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data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">(Top)</div> </a> </li> <li id="toc-Fréchet_derivative" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Fréchet_derivative"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Fréchet derivative</span> </div> </a> <ul id="toc-Fréchet_derivative-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Exterior_derivative_and_Lie_derivative" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Exterior_derivative_and_Lie_derivative"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Exterior derivative and Lie derivative</span> </div> </a> <ul id="toc-Exterior_derivative_and_Lie_derivative-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Differential_topology" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Differential_topology"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Differential topology</span> </div> </a> <ul id="toc-Differential_topology-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Covariant_derivative" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Covariant_derivative"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Covariant derivative</span> </div> </a> <ul id="toc-Covariant_derivative-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Weak_derivatives" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Weak_derivatives"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Weak derivatives</span> </div> </a> <ul id="toc-Weak_derivatives-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Higher-order_and_fractional_derivatives" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Higher-order_and_fractional_derivatives"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Higher-order and fractional derivatives</span> </div> </a> <ul id="toc-Higher-order_and_fractional_derivatives-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Quaternionic_derivatives" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Quaternionic_derivatives"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Quaternionic derivatives</span> </div> </a> <ul id="toc-Quaternionic_derivatives-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Difference_operator,_q-analogues_and_time_scales" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Difference_operator,_q-analogues_and_time_scales"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Difference operator, q-analogues and time scales</span> </div> </a> <ul id="toc-Difference_operator,_q-analogues_and_time_scales-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Derivatives_in_algebra" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Derivatives_in_algebra"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Derivatives in algebra</span> </div> </a> <button 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id="toc-Differential_operators" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Differential_operators"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Differential operators</span> </div> </a> <ul id="toc-Differential_operators-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_generalizations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Other_generalizations"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Other generalizations</span> </div> </a> <ul id="toc-Other_generalizations-sublist" class="vector-toc-list"> </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">12</span> <span>See also</span> </div> </a> <ul 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class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%92%D1%8B%D1%82%D0%B2%D0%BE%D1%80%D0%BD%D0%B0%D1%8F_(%D0%BC%D0%B0%D1%82%D1%8D%D0%BC%D0%B0%D1%82%D1%8B%D0%BA%D0%B0)" title="Вытворная (матэматыка) – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Вытворная (матэматыка)" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A2%C4%83%D1%85%C4%83%D0%BC_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Тăхăм (математика) – Chuvash" lang="cv" hreflang="cv" data-title="Тăхăм (математика)" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Generalizzazioni_della_derivata" title="Generalizzazioni della derivata – Italian" lang="it" hreflang="it" data-title="Generalizzazioni della derivata" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A2%D1%83%D1%8B%D0%BD%D0%B4%D1%8B" title="Туынды – Kazakh" lang="kk" hreflang="kk" data-title="Туынды" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%BE%AE%E5%88%86%E3%81%AE%E4%B8%80%E8%88%AC%E5%8C%96" title="微分の一般化 – Japanese" lang="ja" hreflang="ja" data-title="微分の一般化" 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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">Fundamental construction of differential calculus</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">This article is about the term as used in mathematics. 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rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist"><tbody><tr><td class="sidebar-pretitle">Part of a series of articles about</td></tr><tr><th class="sidebar-title-with-pretitle" style="padding-bottom:0.25em;"><a href="/wiki/Calculus" title="Calculus">Calculus</a></th></tr><tr><td class="sidebar-image"><big><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 \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17d063dc86a53a2efb1fe86f4a5d47d498652766" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.228ex; height:6.343ex;" alt="{\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}"></span></big></td></tr><tr><td class="sidebar-above" style="padding:0.15em 0.25em 0.3em;font-weight:normal;"> <ul><li><a href="/wiki/Fundamental_theorem_of_calculus" title="Fundamental theorem of calculus">Fundamental theorem</a></li></ul> <div class="hlist"> <ul><li><a href="/wiki/Limit_of_a_function" title="Limit of a function">Limits</a></li> <li><a href="/wiki/Continuous_function" title="Continuous function">Continuity</a></li></ul> </div><div class="hlist"> <ul><li><a href="/wiki/Rolle%27s_theorem" title="Rolle's theorem">Rolle's theorem</a></li> <li><a href="/wiki/Mean_value_theorem" title="Mean value theorem">Mean value theorem</a></li> <li><a href="/wiki/Inverse_function_theorem" title="Inverse function theorem">Inverse function theorem</a></li></ul> </div></td></tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base);display:block;margin-top:0.65em;"><span style="font-size:120%"><a href="/wiki/Differential_calculus" title="Differential calculus">Differential</a></span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar-subgroup"><tbody><tr><th class="sidebar-heading"> Definitions</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Derivative" title="Derivative">Derivative</a> (<a class="mw-selflink selflink">generalizations</a>)</li> <li><a href="/wiki/Differential_(mathematics)" title="Differential (mathematics)">Differential</a> <ul><li><a href="/wiki/Differential_(infinitesimal)" class="mw-redirect" title="Differential (infinitesimal)">infinitesimal</a></li> <li><a href="/wiki/Differential_of_a_function" title="Differential of a function">of a function</a></li> <li><a href="/wiki/Differential_of_a_function#Differentials_in_several_variables" title="Differential of a function">total</a></li></ul></li></ul></td> </tr><tr><th class="sidebar-heading"> Concepts</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Notation_for_differentiation" title="Notation for differentiation">Differentiation notation</a></li> <li><a href="/wiki/Second_derivative" title="Second derivative">Second derivative</a></li> <li><a href="/wiki/Implicit_function" title="Implicit function">Implicit differentiation</a></li> <li><a href="/wiki/Logarithmic_differentiation" title="Logarithmic differentiation">Logarithmic differentiation</a></li> <li><a href="/wiki/Related_rates" title="Related rates">Related rates</a></li> <li><a href="/wiki/Taylor%27s_theorem" title="Taylor's theorem">Taylor's theorem</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <a href="/wiki/Differentiation_rules" title="Differentiation rules">Rules and identities</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Sum_rule_in_differentiation" class="mw-redirect" title="Sum rule in differentiation">Sum</a></li> <li><a href="/wiki/Product_rule" title="Product rule">Product</a></li> <li><a href="/wiki/Chain_rule" title="Chain rule">Chain</a></li> <li><a href="/wiki/Power_rule" title="Power rule">Power</a></li> <li><a href="/wiki/Quotient_rule" title="Quotient rule">Quotient</a></li> <li><a href="/wiki/L%27H%C3%B4pital%27s_rule" title="L'Hôpital's rule">L'Hôpital's rule</a></li> <li><a href="/wiki/Inverse_function_rule" title="Inverse function rule">Inverse</a></li> <li><a href="/wiki/General_Leibniz_rule" title="General Leibniz rule">General Leibniz</a></li> <li><a href="/wiki/Fa%C3%A0_di_Bruno%27s_formula" title="Faà di Bruno's formula">Faà di Bruno's formula</a></li> <li><a href="/wiki/Reynolds_transport_theorem" title="Reynolds transport theorem">Reynolds</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><span style="font-size:120%"><a href="/wiki/Integral" title="Integral">Integral</a></span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar-subgroup"><tbody><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Lists_of_integrals" title="Lists of integrals">Lists of integrals</a></li> <li><a href="/wiki/Integral_transform" title="Integral transform">Integral transform</a></li> <li><a href="/wiki/Leibniz_integral_rule" title="Leibniz integral rule">Leibniz integral rule</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Definitions</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Antiderivative" title="Antiderivative">Antiderivative</a></li> <li><a href="/wiki/Integral" title="Integral">Integral</a> (<a href="/wiki/Improper_integral" title="Improper integral">improper</a>)</li> <li><a href="/wiki/Riemann_integral" title="Riemann integral">Riemann integral</a></li> <li><a href="/wiki/Lebesgue_integration" class="mw-redirect" title="Lebesgue integration">Lebesgue integration</a></li> <li><a href="/wiki/Contour_integration" title="Contour integration">Contour integration</a></li> <li><a href="/wiki/Integral_of_inverse_functions" title="Integral of inverse functions">Integral of inverse functions</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Integration by</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Integration_by_parts" title="Integration by parts">Parts</a></li> <li><a href="/wiki/Disc_integration" title="Disc integration">Discs</a></li> <li><a href="/wiki/Shell_integration" title="Shell integration">Cylindrical shells</a></li> <li><a href="/wiki/Integration_by_substitution" title="Integration by substitution">Substitution</a> (<a href="/wiki/Trigonometric_substitution" title="Trigonometric substitution">trigonometric</a>, <a href="/wiki/Tangent_half-angle_substitution" title="Tangent half-angle substitution">tangent half-angle</a>, <a href="/wiki/Euler_substitution" title="Euler substitution">Euler</a>)</li> <li><a href="/wiki/Integration_using_Euler%27s_formula" title="Integration using Euler's formula">Euler's formula</a></li> <li><a href="/wiki/Partial_fractions_in_integration" class="mw-redirect" title="Partial fractions in integration">Partial fractions</a> (<a href="/wiki/Heaviside_cover-up_method" title="Heaviside cover-up method">Heaviside's method</a>)</li> <li><a href="/wiki/Order_of_integration_(calculus)" title="Order of integration (calculus)">Changing order</a></li> <li><a href="/wiki/Integration_by_reduction_formulae" title="Integration by reduction formulae">Reduction formulae</a></li> <li><a href="/wiki/Leibniz_integral_rule#Evaluating_definite_integrals" title="Leibniz integral rule">Differentiating under the integral sign</a></li> <li><a href="/wiki/Risch_algorithm" title="Risch algorithm">Risch algorithm</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><span style="font-size:120%"><a href="/wiki/Series_(mathematics)" title="Series (mathematics)">Series</a></span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar-subgroup"><tbody><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Geometric_series" title="Geometric series">Geometric</a> (<a href="/wiki/Arithmetico%E2%80%93geometric_sequence" class="mw-redirect" title="Arithmetico–geometric sequence">arithmetico-geometric</a>)</li> <li><a href="/wiki/Harmonic_series_(mathematics)" title="Harmonic series (mathematics)">Harmonic</a></li> <li><a href="/wiki/Alternating_series" title="Alternating series">Alternating</a></li> <li><a href="/wiki/Power_series" title="Power series">Power</a></li> <li><a href="/wiki/Binomial_series" title="Binomial series">Binomial</a></li> <li><a href="/wiki/Taylor_series" title="Taylor series">Taylor</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <a href="/wiki/Convergence_tests" title="Convergence tests">Convergence tests</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Term_test" class="mw-redirect" title="Term test">Summand limit (term test)</a></li> <li><a href="/wiki/Ratio_test" title="Ratio test">Ratio</a></li> <li><a href="/wiki/Root_test" title="Root test">Root</a></li> <li><a href="/wiki/Integral_test_for_convergence" title="Integral test for convergence">Integral</a></li> <li><a href="/wiki/Direct_comparison_test" title="Direct comparison test">Direct comparison</a></li> <li><br /><a href="/wiki/Limit_comparison_test" title="Limit comparison test">Limit comparison</a></li> <li><a href="/wiki/Alternating_series_test" title="Alternating series test">Alternating series</a></li> <li><a href="/wiki/Cauchy_condensation_test" title="Cauchy condensation test">Cauchy condensation</a></li> <li><a href="/wiki/Dirichlet%27s_test" title="Dirichlet's test">Dirichlet</a></li> <li><a href="/wiki/Abel%27s_test" title="Abel's test">Abel</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><span style="font-size:120%"><a href="/wiki/Vector_calculus" title="Vector calculus">Vector</a></span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar-subgroup"><tbody><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Gradient" title="Gradient">Gradient</a></li> <li><a href="/wiki/Divergence" title="Divergence">Divergence</a></li> <li><a href="/wiki/Curl_(mathematics)" title="Curl (mathematics)">Curl</a></li> <li><a href="/wiki/Laplace_operator" title="Laplace operator">Laplacian</a></li> <li><a href="/wiki/Directional_derivative" title="Directional derivative">Directional derivative</a></li> <li><a href="/wiki/Vector_calculus_identities" title="Vector calculus identities">Identities</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Theorems</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Gradient_theorem" title="Gradient theorem">Gradient</a></li> <li><a href="/wiki/Green%27s_theorem" title="Green's theorem">Green's</a></li> <li><a href="/wiki/Stokes%27_theorem" title="Stokes' theorem">Stokes'</a></li> <li><a href="/wiki/Divergence_theorem" title="Divergence theorem">Divergence</a></li> <li><a href="/wiki/Generalized_Stokes_theorem" title="Generalized Stokes theorem">generalized Stokes</a></li> <li><a href="/wiki/Helmholtz_decomposition" title="Helmholtz decomposition">Helmholtz decomposition</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><span style="font-size:120%"><a href="/wiki/Multivariable_calculus" title="Multivariable calculus">Multivariable</a></span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar-subgroup"><tbody><tr><th class="sidebar-heading"> Formalisms</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Matrix_calculus" title="Matrix calculus">Matrix</a></li> <li><a href="/wiki/Tensor_calculus" class="mw-redirect" title="Tensor calculus">Tensor</a></li> <li><a href="/wiki/Exterior_derivative" title="Exterior derivative">Exterior</a></li> <li><a href="/wiki/Geometric_calculus" title="Geometric calculus">Geometric</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Definitions</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Partial_derivative" title="Partial derivative">Partial derivative</a></li> <li><a href="/wiki/Multiple_integral" title="Multiple integral">Multiple integral</a></li> <li><a href="/wiki/Line_integral" title="Line integral">Line integral</a></li> <li><a href="/wiki/Surface_integral" title="Surface integral">Surface integral</a></li> <li><a href="/wiki/Volume_integral" title="Volume integral">Volume integral</a></li> <li><a href="/wiki/Jacobian_matrix_and_determinant" title="Jacobian matrix and determinant">Jacobian</a></li> <li><a href="/wiki/Hessian_matrix" title="Hessian matrix">Hessian</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><span style="font-size:120%">Advanced</span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar-subgroup"><tbody><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Calculus_on_Euclidean_space" title="Calculus on Euclidean space">Calculus on Euclidean space</a></li> <li><a href="/wiki/Generalized_function" title="Generalized function">Generalized functions</a></li> <li><a href="/wiki/Limit_of_distributions" title="Limit of distributions">Limit of distributions</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><span style="font-size:120%">Specialized</span></div><div class="sidebar-list-content mw-collapsible-content" 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template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Calculus" title="Template talk:Calculus"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Calculus" title="Special:EditPage/Template:Calculus"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, the <a href="/wiki/Derivative" title="Derivative">derivative</a> is a fundamental construction of <a href="/wiki/Differential_calculus" title="Differential calculus">differential calculus</a> and admits many possible generalizations within the fields of <a href="/wiki/Mathematical_analysis" title="Mathematical analysis">mathematical analysis</a>, <a href="/wiki/Combinatorics" title="Combinatorics">combinatorics</a>, <a href="/wiki/Algebra" title="Algebra">algebra</a>, <a href="/wiki/Geometry" title="Geometry">geometry</a>, etc. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Fréchet_derivative"><span id="Fr.C3.A9chet_derivative"></span>Fréchet derivative</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Generalizations_of_the_derivative&action=edit&section=1" title="Edit section: Fréchet derivative"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Fr%C3%A9chet_derivative" title="Fréchet derivative">Fréchet derivative</a> defines the derivative for general <a href="/wiki/Normed_vector_space" title="Normed vector space">normed vector spaces</a> <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 V,W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>,</mo> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V,W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a40b0deabeee6e15bff1e3079b601986d8fe337" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.256ex; height:2.509ex;" alt="{\displaystyle V,W}"></span>. Briefly, a function <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 f:U\to W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>U</mi> <mo stretchy="false">→</mo> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:U\to W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec1eca5d64b33b33cefc08de947070177c8f4261" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.048ex; height:2.509ex;" alt="{\displaystyle f:U\to W}"></span>, where <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 U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></span> is an open subset of <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 V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span>, is called <i>Fréchet differentiable</i> at <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 x\in U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c32ddcb2941216f2980b950ce969dc15cba26906" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.953ex; height:2.176ex;" alt="{\displaystyle x\in U}"></span> if there exists a <a href="/wiki/Bounded_linear_operator" class="mw-redirect" title="Bounded linear operator">bounded linear operator</a> <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 A:V\to W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>:</mo> <mi>V</mi> <mo stretchy="false">→</mo> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A:V\to W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97b4b6380b2f4700cea977536a2af33d51efb89f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.517ex; height:2.176ex;" alt="{\displaystyle A:V\to W}"></span> such that <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 \lim _{\|h\|\to 0}{\frac {\|f(x+h)-f(x)-Ah\|_{W}}{\|h\|_{V}}}=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">‖</mo> <mi>h</mi> <mo fence="false" stretchy="false">‖</mo> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>A</mi> <mi>h</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>W</mi> </mrow> </msub> </mrow> <mrow> <mo fence="false" stretchy="false">‖</mo> <mi>h</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{\|h\|\to 0}{\frac {\|f(x+h)-f(x)-Ah\|_{W}}{\|h\|_{V}}}=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23e3f5b486b55dddd230afe8a27485e4e9b1cf9b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:37.243ex; height:6.509ex;" alt="{\displaystyle \lim _{\|h\|\to 0}{\frac {\|f(x+h)-f(x)-Ah\|_{W}}{\|h\|_{V}}}=0.}"></span> </p><p>Functions are defined as being differentiable in some open <a href="/wiki/Neighbourhood_(mathematics)" title="Neighbourhood (mathematics)">neighbourhood</a> of <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 x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>, rather than at individual points, as not doing so tends to lead to many <a href="/wiki/Pathological_(mathematics)" title="Pathological (mathematics)">pathological</a> <a href="/wiki/Counterexamples" class="mw-redirect" title="Counterexamples">counterexamples</a>. </p><p>The Fréchet derivative is quite similar to the formula for the <a href="/wiki/Derivative" title="Derivative">derivative</a> found in elementary one-variable calculus, <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 \lim _{h\to 0}{\frac {f(x+h)-f(x)}{h}}=A,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mi>h</mi> </mfrac> </mrow> <mo>=</mo> <mi>A</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{h\to 0}{\frac {f(x+h)-f(x)}{h}}=A,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f05fc078a2061026f5222ca12e2aebc1ccca8f48" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:25.978ex; height:5.843ex;" alt="{\displaystyle \lim _{h\to 0}{\frac {f(x+h)-f(x)}{h}}=A,}"></span> and simply moves <i>A</i> to the left hand side. However, the Fréchet derivative <i>A</i> denotes the function <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 t\mapsto f'(x)\cdot t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo stretchy="false">↦</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t\mapsto f'(x)\cdot t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab373647b0e2e54965e010ec6ac1c7f4f5775a5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.117ex; height:3.009ex;" alt="{\displaystyle t\mapsto f'(x)\cdot t}"></span>. </p><p>In <a href="/wiki/Multivariable_calculus" title="Multivariable calculus">multivariable calculus</a>, in the context of differential equations defined by a vector valued function <b>R</b><sup><i>n</i></sup> to <b>R</b><sup><i>m</i></sup>, the Fréchet derivative <i>A</i> is a <a href="/wiki/Linear_operator" class="mw-redirect" title="Linear operator">linear operator</a> on <b>R</b> considered as a vector space over itself, and corresponds to the <i>best linear approximation</i> of a function. If such an operator exists, then it is unique, and can be represented by an <i>m</i> by <i>n</i> <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a> known as the <a href="/wiki/Jacobian_matrix_and_determinant" title="Jacobian matrix and determinant">Jacobian matrix</a> J<sub><i>x</i></sub>(ƒ) of the mapping ƒ at point <i>x</i>. Each entry of this matrix represents a <a href="/wiki/Partial_derivative" title="Partial derivative">partial derivative</a>, specifying the rate of change of one range coordinate with respect to a change in a domain coordinate. Of course, the Jacobian matrix of the composition <i>g<sub>°</sub>f</i> is a product of corresponding Jacobian matrices: J<sub><i>x</i></sub>(<i>g<sub>°</sub>f</i>) =J<sub>ƒ(<i>x</i>)</sub>(<i>g</i>)J<sub><i>x</i></sub>(ƒ). This is a higher-dimensional statement of the <a href="/wiki/Chain_rule" title="Chain rule">chain rule</a>. </p><p>For real valued functions from <b>R</b><sup><i>n</i></sup> to <b>R</b> (<a href="/wiki/Scalar_field" title="Scalar field">scalar fields</a>), the Fréchet derivative corresponds to a <a href="/wiki/Vector_field" title="Vector field">vector field</a> called the <a href="/wiki/Total_derivative" title="Total derivative">total derivative</a>. This can be interpreted as the <a href="/wiki/Gradient" title="Gradient">gradient</a> but it is more natural to use the <a href="/wiki/Exterior_derivative" title="Exterior derivative">exterior derivative</a>. </p><p>The <a href="/wiki/Convective_derivative" class="mw-redirect" title="Convective derivative">convective derivative</a> takes into account changes due to time dependence and motion through space along a vector field, and is a special case of the total derivative. </p><p>For <a href="/wiki/Vector-valued_functions" class="mw-redirect" title="Vector-valued functions">vector-valued functions</a> from <b>R</b> to <b>R</b><sup><i>n</i></sup> (i.e., <a href="/wiki/Parametric_curve" class="mw-redirect" title="Parametric curve">parametric curves</a>), the Fréchet derivative corresponds to taking the derivative of each component separately. The resulting derivative can be mapped to a vector. This is useful, for example, if the vector-valued function is the position vector of a particle through time, then the derivative is the velocity vector of the particle through time. </p><p>In <a href="/wiki/Complex_analysis" title="Complex analysis">complex analysis</a>, the central objects of study are <a href="/wiki/Holomorphic_function" title="Holomorphic function">holomorphic functions</a>, which are complex-valued functions on the <a href="/wiki/Complex_numbers" class="mw-redirect" title="Complex numbers">complex numbers</a> where the Fréchet derivative exists. </p><p>In <a href="/wiki/Geometric_calculus" title="Geometric calculus">geometric calculus</a>, the <a href="/wiki/Geometric_calculus#Differentiation" title="Geometric calculus">geometric derivative</a> satisfies a weaker form of the Leibniz (product) rule. It specializes the Fréchet derivative to the objects of geometric algebra. Geometric calculus is a powerful formalism that has been shown to encompass the similar frameworks of differential forms and differential geometry.<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> </p> <div class="mw-heading mw-heading2"><h2 id="Exterior_derivative_and_Lie_derivative">Exterior derivative and Lie derivative</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Generalizations_of_the_derivative&action=edit&section=2" title="Edit section: Exterior derivative and Lie derivative"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>On the <a href="/wiki/Exterior_algebra" title="Exterior algebra">exterior algebra</a> of <a href="/wiki/Differential_forms" class="mw-redirect" title="Differential forms">differential forms</a> over a <a href="/wiki/Smooth_manifold" class="mw-redirect" title="Smooth manifold">smooth manifold</a>, the <a href="/wiki/Exterior_derivative" title="Exterior derivative">exterior derivative</a> is the unique linear map which satisfies a <a href="/wiki/Graded_Leibniz_rule" class="mw-redirect" title="Graded Leibniz rule">graded version of the Leibniz law</a> and squares to zero. It is a grade 1 derivation on the exterior algebra. In <b>R</b><sup>3</sup>, the <a href="/wiki/Gradient" title="Gradient">gradient</a>, <a href="/wiki/Curl_(mathematics)" title="Curl (mathematics)">curl</a>, and <a href="/wiki/Divergence" title="Divergence">divergence</a> are special cases of the exterior derivative. An intuitive interpretation of the gradient is that it points "up": in other words, it points in the direction of fastest increase of the function. It can be used to calculate <a href="/wiki/Directional_derivative" title="Directional derivative">directional derivatives</a> of <a href="/wiki/Scalar_(mathematics)" title="Scalar (mathematics)">scalar</a> functions or normal directions. Divergence gives a measure of how much "source" or "sink" near a point there is. It can be used to calculate <a href="/wiki/Flux" title="Flux">flux</a> by <a href="/wiki/Divergence_theorem" title="Divergence theorem">divergence theorem</a>. Curl measures how much "<a href="/wiki/Rotation" title="Rotation">rotation</a>" a vector field has near a point. </p><p>The <a href="/wiki/Lie_derivative" title="Lie derivative">Lie derivative</a> is the rate of change of a vector or tensor field along the flow of another vector field. On vector fields, it is an example of a <a href="/wiki/Lie_bracket" class="mw-redirect" title="Lie bracket">Lie bracket</a> (vector fields form the <a href="/wiki/Lie_algebra" title="Lie algebra">Lie algebra</a> of the <a href="/wiki/Diffeomorphism_group" class="mw-redirect" title="Diffeomorphism group">diffeomorphism group</a> of the manifold). It is a grade 0 derivation on the algebra. </p><p>Together with the <a href="/wiki/Interior_product" title="Interior product">interior product</a> (a degree -1 derivation on the exterior algebra defined by contraction with a vector field), the exterior derivative and the Lie derivative form a <a href="/wiki/Lie_superalgebra" title="Lie superalgebra">Lie superalgebra</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Differential_topology">Differential topology</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Generalizations_of_the_derivative&action=edit&section=3" title="Edit section: Differential topology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Differential_topology" title="Differential topology">differential topology</a>, a <a href="/wiki/Vector_field" title="Vector field">vector field</a> may be defined as a derivation on the ring of <a href="/wiki/Smooth_function" class="mw-redirect" title="Smooth function">smooth functions</a> on a <a href="/wiki/Manifold" title="Manifold">manifold</a>, and a <a href="/wiki/Tangent_vector" title="Tangent vector">tangent vector</a> may be defined as a derivation at a point. This allows the abstraction of the notion of a <a href="/wiki/Directional_derivative" title="Directional derivative">directional derivative</a> of a scalar function to general manifolds. For manifolds that are <a href="/wiki/Subset" title="Subset">subsets</a> of <b>R</b><sup><i>n</i></sup>, this tangent vector will agree with the <a href="/wiki/Directional_derivative" title="Directional derivative">directional derivative</a>. </p><p>The <a href="/wiki/Pushforward_(differential)" title="Pushforward (differential)">differential or pushforward</a> of a map between manifolds is the induced map between tangent spaces of those maps. It abstracts the <a href="/wiki/Jacobian_matrix" class="mw-redirect" title="Jacobian matrix">Jacobian matrix</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Covariant_derivative">Covariant derivative</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Generalizations_of_the_derivative&action=edit&section=4" title="Edit section: Covariant derivative"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Differential_geometry" title="Differential geometry">differential geometry</a>, the <a href="/wiki/Covariant_derivative" title="Covariant derivative">covariant derivative</a> makes a choice for taking directional derivatives of vector fields along <a href="/wiki/Curve" title="Curve">curves</a>. This extends the directional derivative of scalar functions to sections of <a href="/wiki/Vector_bundle" title="Vector bundle">vector bundles</a> or <a href="/wiki/Principal_bundle" title="Principal bundle">principal bundles</a>. In <a href="/wiki/Riemannian_geometry" title="Riemannian geometry">Riemannian geometry</a>, the existence of a metric chooses a unique preferred <a href="/wiki/Torsion_tensor" title="Torsion tensor">torsion</a>-free covariant derivative, known as the <a href="/wiki/Levi-Civita_connection" title="Levi-Civita connection">Levi-Civita connection</a>. See also <a href="/wiki/Gauge_covariant_derivative" title="Gauge covariant derivative">gauge covariant derivative</a> for a treatment oriented to physics. </p><p>The <a href="/wiki/Exterior_covariant_derivative" title="Exterior covariant derivative">exterior covariant derivative</a> extends the exterior derivative to vector valued forms. </p> <div class="mw-heading mw-heading2"><h2 id="Weak_derivatives">Weak derivatives</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Generalizations_of_the_derivative&action=edit&section=5" title="Edit section: Weak derivatives"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Given a function <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 u:\mathbb {R} ^{n}\to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>:</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u:\mathbb {R} ^{n}\to \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91c7fddd01e783ba4cc25f4d201e9570ee3b9acc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.456ex; height:2.343ex;" alt="{\displaystyle u:\mathbb {R} ^{n}\to \mathbb {R} }"></span> which is <a href="/wiki/Locally_integrable_function" title="Locally integrable function">locally integrable</a>, but not necessarily classically differentiable, a <a href="/wiki/Weak_derivative" title="Weak derivative">weak derivative</a> may be defined by means of <a href="/wiki/Integration_by_parts" title="Integration by parts">integration by parts</a>. First define test functions, which are infinitely differentiable and compactly supported functions <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 \varphi \in C_{c}^{\infty }\left(\mathbb {R} ^{n}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>∈</mo> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi \in C_{c}^{\infty }\left(\mathbb {R} ^{n}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5ed28ab84714acfbe4472033a7c72f809ee5f1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.127ex; height:2.843ex;" alt="{\displaystyle \varphi \in C_{c}^{\infty }\left(\mathbb {R} ^{n}\right)}"></span>, and <a href="/wiki/Multi-index_notation" title="Multi-index notation">multi-indices</a>, which are length <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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> lists of integers <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 \alpha =(\alpha _{1},\dots ,\alpha _{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =(\alpha _{1},\dots ,\alpha _{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97ad3d26116200a07d3300c771813cf4af907361" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.822ex; height:2.843ex;" alt="{\displaystyle \alpha =(\alpha _{1},\dots ,\alpha _{n})}"></span> 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 |\alpha |:=\sum _{1}^{n}\alpha _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>:=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle |\alpha |:=\sum _{1}^{n}\alpha _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c28d640166cf5106ea7ed5cafe13d72b79cf970f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.873ex; height:3.176ex;" alt="{\textstyle |\alpha |:=\sum _{1}^{n}\alpha _{i}}"></span>. Applied to test functions, <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 D^{\alpha }\varphi :={\frac {\partial ^{|\alpha |}\varphi }{\partial x_{1}^{\alpha _{1}}\dotsm x_{n}^{\alpha _{n}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mi>φ</mi> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </msup> <mi>φ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msubsup> <mo>⋯</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msubsup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle D^{\alpha }\varphi :={\frac {\partial ^{|\alpha |}\varphi }{\partial x_{1}^{\alpha _{1}}\dotsm x_{n}^{\alpha _{n}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aee0b1670ee4582e332e24599f895c9462fb4c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:17.819ex; height:5.676ex;" alt="{\textstyle D^{\alpha }\varphi :={\frac {\partial ^{|\alpha |}\varphi }{\partial x_{1}^{\alpha _{1}}\dotsm x_{n}^{\alpha _{n}}}}}"></span>. Then the <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 \alpha ^{\text{th}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>th</mtext> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \alpha ^{\text{th}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e05bfd9ff02b84bde4f6b7bfd4b3dab4ae837c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.274ex; height:2.509ex;" alt="{\textstyle \alpha ^{\text{th}}}"></span> weak derivative of <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 u}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}"></span> exists if there is a function <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 v:\mathbb {R} ^{n}\to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>:</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v:\mathbb {R} ^{n}\to \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99bfc44041d829837d3bb7cd3fe2117b69d53091" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.253ex; height:2.343ex;" alt="{\displaystyle v:\mathbb {R} ^{n}\to \mathbb {R} }"></span> such that for <i>all</i> test functions <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 \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"></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 \int _{\mathbb {R} ^{n}}u\ D^{\alpha }\!\varphi \ dx=(-1)^{|\alpha |}\int _{\mathbb {R} ^{n}}v\ \varphi \ dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msub> <mi>u</mi> <mtext> </mtext> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mspace width="negativethinmathspace" /> <mi>φ</mi> <mtext> </mtext> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </msup> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msub> <mi>v</mi> <mtext> </mtext> <mi>φ</mi> <mtext> </mtext> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{\mathbb {R} ^{n}}u\ D^{\alpha }\!\varphi \ dx=(-1)^{|\alpha |}\int _{\mathbb {R} ^{n}}v\ \varphi \ dx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c66b3bfb1735b9382d12fb2ebf4f66a96667a9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:34.324ex; height:5.676ex;" alt="{\displaystyle \int _{\mathbb {R} ^{n}}u\ D^{\alpha }\!\varphi \ dx=(-1)^{|\alpha |}\int _{\mathbb {R} ^{n}}v\ \varphi \ dx}"></span></dd></dl> <p>If such a function exists, then <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 D^{\alpha }u:=v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> <mi>u</mi> <mo>:=</mo> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D^{\alpha }u:=v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ff6c0eb32ca56c2e95d182c250aa11f4d1547dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.411ex; height:2.343ex;" alt="{\displaystyle D^{\alpha }u:=v}"></span>, which is unique <a href="/wiki/Almost_everywhere" title="Almost everywhere">almost everywhere</a>. This definition coincides with the classical derivative for functions <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 u\in C^{|\alpha |}\left(\mathbb {R} ^{n}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>∈</mo> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </msup> <mrow> <mo>(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u\in C^{|\alpha |}\left(\mathbb {R} ^{n}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e950676da58640880050ce61eb830882e3c69031" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.26ex; height:3.343ex;" alt="{\displaystyle u\in C^{|\alpha |}\left(\mathbb {R} ^{n}\right)}"></span>, and can be extended to a type of generalized functions called <a href="/wiki/Distribution_(mathematics)" title="Distribution (mathematics)">distributions</a>, the dual space of test functions. Weak derivatives are particularly useful in the study of partial differential equations, and within parts of functional analysis. </p> <div class="mw-heading mw-heading2"><h2 id="Higher-order_and_fractional_derivatives">Higher-order and fractional derivatives</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Generalizations_of_the_derivative&action=edit&section=6" title="Edit section: Higher-order and fractional derivatives"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the real numbers one can iterate the differentiation process, that is, apply derivatives more than once, obtaining derivatives of second and higher order. Higher derivatives can also be defined for functions of several variables, studied in <a href="/wiki/Multivariable_calculus" title="Multivariable calculus">multivariable calculus</a>. In this case, instead of repeatedly applying the derivative, one repeatedly applies <a href="/wiki/Partial_derivative" title="Partial derivative">partial derivatives</a> with respect to different variables. For example, the second order partial derivatives of a scalar function of <i>n</i> variables can be organized into an <i>n</i> by <i>n</i> matrix, the <a href="/wiki/Hessian_matrix" title="Hessian matrix">Hessian matrix</a>. One of the subtle points is that the higher derivatives are not intrinsically defined, and depend on the choice of the coordinates in a complicated fashion (in particular, the Hessian matrix of a function is not a <a href="/wiki/Tensor" title="Tensor">tensor</a>). Nevertheless, higher derivatives have important applications to analysis of <a href="/wiki/Maxima_and_minima" class="mw-redirect" title="Maxima and minima">local extrema</a> of a function at its <a href="/wiki/Critical_point_(mathematics)" title="Critical point (mathematics)">critical points</a>. For an advanced application of this analysis to topology of <a href="/wiki/Manifold" title="Manifold">manifolds</a>, see <a href="/wiki/Morse_theory" title="Morse theory">Morse theory</a>. </p><p>In addition to <i>n</i> th derivatives for any natural number <i>n</i>, there are various ways to define derivatives of fractional or negative orders, which are studied in <a href="/wiki/Fractional_calculus" title="Fractional calculus">fractional calculus</a>. The −1 order derivative corresponds to the integral, whence the term <a href="/wiki/Differintegral" title="Differintegral">differintegral</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Quaternionic_derivatives">Quaternionic derivatives</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Generalizations_of_the_derivative&action=edit&section=7" title="Edit section: Quaternionic derivatives"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Quaternionic_analysis" title="Quaternionic analysis">quaternionic analysis</a>, derivatives can be defined in a similar way to real and complex functions. Since the <a href="/wiki/Quaternion" title="Quaternion">quaternions</a> <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 {H} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">H</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {H} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e050965453c42bcc6bd544546703c836bdafeac9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {H} }"></span> are not commutative, the limit of the difference quotient yields two different derivatives: A left derivative </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 \lim _{h\to 0}\left[h^{-1}\left(f(a+h)-f(a)\right)\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mrow> <mo>[</mo> <mrow> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{h\to 0}\left[h^{-1}\left(f(a+h)-f(a)\right)\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22049379c8a629164b6ba68c9380b1cc5de45a1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:27.262ex; height:4.343ex;" alt="{\displaystyle \lim _{h\to 0}\left[h^{-1}\left(f(a+h)-f(a)\right)\right]}"></span></dd></dl> <p>and a right derivative </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 \lim _{h\to 0}\left[\left(f(a+h)-f(a)\right)h^{-1}\right].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mrow> <mo>[</mo> <mrow> <mrow> <mo>(</mo> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{h\to 0}\left[\left(f(a+h)-f(a)\right)h^{-1}\right].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2159362c93325e84348f4d263be3fc4dcd55a93b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:28.296ex; height:4.343ex;" alt="{\displaystyle \lim _{h\to 0}\left[\left(f(a+h)-f(a)\right)h^{-1}\right].}"></span></dd></dl> <p>The existence of these limits are very restrictive conditions. For example, 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 f:\mathbb {H} \to \mathbb {H} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">H</mi> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">H</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:\mathbb {H} \to \mathbb {H} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ecaa43c5fb298076c7d43fc0ccc57146c688a37d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.446ex; height:2.509ex;" alt="{\displaystyle f:\mathbb {H} \to \mathbb {H} }"></span> has left-derivatives at every point on an open connected set <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 U\subset \mathbb {H} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>⊂</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">H</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U\subset \mathbb {H} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8339e259cea8d540698935fe234e955fe4126552" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.689ex; height:2.176ex;" alt="{\displaystyle U\subset \mathbb {H} }"></span>, then <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 f(q)=a+qb}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mi>q</mi> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(q)=a+qb}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9929d6654cce3ece2b27bea45539c85041608a00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.393ex; height:2.843ex;" alt="{\displaystyle f(q)=a+qb}"></span> 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 a,b\in \mathbb {H} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">H</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b\in \mathbb {H} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3069a5d4050c7825ef91b5e1a87fafab816af556" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.91ex; height:2.509ex;" alt="{\displaystyle a,b\in \mathbb {H} }"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Difference_operator,_q-analogues_and_time_scales"><span id="Difference_operator.2C_q-analogues_and_time_scales"></span>Difference operator, q-analogues and time scales</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Generalizations_of_the_derivative&action=edit&section=8" title="Edit section: Difference operator, q-analogues and time scales"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>The <a href="/wiki/Q-derivative" title="Q-derivative">q-derivative</a> of a function is defined by the formula <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 D_{q}f(x)={\frac {f(qx)-f(x)}{(q-1)x}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>q</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>q</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{q}f(x)={\frac {f(qx)-f(x)}{(q-1)x}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a6fc007a93230248708d853585dd5d175d723c3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:24.657ex; height:6.509ex;" alt="{\displaystyle D_{q}f(x)={\frac {f(qx)-f(x)}{(q-1)x}}.}"></span> For <i>x</i> nonzero, if <i>f</i> is a differentiable function of <i>x</i> then in the limit as <span class="texhtml"><i>q</i> → 1</span> we obtain the ordinary derivative, thus the <i>q</i>-derivative may be viewed as its <a href="/wiki/Q-deformation" class="mw-redirect" title="Q-deformation">q-deformation</a>. A large body of results from ordinary differential calculus, such as <a href="/wiki/Binomial_formula" class="mw-redirect" title="Binomial formula">binomial formula</a> and <a href="/wiki/Taylor_expansion" class="mw-redirect" title="Taylor expansion">Taylor expansion</a>, have natural <i>q</i>-analogues that were discovered in the 19th century, but remained relatively obscure for a big part of the 20th century, outside of the theory of <a href="/wiki/Special_functions" title="Special functions">special functions</a>. The progress of <a href="/wiki/Combinatorics" title="Combinatorics">combinatorics</a> and the discovery of <a href="/wiki/Quantum_group" title="Quantum group">quantum groups</a> have changed the situation dramatically, and the popularity of <i>q</i>-analogues is on the rise.</li> <li>The <a href="/wiki/Difference_operator" class="mw-redirect" title="Difference operator">difference operator</a> of <a href="/wiki/Difference_equations" class="mw-redirect" title="Difference equations">difference equations</a> is another discrete analog of the standard derivative. <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 \Delta f(x)=f(x+1)-f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta f(x)=f(x+1)-f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f476f29e065f1a1e0f67e13e89fb2a8f07cad235" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.13ex; height:2.843ex;" alt="{\displaystyle \Delta f(x)=f(x+1)-f(x)}"></span></li> <li>The q-derivative, the difference operator and the standard derivative can all be viewed as the same thing on different <a href="/wiki/Time_scale_calculus" class="mw-redirect" title="Time scale calculus">time scales</a>. For example, taking <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 =(q-1)x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>q</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon =(q-1)x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e94cebc57ce9fb2d041967188cb362653715c067" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.393ex; height:2.843ex;" alt="{\displaystyle \varepsilon =(q-1)x}"></span>, we may 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 {\frac {f(qx)-f(x)}{(q-1)x}}={\frac {f(x+\varepsilon )-f(x)}{\varepsilon }}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>q</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>q</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>ε</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mi>ε</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {f(qx)-f(x)}{(q-1)x}}={\frac {f(x+\varepsilon )-f(x)}{\varepsilon }}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/719e557b34d30c4d10d7d5163b04417732538a01" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:33.762ex; height:6.509ex;" alt="{\displaystyle {\frac {f(qx)-f(x)}{(q-1)x}}={\frac {f(x+\varepsilon )-f(x)}{\varepsilon }}.}"></span> The q-derivative is a special case of the <a href="/wiki/Wolfgang_Hahn" title="Wolfgang Hahn"> Hahn</a> difference,<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> <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 {\frac {f(qx+\omega )-f(x)}{qx+\omega -x}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>q</mi> <mi>x</mi> <mo>+</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>q</mi> <mi>x</mi> <mo>+</mo> <mi>ω</mi> <mo>−</mo> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {f(qx+\omega )-f(x)}{qx+\omega -x}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c71ed167ec4333871356810339da0a80866f6fd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.514ex; height:6.176ex;" alt="{\displaystyle {\frac {f(qx+\omega )-f(x)}{qx+\omega -x}}.}"></span>The Hahn difference is not only a generalization of the q-derivative but also an extension of the forward difference.</li> <li>Also note that the q-derivative is nothing but a special case of the familiar derivative. Take <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 z=qx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>q</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=qx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/221cc28e9ae53a6be8987ce1590c0f293c60aa7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.586ex; height:2.009ex;" alt="{\displaystyle z=qx}"></span>. Then 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 \lim _{z\to x}{\frac {f(z)-f(x)}{z-x}}=\lim _{q\to 1}{\frac {f(qx)-f(x)}{qx-x}}=\lim _{q\to 1}{\frac {f(qx)-f(x)}{(q-1)x}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mo stretchy="false">→</mo> <mi>x</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>z</mi> <mo>−</mo> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> <mo stretchy="false">→</mo> <mn>1</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>q</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>q</mi> <mi>x</mi> <mo>−</mo> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> <mo stretchy="false">→</mo> <mn>1</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>q</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>q</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{z\to x}{\frac {f(z)-f(x)}{z-x}}=\lim _{q\to 1}{\frac {f(qx)-f(x)}{qx-x}}=\lim _{q\to 1}{\frac {f(qx)-f(x)}{(q-1)x}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e77d5dee0be29119454b356daf3074bf0005b4c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:57.25ex; height:6.509ex;" alt="{\displaystyle \lim _{z\to x}{\frac {f(z)-f(x)}{z-x}}=\lim _{q\to 1}{\frac {f(qx)-f(x)}{qx-x}}=\lim _{q\to 1}{\frac {f(qx)-f(x)}{(q-1)x}}.}"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Derivatives_in_algebra">Derivatives in algebra</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Generalizations_of_the_derivative&action=edit&section=9" title="Edit section: Derivatives in algebra"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In algebra, generalizations of the derivative can be obtained by imposing the <a href="/wiki/Product_rule" title="Product rule">Leibniz rule of differentiation</a> in an algebraic structure, such as a <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">ring</a> or a <a href="/wiki/Lie_algebra" title="Lie algebra">Lie algebra</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Derivations">Derivations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Generalizations_of_the_derivative&action=edit&section=10" title="Edit section: Derivations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <a href="/wiki/Derivation_(abstract_algebra)" class="mw-redirect" title="Derivation (abstract algebra)">derivation</a> is a linear map on a ring or <a href="/wiki/Algebra_over_a_field" title="Algebra over a field">algebra</a> which satisfies the Leibniz law (the product rule). Higher derivatives and <a href="/wiki/Algebraic_differential_equation" title="Algebraic differential equation">algebraic differential operators</a> can also be defined. They are studied in a purely algebraic setting in <a href="/wiki/Differential_Galois_theory" title="Differential Galois theory">differential Galois theory</a> and the theory of <a href="/wiki/D-module" title="D-module">D-modules</a>, but also turn up in many other areas, where they often agree with less algebraic definitions of derivatives. </p><p>For example, the <a href="/wiki/Differential_algebra" title="Differential algebra">formal derivative</a> of a <a href="/wiki/Polynomial" title="Polynomial">polynomial</a> over a commutative ring <i>R</i> is defined by </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 \left(a_{d}x^{d}+a_{d-1}x^{d-1}+\cdots +a_{1}x+a_{0}\right)'=da_{d}x^{d-1}+(d-1)a_{d-1}x^{d-2}+\cdots +a_{1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msup> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>x</mi> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>′</mo> </msup> <mo>=</mo> <mi>d</mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <mi>d</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(a_{d}x^{d}+a_{d-1}x^{d-1}+\cdots +a_{1}x+a_{0}\right)'=da_{d}x^{d-1}+(d-1)a_{d-1}x^{d-2}+\cdots +a_{1}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2b29f724a28cffe1a4dc0f8dabab68a17b95d21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:77.792ex; height:3.676ex;" alt="{\displaystyle \left(a_{d}x^{d}+a_{d-1}x^{d-1}+\cdots +a_{1}x+a_{0}\right)'=da_{d}x^{d-1}+(d-1)a_{d-1}x^{d-2}+\cdots +a_{1}.}"></span></dd></dl> <p>The mapping <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 f\mapsto f'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">↦</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\mapsto f'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ca5feb4bd63a6547092b5c07b8b66535eaf7441" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.898ex; height:2.843ex;" alt="{\displaystyle f\mapsto f'}"></span> is then a derivation on the <a href="/wiki/Polynomial_ring" title="Polynomial ring">polynomial ring</a> <i>R</i>[<i>X</i>]. This definition can be extended to <a href="/wiki/Rational_function" title="Rational function">rational functions</a> as well. </p><p>The notion of derivation applies to noncommutative as well as commutative rings, and even to non-associative algebraic structures, such as Lie algebras. </p> <div class="mw-heading mw-heading3"><h3 id="Derivative_of_a_type">Derivative of a type</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Generalizations_of_the_derivative&action=edit&section=11" title="Edit section: Derivative of a type"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Type_theory" title="Type theory">type theory</a>, many <a href="/wiki/Abstract_data_type" title="Abstract data type">abstract data types</a> can be described as the <a href="/wiki/Universal_algebra" title="Universal algebra">algebra</a> generated by a transformation that maps structures based on the type back into the type. For example, the type T of <a href="/wiki/Binary_tree" title="Binary tree">binary trees</a> containing values of type A can be represented as the algebra generated by the transformation 1+A×T<sup>2</sup>→T. The "1" represents the construction of an empty tree, and the second term represents the construction of a tree from a value and two subtrees. The "+" indicates that a tree can be constructed either way. </p><p>The derivative of such a type is the type that describes the context of a particular substructure with respect to its next outer containing structure. Put another way, it is the type representing the "difference" between the two. In the tree example, the derivative is a type that describes the information needed, given a particular subtree, to construct its parent tree. This information is a tuple that contains a binary indicator of whether the child is on the left or right, the value at the parent, and the sibling subtree. This type can be represented as 2×A×T, which looks very much like the derivative of the transformation that generated the tree type. </p><p>This concept of a derivative of a type has practical applications, such as the <a href="/wiki/Zipper_(data_structure)" title="Zipper (data structure)">zipper</a> technique used in <a href="/wiki/Functional_programming_language" class="mw-redirect" title="Functional programming language">functional programming languages</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Differential_operators">Differential operators</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Generalizations_of_the_derivative&action=edit&section=12" title="Edit section: Differential operators"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <a href="/wiki/Differential_operator" title="Differential operator">differential operator</a> combines several derivatives, possibly of different orders, in one algebraic expression. This is especially useful in considering ordinary <a href="/wiki/Linear_differential_equation" title="Linear differential equation">linear differential equations</a> with constant coefficients. For example, if <i>f</i>(<i>x</i>) is a twice differentiable function of one variable, the differential equation <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 f''+2f'-3f=4x-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mo>″</mo> </msup> <mo>+</mo> <mn>2</mn> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo>−</mo> <mn>3</mn> <mi>f</mi> <mo>=</mo> <mn>4</mn> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f''+2f'-3f=4x-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af15b1d5ef0d7bd89b7178c3191b1a1abcef8f2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.341ex; height:2.843ex;" alt="{\displaystyle f''+2f'-3f=4x-1}"></span> may be rewritten in the form <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 L(f)=4x-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>4</mn> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L(f)=4x-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50cebd95780694c23ba41a854e4bc74c1b1ec6b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.264ex; height:2.843ex;" alt="{\displaystyle L(f)=4x-1}"></span>, where <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 L={\frac {d^{2}}{dx^{2}}}+2{\frac {d}{dx}}-3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L={\frac {d^{2}}{dx^{2}}}+2{\frac {d}{dx}}-3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02c07f6d64018425ec213a2156c6c4af1031f574" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:20.505ex; height:6.009ex;" alt="{\displaystyle L={\frac {d^{2}}{dx^{2}}}+2{\frac {d}{dx}}-3}"></span> is a <i>second order linear constant coefficient differential operator</i> acting on functions of <i>x</i>. The key idea here is that we consider a particular <a href="/wiki/Linear_combination" title="Linear combination">linear combination</a> of zeroth, first and second order derivatives "all at once". This allows us to think of the set of solutions of this differential equation as a "generalized antiderivative" of its right hand side 4<i>x</i> − 1, by analogy with ordinary <a href="/wiki/Integral" title="Integral">integration</a>, and formally write <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 f(x)=L^{-1}(4x-1).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>4</mn> <mi>x</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=L^{-1}(4x-1).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/186213116de4d3525543bde3754863b9d8c6913e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.383ex; height:3.176ex;" alt="{\displaystyle f(x)=L^{-1}(4x-1).}"></span> </p><p>Combining derivatives of different variables results in a notion of a <a href="/wiki/Partial_differential_operator" class="mw-redirect" title="Partial differential operator">partial differential operator</a>. The <a href="/wiki/Linear_operator" class="mw-redirect" title="Linear operator">linear operator</a> which assigns to each function its derivative is an example of a differential operator on a <a href="/wiki/Function_space" title="Function space">function space</a>. By means of the <a href="/wiki/Fourier_transform" title="Fourier transform">Fourier transform</a>, <a href="/wiki/Pseudo-differential_operator" title="Pseudo-differential operator">pseudo-differential operators</a> can be defined which allow for fractional calculus. </p><p>Some of these operators are so important that they have their own names: </p> <ul><li>The <a href="/wiki/Laplace_operator" title="Laplace operator">Laplace operator</a> or Laplacian on <b>R</b><sup>3</sup> is a second-order partial differential operator <span class="texhtml">Δ</span> given by the <a href="/wiki/Divergence" title="Divergence">divergence</a> of the <a href="/wiki/Gradient" title="Gradient">gradient</a> of a scalar function of three variables, or explicitly 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 \Delta ={\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}+{\frac {\partial ^{2}}{\partial z^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta ={\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}+{\frac {\partial ^{2}}{\partial z^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f575b1beb7d81d5ba8a6e0f3b8a66039b5640265" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:24.568ex; height:6.343ex;" alt="{\displaystyle \Delta ={\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}+{\frac {\partial ^{2}}{\partial z^{2}}}.}"></span> Analogous operators can be defined for functions of any number of variables.</li> <li>The <a href="/wiki/D%27Alembertian" class="mw-redirect" title="D'Alembertian">d'Alembertian</a> or wave operator is similar to the Laplacian, but acts on functions of four variables. Its definition uses the indefinite <a href="/wiki/Metric_tensor" title="Metric tensor">metric tensor</a> of <a href="/wiki/Minkowski_space" title="Minkowski space">Minkowski space</a>, instead of the <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean</a> <a href="/wiki/Dot_product" title="Dot product">dot product</a> of <b>R</b><sup>3</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 \square ={\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}+{\frac {\partial ^{2}}{\partial z^{2}}}-{\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>◻</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \square ={\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}+{\frac {\partial ^{2}}{\partial z^{2}}}-{\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/806426dfe9ae9d2c802a2f207e2af14d68622152" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:34.226ex; height:6.343ex;" alt="{\displaystyle \square ={\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}+{\frac {\partial ^{2}}{\partial z^{2}}}-{\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}.}"></span></li> <li>The <a href="/wiki/Schwarzian_derivative" title="Schwarzian derivative">Schwarzian derivative</a> is a non-linear differential operator which describes how a complex function is approximated by a <a href="/wiki/Fractional-linear_map" class="mw-redirect" title="Fractional-linear map">fractional-linear map</a>, in much the same way that a normal derivative describes how a function is approximated by a linear map.</li> <li>The <a href="/wiki/Wirtinger_derivatives" title="Wirtinger derivatives">Wirtinger derivatives</a> are a set of differential operators that permit the construction of a differential calculus for complex functions that is entirely analogous to the ordinary differential calculus for functions of real variables.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Other_generalizations">Other generalizations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Generalizations_of_the_derivative&action=edit&section=13" title="Edit section: Other generalizations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Functional_analysis" title="Functional analysis">functional analysis</a>, the <a href="/wiki/Functional_derivative" title="Functional derivative">functional derivative</a> defines the derivative with respect to a function of a functional on a space of functions. This is an extension of the directional derivative to an infinite <a href="/wiki/Dimension" title="Dimension">dimensional</a> vector space. An important case is the variational derivative in the <a href="/wiki/Calculus_of_variations" title="Calculus of variations">calculus of variations</a>. </p><p>The <a href="/wiki/Subderivative" title="Subderivative">subderivative</a> and <a href="/wiki/Subgradient" class="mw-redirect" title="Subgradient">subgradient</a> are generalizations of the derivative to <a href="/wiki/Convex_function" title="Convex function">convex functions</a> used in convex analysis. </p><p>In <a href="/wiki/Commutative_algebra" title="Commutative algebra">commutative algebra</a>, <a href="/wiki/K%C3%A4hler_differential" title="Kähler differential">Kähler differentials</a> are universal derivations of a <a href="/wiki/Commutative_ring" title="Commutative ring">commutative ring</a> or <a href="/wiki/Module_(algebra)" class="mw-redirect" title="Module (algebra)">module</a>. They can be used to define an analogue of exterior derivative from differential geometry that applies to arbitrary <a href="/wiki/Algebraic_varieties" class="mw-redirect" title="Algebraic varieties">algebraic varieties</a>, instead of just smooth manifolds. </p><p>In <a href="/wiki/P-adic_analysis" title="P-adic analysis">p-adic analysis</a>, the usual definition of derivative is not quite strong enough, and one requires <a href="/wiki/Strictly_differentiable" class="mw-redirect" title="Strictly differentiable">strict differentiability</a> instead. </p><p>The <a href="/wiki/Gateaux_derivative" title="Gateaux derivative">Gateaux derivative</a> extends the Fréchet derivative to <a href="/wiki/Locally_convex" class="mw-redirect" title="Locally convex">locally convex</a> <a href="/wiki/Topological_vector_space" title="Topological vector space">topological vector spaces</a>. Fréchet differentiability is a strictly stronger condition than Gateaux differentiability, even in finite dimensions. Between the two extremes is the <a href="/wiki/Quasi-derivative" title="Quasi-derivative">quasi-derivative</a>. </p><p>In <a href="/wiki/Measure_theory" class="mw-redirect" title="Measure theory">measure theory</a>, the <a href="/wiki/Radon%E2%80%93Nikodym_derivative" class="mw-redirect" title="Radon–Nikodym derivative">Radon–Nikodym derivative</a> generalizes the <a href="/wiki/Jacobian_matrix_and_determinant" title="Jacobian matrix and determinant">Jacobian</a>, used for changing variables, to measures. It expresses one measure μ in terms of another measure ν (under certain conditions). </p><p>The <a href="/wiki/H-derivative" title="H-derivative"><i>H</i>-derivative</a> is a notion of derivative in the study of <a href="/wiki/Abstract_Wiener_space" title="Abstract Wiener space">abstract Wiener spaces</a> and the <a href="/wiki/Malliavin_calculus" title="Malliavin calculus">Malliavin calculus</a>. It is used in the study of <a href="/wiki/Stochastic_processes" class="mw-redirect" title="Stochastic processes">stochastic processes</a>. </p><p>Laplacians and differential equations using the Laplacian can be <a href="/wiki/Analysis_on_fractals" title="Analysis on fractals">defined on fractals</a>. There is no completely satisfactory analog of the first-order derivative or gradient.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p><span class="anchor" id="Analogues_of_derivatives_in_fields_of_positive_characteristic"></span> The <b>Carlitz derivative</b> is an operation similar to usual differentiation but with the usual context of real or complex numbers changed to <a href="/wiki/Local_fields" class="mw-redirect" title="Local fields">local fields</a> of positive <a href="/wiki/Characteristic_(algebra)" title="Characteristic (algebra)">characteristic</a> in the form of <a href="/wiki/Formal_Laurent_series" class="mw-redirect" title="Formal Laurent series">formal Laurent series</a> with coefficients in some <a href="/wiki/Finite_field" title="Finite field">finite field</a> F<sub><i>q</i></sub> (it is known that any local field of positive characteristic is isomorphic to a Laurent series field). Along with suitably defined analogs to the <a href="/wiki/Exponential_function" title="Exponential function">exponential function</a>, <a href="/wiki/Logarithms" class="mw-redirect" title="Logarithms">logarithms</a> and others the derivative can be used to develop notions of smoothness, analycity, integration, Taylor series as well as a theory of differential equations.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>It may be possible to combine two or more of the above different notions of extension or abstraction of the original derivative. For example, in <a href="/wiki/Finsler_geometry" class="mw-redirect" title="Finsler geometry">Finsler geometry</a>, one studies spaces which look <a href="/wiki/Locally" class="mw-redirect" title="Locally">locally</a> like <a href="/wiki/Banach_space" title="Banach space">Banach spaces</a>. Thus one might want a derivative with some of the features of a <a href="/wiki/Functional_derivative" title="Functional derivative">functional derivative</a> and the <a href="/wiki/Covariant_derivative" title="Covariant derivative">covariant derivative</a>. </p><p>Multiplicative calculus replaces addition with multiplication, and hence rather than dealing with the limit of a ratio of differences, it deals with the limit of an exponentiation of ratios. This allows the development of the geometric derivative and bigeometric derivative. Moreover, just like the classical differential operator has a discrete analog, the difference operator, there are also <a href="/wiki/List_of_derivatives_and_integrals_in_alternative_calculi" title="List of derivatives and integrals in alternative calculi">discrete analogs of these multiplicative derivatives</a>. </p> <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=Generalizations_of_the_derivative&action=edit&section=14" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Arithmetic_derivative" title="Arithmetic derivative">Arithmetic derivative</a> – Function defined on integers in number theory</li> <li><a href="/wiki/Automatic_differentiation" title="Automatic differentiation">Automatic differentiation</a> – Numerical calculations carrying along derivatives</li> <li><a href="/wiki/Brzozowski_derivative" title="Brzozowski derivative">Brzozowski derivative</a> – Function defined on formal languages in computer science</li> <li><a href="/wiki/Dini_derivative" title="Dini derivative">Dini derivative</a> – Class of generalisations of the derivative</li> <li><a href="/wiki/Fractal_derivative" title="Fractal derivative">Fractal derivative</a> – Generalization of derivative to fractals</li> <li><a href="/wiki/Hasse_derivative" title="Hasse derivative">Hasse derivative</a> – Mathematical concept</li> <li><a href="/wiki/Logarithmic_derivative" title="Logarithmic derivative">Logarithmic derivative</a> – Mathematical operation in calculus</li> <li><a href="/wiki/Logarithmic_differentiation" title="Logarithmic differentiation">Logarithmic differentiation</a> – Method of mathematical differentiation</li> <li><a href="/wiki/Non-classical_analysis" class="mw-redirect" title="Non-classical analysis">Non-classical analysis</a> – Branch of mathematics<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li> <li><a href="/wiki/Numerical_differentiation" title="Numerical differentiation">Numerical differentiation</a> – Use of numerical analysis to estimate derivatives of functions</li> <li><a href="/wiki/Pincherle_derivative" title="Pincherle derivative">Pincherle derivative</a> – Type of derivative of a linear operator</li> <li><a href="/wiki/Q-derivative" title="Q-derivative">q-derivative</a> – Q-analog of the ordinary derivative</li> <li><a href="/wiki/Semi-differentiability" title="Semi-differentiability">Semi-differentiability</a></li> <li><a href="/wiki/Symmetric_derivative" title="Symmetric derivative">Symmetric derivative</a> – generalization of the derivative<span style="display:none" class="category-wikidata-fallback-annotation">Pages displaying wikidata descriptions as a fallback</span></li> <li><a href="/wiki/Topological_derivative" title="Topological derivative">Topological derivative</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=Generalizations_of_the_derivative&action=edit&section=15" 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"><a href="/wiki/David_Hestenes" title="David Hestenes">David Hestenes</a>, Garrett Sobczyk: Clifford Algebra to Geometric Calculus, a Unified Language for mathematics and Physics (Dordrecht/Boston:G.Reidel Publ.Co., 1984, <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/90-277-2561-6" title="Special:BookSources/90-277-2561-6">90-277-2561-6</a></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="CITEREFHahn1949" class="citation journal cs1"><a href="/wiki/Wolfgang_Hahn" title="Wolfgang Hahn">Hahn, Wolfgang</a> (1949). "Über Orthogonalpolynome, die q-Differenzengleichungen genügen". <i><a href="/wiki/Mathematische_Nachrichten" title="Mathematische Nachrichten">Mathematische Nachrichten</a></i>. <b>2</b> (1–2): 4–34. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fmana.19490020103">10.1002/mana.19490020103</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0025-584X">0025-584X</a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0030647">0030647</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematische+Nachrichten&rft.atitle=%C3%9Cber+Orthogonalpolynome%2C+die+q-Differenzengleichungen+gen%C3%BCgen&rft.volume=2&rft.issue=1%E2%80%932&rft.pages=4-34&rft.date=1949&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0030647%23id-name%3DMR&rft.issn=0025-584X&rft_id=info%3Adoi%2F10.1002%2Fmana.19490020103&rft.aulast=Hahn&rft.aufirst=Wolfgang&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneralizations+of+the+derivative" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://www.ams.org/notices/199910/fea-strichartz.pdf">Analysis on Fractals</a>, Robert S. Strichartz - Article in Notices of the AMS</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKochubei2009" class="citation book cs1">Kochubei, Anatoly N. (2009). <i>Analysis in Positive Characteristic</i>. New York: Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-50977-0" title="Special:BookSources/978-0-521-50977-0"><bdi>978-0-521-50977-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Analysis+in+Positive+Characteristic&rft.place=New+York&rft.pub=Cambridge+University+Press&rft.date=2009&rft.isbn=978-0-521-50977-0&rft.aulast=Kochubei&rft.aufirst=Anatoly+N.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneralizations+of+the+derivative" class="Z3988"></span></span> </li> </ol></div></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 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.navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Analysis_in_topological_vector_spaces" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Analysis_in_topological_vector_spaces" title="Template:Analysis in topological vector spaces"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Analysis_in_topological_vector_spaces" title="Template talk:Analysis in topological vector spaces"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Analysis_in_topological_vector_spaces" title="Special:EditPage/Template:Analysis in topological vector spaces"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Analysis_in_topological_vector_spaces" style="font-size:114%;margin:0 4em"><a href="/wiki/Mathematical_analysis" title="Mathematical analysis">Analysis</a> in <a href="/wiki/Topological_vector_space" title="Topological vector space">topological vector spaces</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Basic concepts</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/Abstract_Wiener_space" title="Abstract Wiener space">Abstract Wiener space</a> <ul><li><a href="/wiki/Classical_Wiener_space" title="Classical Wiener space">Classical Wiener space</a></li></ul></li> <li><a href="/wiki/Bochner_space" title="Bochner space">Bochner space</a></li> <li><a href="/wiki/Convex_series" title="Convex series">Convex series</a></li> <li><a href="/wiki/Cylinder_set_measure" title="Cylinder set measure">Cylinder set measure</a></li> <li><a href="/wiki/Infinite-dimensional_vector_function" title="Infinite-dimensional vector function">Infinite-dimensional vector function</a></li> <li><a href="/wiki/Matrix_calculus" title="Matrix calculus">Matrix calculus</a></li> <li><a href="/wiki/Vector_calculus" title="Vector calculus">Vector calculus</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Derivative" title="Derivative">Derivatives</a></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/Differentiable_vector%E2%80%93valued_functions_from_Euclidean_space" title="Differentiable vector–valued functions from Euclidean space">Differentiable vector–valued functions from Euclidean space</a></li> <li><a href="/wiki/Differentiation_in_Fr%C3%A9chet_spaces" title="Differentiation in Fréchet spaces">Differentiation in Fréchet spaces</a></li> <li><a href="/wiki/Fr%C3%A9chet_derivative" title="Fréchet derivative">Fréchet derivative</a> <ul><li><a href="/wiki/Total_derivative" title="Total derivative">Total</a></li></ul></li> <li><a href="/wiki/Functional_derivative" title="Functional derivative">Functional derivative</a></li> <li><a href="/wiki/Gateaux_derivative" title="Gateaux derivative">Gateaux derivative</a> <ul><li><a href="/wiki/Directional_derivative" title="Directional derivative">Directional</a></li></ul></li> <li><a class="mw-selflink selflink">Generalizations of the derivative</a></li> <li><a href="/wiki/Hadamard_derivative" title="Hadamard derivative">Hadamard derivative</a></li> <li><a href="/wiki/Infinite-dimensional_holomorphy" title="Infinite-dimensional holomorphy">Holomorphic</a></li> <li><a href="/wiki/Quasi-derivative" title="Quasi-derivative">Quasi-derivative</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Measurability</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/Besov_measure" title="Besov measure">Besov measure</a></li> <li><a href="/wiki/Cylinder_set_measure" title="Cylinder set measure">Cylinder set measure</a> <ul><li><a href="/wiki/Canonical_Gaussian_cylinder_set_measure" class="mw-redirect" title="Canonical Gaussian cylinder set measure">Canonical Gaussian</a></li> <li><a href="/wiki/Classical_Wiener_measure" class="mw-redirect" title="Classical Wiener measure">Classical Wiener measure</a></li></ul></li> <li><a href="/wiki/Measure_(mathematics)" title="Measure (mathematics)">Measure</a> like <a href="/wiki/Set_function" title="Set function">set functions</a> <ul><li><a href="/wiki/Gaussian_measure#Infinite-dimensional_spaces" title="Gaussian measure">infinite-dimensional Gaussian measure</a></li> <li><a href="/wiki/Projection-valued_measure" title="Projection-valued measure">Projection-valued</a></li> <li><a href="/wiki/Vector_measure" title="Vector measure">Vector</a></li></ul></li> <li><a href="/wiki/Bochner_measurable_function" title="Bochner measurable function">Bochner</a> / <a href="/wiki/Weakly_measurable_function" title="Weakly measurable function">Weakly</a> / <a href="/wiki/Strongly_measurable_function" title="Strongly measurable function">Strongly</a> <a href="/wiki/Measurable_function" title="Measurable function">measurable function</a></li> <li><a href="/wiki/Radonifying_function" title="Radonifying function">Radonifying function</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Integral" title="Integral">Integrals</a></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/Bochner_integral" title="Bochner integral">Bochner</a></li> <li><a href="/wiki/Direct_integral" title="Direct integral">Direct integral</a></li> <li><a href="/wiki/Dunford_integral" class="mw-redirect" title="Dunford integral">Dunford</a></li> <li><a href="/wiki/Pettis_integral" title="Pettis integral">Gelfand–Pettis/Weak</a></li> <li><a href="/wiki/Regulated_integral" title="Regulated integral">Regulated</a></li> <li><a href="/wiki/Paley%E2%80%93Wiener_integral" title="Paley–Wiener integral">Paley–Wiener</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Results</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/Cameron%E2%80%93Martin_theorem" title="Cameron–Martin theorem">Cameron–Martin theorem</a></li> <li><a href="/wiki/Inverse_function_theorem" title="Inverse function theorem">Inverse function theorem</a> <ul><li><a href="/wiki/Nash%E2%80%93Moser_theorem" title="Nash–Moser theorem">Nash–Moser theorem</a></li></ul></li> <li><a href="/wiki/Feldman%E2%80%93H%C3%A1jek_theorem" title="Feldman–Hájek theorem">Feldman–Hájek theorem</a></li> <li><a href="/wiki/Infinite-dimensional_Lebesgue_measure" title="Infinite-dimensional Lebesgue measure">No infinite-dimensional Lebesgue measure</a></li> <li><a href="/wiki/Sazonov%27s_theorem" title="Sazonov's theorem">Sazonov's theorem</a></li> <li><a href="/wiki/Structure_theorem_for_Gaussian_measures" title="Structure theorem for Gaussian measures">Structure theorem for Gaussian measures</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</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/Crinkled_arc" title="Crinkled arc">Crinkled arc</a></li> <li><a href="/wiki/Covariance_operator" title="Covariance operator">Covariance operator</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Functional_calculus" title="Functional calculus">Functional calculus</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/Borel_functional_calculus" title="Borel functional calculus">Borel functional calculus</a></li> <li><a href="/wiki/Continuous_functional_calculus" title="Continuous functional calculus">Continuous functional calculus</a></li> <li><a href="/wiki/Holomorphic_functional_calculus" title="Holomorphic functional calculus">Holomorphic functional calculus</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Applications</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/Banach_manifold" title="Banach manifold">Banach manifold</a> (<a href="/wiki/Banach_bundle" title="Banach bundle">bundle</a>)</li> <li><a href="/wiki/Convenient_vector_space" title="Convenient vector space">Convenient vector space</a></li> <li><a href="/wiki/Choquet_theory" title="Choquet theory">Choquet theory</a></li> <li><a href="/wiki/Fr%C3%A9chet_manifold" title="Fréchet manifold">Fréchet manifold</a></li> <li><a href="/wiki/Hilbert_manifold" title="Hilbert manifold">Hilbert manifold</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Generalizations_of_the_derivative&oldid=1217657712">https://en.wikipedia.org/w/index.php?title=Generalizations_of_the_derivative&oldid=1217657712</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" 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