" title="Gluon field strength tensor">Gluon field strength tensor</a></li> <li><a href="/wiki/Metric_tensor_(general_relativity)" title="Metric tensor (general relativity)">Metric tensor (GR)</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Mathematician" title="Mathematician">Mathematicians</a></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 href="/wiki/%C3%89lie_Cartan" title="Élie Cartan">Élie Cartan</a></li> <li><a href="/wiki/Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Augustin-Louis Cauchy</a></li> <li><a href="/wiki/Elwin_Bruno_Christoffel" title="Elwin Bruno Christoffel">Elwin Bruno Christoffel</a></li> <li><a href="/wiki/Albert_Einstein" title="Albert Einstein">Albert Einstein</a></li> <li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a></li> <li><a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a></li> <li><a href="/wiki/Hermann_Grassmann" title="Hermann Grassmann">Hermann Grassmann</a></li> <li><a href="/wiki/Tullio_Levi-Civita" title="Tullio Levi-Civita">Tullio Levi-Civita</a></li> <li><a href="/wiki/Gregorio_Ricci-Curbastro" 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Levi-Civita symbol - Wikipedia