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style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Differentiable maps</title></head> <body> <blockquote> <p>See also <em><a class="existingWikiWord" href="/nlab/show/differentiation">differentiation</a></em> and <em><a class="existingWikiWord" href="/nlab/show/derivative">derivative</a></em>.</p> </blockquote> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="differential_geometry">Differential geometry</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/synthetic+differential+geometry">synthetic</a> <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a></strong></p> <p><strong>Introductions</strong></p> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology+--+1">from point-set topology to differentiable manifolds</a></p> <p><a class="existingWikiWord" href="/nlab/show/geometry+of+physics">geometry of physics</a>: <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+coordinate+systems">coordinate systems</a>, <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+smooth+spaces">smooth spaces</a>, <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+manifolds+and+orbifolds">manifolds</a>, <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+smooth+homotopy+types">smooth homotopy types</a>, <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+supergeometry">supergeometry</a></p> <p><strong>Differentials</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differentiation">differentiation</a>, <a class="existingWikiWord" href="/nlab/show/chain+rule">chain rule</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differentiable+function">differentiable function</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infinitesimal+space">infinitesimal space</a>, <a class="existingWikiWord" href="/nlab/show/infinitesimally+thickened+point">infinitesimally thickened point</a>, <a class="existingWikiWord" href="/nlab/show/amazing+right+adjoint">amazing right adjoint</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/V-manifolds">V-manifolds</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differentiable+manifold">differentiable manifold</a>, <a class="existingWikiWord" href="/nlab/show/coordinate+chart">coordinate chart</a>, <a class="existingWikiWord" href="/nlab/show/atlas">atlas</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a>, <a class="existingWikiWord" href="/nlab/show/smooth+structure">smooth structure</a>, <a class="existingWikiWord" href="/nlab/show/exotic+smooth+structure">exotic smooth structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/analytic+manifold">analytic manifold</a>, <a class="existingWikiWord" href="/nlab/show/complex+manifold">complex manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/formal+smooth+manifold">formal smooth manifold</a>, <a class="existingWikiWord" href="/nlab/show/derived+smooth+manifold">derived smooth manifold</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/smooth+space">smooth space</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/diffeological+space">diffeological space</a>, <a class="existingWikiWord" href="/nlab/show/Fr%C3%B6licher+space">Frölicher space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/manifold+structure+of+mapping+spaces">manifold structure of mapping spaces</a></p> </li> </ul> <p><strong>Tangency</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a>, <a class="existingWikiWord" href="/nlab/show/frame+bundle">frame bundle</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/vector+field">vector field</a>, <a class="existingWikiWord" href="/nlab/show/multivector+field">multivector field</a>, <a class="existingWikiWord" href="/nlab/show/tangent+Lie+algebroid">tangent Lie algebroid</a>;</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+forms+in+synthetic+differential+geometry">differential forms</a>, <a class="existingWikiWord" href="/nlab/show/de+Rham+complex">de Rham complex</a>, <a class="existingWikiWord" href="/nlab/show/Dolbeault+complex">Dolbeault complex</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/pullback+of+differential+forms">pullback of differential forms</a>, <a class="existingWikiWord" href="/nlab/show/invariant+differential+form">invariant differential form</a>, <a class="existingWikiWord" href="/nlab/show/Maurer-Cartan+form">Maurer-Cartan form</a>, <a class="existingWikiWord" href="/nlab/show/horizontal+differential+form">horizontal differential form</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cogerm+differential+form">cogerm differential form</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integration+of+differential+forms">integration of differential forms</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+diffeomorphism">local diffeomorphism</a>, <a class="existingWikiWord" href="/nlab/show/formally+%C3%A9tale+morphism">formally étale morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/submersion">submersion</a>, <a class="existingWikiWord" href="/nlab/show/formally+smooth+morphism">formally smooth morphism</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/immersion">immersion</a>, <a class="existingWikiWord" href="/nlab/show/formally+unramified+morphism">formally unramified morphism</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/de+Rham+space">de Rham space</a>, <a class="existingWikiWord" href="/nlab/show/crystal">crystal</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infinitesimal+disk+bundle">infinitesimal disk bundle</a></p> </li> </ul> <p><strong>The magic algebraic facts</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/embedding+of+smooth+manifolds+into+formal+duals+of+R-algebras">embedding of smooth manifolds into formal duals of R-algebras</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+Serre-Swan+theorem">smooth Serre-Swan theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derivations+of+smooth+functions+are+vector+fields">derivations of smooth functions are vector fields</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hadamard+lemma">Hadamard lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Borel%27s+theorem">Borel's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Boman%27s+theorem">Boman's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitney+extension+theorem">Whitney extension theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Steenrod-Wockel+approximation+theorem">Steenrod-Wockel approximation theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitney+embedding+theorem">Whitney embedding theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Poincare+lemma">Poincare lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Stokes+theorem">Stokes theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/de+Rham+theorem">de Rham theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hochschild-Kostant-Rosenberg+theorem">Hochschild-Kostant-Rosenberg theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cohomology+hexagon">differential cohomology hexagon</a></p> </li> </ul> <p><strong>Axiomatics</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Kock-Lawvere+axiom">Kock-Lawvere axiom</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+topos">smooth topos</a>, <a class="existingWikiWord" href="/nlab/show/super+smooth+topos">super smooth topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/microlinear+space">microlinear space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integration+axiom">integration axiom</a></p> </li> </ul> </li> </ul> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/cohesion">cohesion</a></strong></p> <ul> <li> <p>(<a class="existingWikiWord" href="/nlab/show/shape+modality">shape modality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/flat+modality">flat modality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/sharp+modality">sharp modality</a>)</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="0em" rspace="thinmathspace">ʃ</mo><mo>⊣</mo><mo>♭</mo><mo>⊣</mo><mo>♯</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\esh \dashv \flat \dashv \sharp )</annotation></semantics></math></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/discrete+object">discrete object</a>, <a class="existingWikiWord" href="/nlab/show/codiscrete+object">codiscrete object</a>, <a class="existingWikiWord" href="/nlab/show/concrete+object">concrete object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/points-to-pieces+transform">points-to-pieces transform</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohesive+%28infinity%2C1%29-topos+--+structures">structures in cohesion</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dR-shape+modality">dR-shape modality</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/dR-flat+modality">dR-flat modality</a></p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="0em" rspace="thinmathspace">ʃ</mo> <mi>dR</mi></msub><mo>⊣</mo><msub><mo>♭</mo> <mi>dR</mi></msub></mrow><annotation encoding="application/x-tex">\esh_{dR} \dashv \flat_{dR}</annotation></semantics></math></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/infinitesimal+cohesion">infinitesimal cohesion</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/classical+modality">classical modality</a></li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/tangent+cohesive+%28%E2%88%9E%2C1%29-topos">tangent cohesion</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/differential+cohomology+diagram">differential cohomology diagram</a></li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/differential+cohesion">differential cohesion</a></strong></p> <ul> <li> <p>(<a class="existingWikiWord" href="/nlab/show/reduction+modality">reduction modality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/infinitesimal+shape+modality">infinitesimal shape modality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/infinitesimal+flat+modality">infinitesimal flat modality</a>)</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>ℜ</mi><mo>⊣</mo><mi>ℑ</mi><mo>⊣</mo><mi>&</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\Re \dashv \Im \dashv \&)</annotation></semantics></math></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/reduced+object">reduced object</a>, <a class="existingWikiWord" href="/nlab/show/coreduced+object">coreduced object</a>, <a class="existingWikiWord" href="/nlab/show/formally+smooth+object">formally smooth object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/formally+%C3%A9tale+map">formally étale map</a></p> </li> <li> <p><a href="cohesive+%28infinity%2C1%29-topos+--+infinitesimal+cohesion#StructuresInDifferentialCohesion">structures in differential cohesion</a></p> </li> </ul> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/super+smooth+infinity-groupoid">graded differential cohesion</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fermionic+modality">fermionic modality</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/bosonic+modality">bosonic modality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/rheonomy+modality">rheonomy modality</a></p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo>⇉</mo><mo>⊣</mo><mo>⇝</mo><mo>⊣</mo><mi>Rh</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)</annotation></semantics></math></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/orbifold+cohomology">singular cohesion</a></strong></p> <div id="Diagram" class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>id</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi>id</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>fermionic</mi></mover></mtd> <mtd><mo>⇉</mo></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>⇝</mo></mtd> <mtd><mover><mrow></mrow><mi>bosonic</mi></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>bosonic</mi></mover></mtd> <mtd><mo>⇝</mo></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi mathvariant="normal">R</mi><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mi mathvariant="normal">h</mi></mtd> <mtd><mover><mrow></mrow><mi>rheonomic</mi></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>reduced</mi></mover></mtd> <mtd><mi>ℜ</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi>ℑ</mi></mtd> <mtd><mover><mrow></mrow><mi>infinitesimal</mi></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>infinitesimal</mi></mover></mtd> <mtd><mi>ℑ</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi>&</mi></mtd> <mtd><mover><mrow></mrow><mtext>étale</mtext></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>cohesive</mi></mover></mtd> <mtd><mo lspace="0em" rspace="thinmathspace">ʃ</mo></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>♭</mo></mtd> <mtd><mover><mrow></mrow><mi>discrete</mi></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>discrete</mi></mover></mtd> <mtd><mo>♭</mo></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>♯</mo></mtd> <mtd><mover><mrow></mrow><mi>continuous</mi></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>∅</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>*</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{&#233;tale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast } </annotation></semantics></math></div></div> <p id="models_2"><strong>Models</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Models+for+Smooth+Infinitesimal+Analysis">Models for Smooth Infinitesimal Analysis</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+algebra">smooth algebra</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup></mrow><annotation encoding="application/x-tex">C^\infty</annotation></semantics></math>-ring)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+locus">smooth locus</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fermat+theory">Fermat theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cahiers+topos">Cahiers topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth ∞-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/formal+smooth+%E2%88%9E-groupoid">formal smooth ∞-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+formal+smooth+%E2%88%9E-groupoid">super formal smooth ∞-groupoid</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/Lie+theory">Lie theory</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+theory">∞-Lie theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/Lie+n-algebra">Lie n-algebra</a>, <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra">L-∞ algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a>, <a class="existingWikiWord" href="/nlab/show/Lie+2-group">Lie 2-group</a>, <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-group">smooth ∞-group</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/differential+equations">differential equations</a>, <a class="existingWikiWord" href="/nlab/show/variational+calculus">variational calculus</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/D-geometry">D-geometry</a>, <a class="existingWikiWord" href="/nlab/show/D-module">D-module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/jet+bundle">jet bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/variational+bicomplex">variational bicomplex</a>, <a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+complex">Euler-Lagrange complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+equation">Euler-Lagrange equation</a>, <a class="existingWikiWord" href="/nlab/show/de+Donder-Weyl+formalism">de Donder-Weyl formalism</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Weil+theory">∞-Chern-Weil theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connection on a bundle</a>, <a class="existingWikiWord" href="/nlab/show/connection+on+an+%E2%88%9E-bundle">connection on an ∞-bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/ordinary+differential+cohomology">ordinary differential cohomology</a>, <a class="existingWikiWord" href="/nlab/show/Deligne+complex">Deligne complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+K-theory">differential K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cobordism+cohomology">differential cobordism cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/parallel+transport">parallel transport</a>, <a class="existingWikiWord" href="/nlab/show/higher+parallel+transport">higher parallel transport</a>, <a class="existingWikiWord" href="/nlab/show/fiber+integration+in+differential+cohomology">fiber integration in differential cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/holonomy">holonomy</a>, <a class="existingWikiWord" href="/nlab/show/higher+holonomy">higher holonomy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a>, <a class="existingWikiWord" href="/nlab/show/higher+gauge+theory">higher gauge theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wilson+line">Wilson line</a>, <a class="existingWikiWord" href="/nlab/show/Wilson+surface">Wilson surface</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/Cartan+geometry">Cartan geometry</a> (<a class="existingWikiWord" href="/nlab/show/super+Cartan+geometry">super</a>, <a class="existingWikiWord" href="/nlab/show/higher+Cartan+geometry">higher</a>)</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Klein+geometry">Klein geometry</a>, (<a class="existingWikiWord" href="/nlab/show/higher+Klein+geometry">higher</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/G-structure">G-structure</a>, <a class="existingWikiWord" href="/nlab/show/torsion+of+a+G-structure">torsion of a G-structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Euclidean+geometry">Euclidean geometry</a>, <a class="existingWikiWord" href="/nlab/show/hyperbolic+geometry">hyperbolic geometry</a>, <a class="existingWikiWord" href="/nlab/show/elliptic+geometry">elliptic geometry</a></p> </li> <li> <p>(<a class="existingWikiWord" href="/nlab/show/pseudo-Riemannian+geometry">pseudo</a>-)<a class="existingWikiWord" href="/nlab/show/Riemannian+geometry">Riemannian geometry</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/orthogonal+structure">orthogonal structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/isometry">isometry</a>, <a class="existingWikiWord" href="/nlab/show/Killing+vector+field">Killing vector field</a>, <a class="existingWikiWord" href="/nlab/show/Killing+spinor">Killing spinor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a>, <a class="existingWikiWord" href="/nlab/show/super-spacetime">super-spacetime</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex+geometry">complex geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conformal+geometry">conformal geometry</a></p> </li> </ul> </div></div> </div> </div> <h1 id="differentiable_maps">Differentiable maps</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definitions'>Definitions</a></li> <ul> <li><a href='#in_the_real_numbers'>In the real numbers</a></li> <ul> <li><a href='#epsilondelta_definition'>Epsilon-delta definition</a></li> <li><a href='#infinitesimal_definition'>Infinitesimal definition</a></li> </ul> <li><a href='#from_a_cartesian_space_to_the_real_numbers'>From a Cartesian space to the real numbers</a></li> <li><a href='#partial_and_directional_derivatives'>Partial and directional derivatives</a></li> <li><a href='#functions_between_cartesian_spaces'>Functions between Cartesian spaces</a></li> <li><a href='#functions_between_differentiable_manifolds'>Functions between differentiable manifolds</a></li> </ul> <li><a href='#higher_differentiability'>Higher differentiability</a></li> <ul> <li><a href='#iterated_differentiability'>Iterated differentiability</a></li> <li><a href='#uniform_differentiability'>Uniform differentiability</a></li> <li><a href='#symmetry'>Symmetry of higher derivatives</a></li> <li><a href='#a_direct_definition_of_the_second_derivative'>A direct definition of the second derivative</a></li> <li><a href='#higher_differentiability_on_tangent_spaces_and_the_chain_rule'>Higher differentiability on tangent spaces and the chain rule</a></li> <li><a href='#for_maps_between_manifolds'>For maps between manifolds</a></li> </ul> <li><a href='#examples_and_nonexamples'>Examples and non-examples</a></li> <ul> <li><a href='#DifferentiabilityOpposedToPartialDifferentiability'>Differentiability versus partial differentiability</a></li> <li><a href='#differentiability_versus_continuous_and_higher_differentiability'>Differentiability versus continuous and higher differentiability</a></li> <li><a href='#symmetry_of_the_second_partial_derivatives'>Symmetry of the second partial derivatives</a></li> <li><a href='#twice_differentiability_versus_quadratic_approximation'>Twice differentiability versus quadratic approximation</a></li> <li><a href='#the_second_derivative_as_a_quadratic_form'>The second derivative as a quadratic form</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>A <a class="existingWikiWord" href="/nlab/show/function">function</a> (map) is <em>differentiable</em> at some point if it can be well approximated by a <a class="existingWikiWord" href="/nlab/show/linear+map">linear map</a> near that point. The approximating linear maps at different points together form the <em>derivative</em> of the map.</p> <p>One may then ask whether the derivative itself is differentiable, and so on. This leads to a hierarchy of ever more differentiable maps, starting with <a class="existingWikiWord" href="/nlab/show/continuous+maps">continuous maps</a> and progressing through maps that are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> times (continuously) differentiable to those that are infinitely differentiable, and finally to those that are <a class="existingWikiWord" href="/nlab/show/analytic+map">analytic</a>. Infinitely differentiable maps are sometimes called <em>smooth</em>.</p> <p>Differentiability is first defined directly for maps between (open subsets of) a <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a>. These differentiable maps can then be used to define the notion of <a class="existingWikiWord" href="/nlab/show/differentiable+manifold">differentiable manifold</a>, and then a more general notion of differentiable map <em>between</em> differentiable manifolds, forming a <a class="existingWikiWord" href="/nlab/show/category">category</a> called <a class="existingWikiWord" href="/nlab/show/Diff">Diff</a>. We have a parallel hierarchy of ever more differentiable manifolds and ever more differentiable maps between them. Since every more differentiable manifold has an <a class="existingWikiWord" href="/nlab/show/underlying">underlying</a> less differentiable manifold, we may always consider maps that are less differentiable than the manifolds between which they run.</p> <h2 id="definitions">Definitions</h2> <p>In all of the following <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math> denotes the <a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math> their <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-fold <a class="existingWikiWord" href="/nlab/show/Cartesian+product">Cartesian product</a>. For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>∈</mo><mo stretchy="false">{</mo><mn>1</mn><mo>,</mo><mi>⋯</mi><mo>,</mo><mi>n</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">i \in \{1, \cdots, n\}</annotation></semantics></math> we write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>ℝ</mi> <mi>n</mi></msup></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>pr</mi> <mi>i</mi></msub></mrow></mover></mtd> <mtd><mi>ℝ</mi></mtd></mtr> <mtr><mtd><mover><mi>v</mi><mo stretchy="false">→</mo></mover><mo>=</mo><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>v</mi> <mi>n</mi></msub><mo stretchy="false">)</mo></mtd> <mtd><mo>↦</mo></mtd> <mtd><msub><mi>v</mi> <mi>i</mi></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathbb{R}^n &\overset{pr_i}{\longrightarrow}& \mathbb{R} \\ \vec v = (v_1, \cdots, v_n) &\mapsto& v_i } </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/projection">projection</a> map onto the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math>th factor.</p> <p>When considering <a class="existingWikiWord" href="/nlab/show/convergence">convergence</a> of <a class="existingWikiWord" href="/nlab/show/sequences">sequences</a> of elements of these sets we regard them as <a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean</a> <a class="existingWikiWord" href="/nlab/show/metric+spaces">metric spaces</a> with the <a class="existingWikiWord" href="/nlab/show/Euclidean+norm">Euclidean norm</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">‖</mo><mo>−</mo><mo stretchy="false">‖</mo></mrow><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>⟶</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo><mo>⊂</mo><mi>ℝ</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> {\Vert -\Vert} \;\colon\; \mathbb{R}^n \longrightarrow [0,\infty) \subset \mathbb{R} \,. </annotation></semantics></math></div> <p>The <a class="existingWikiWord" href="/nlab/show/open+subsets">open subsets</a> of the corresponding <a class="existingWikiWord" href="/nlab/show/metric+topology">metric topology</a> are the <a class="existingWikiWord" href="/nlab/show/unions">unions</a> of <a class="existingWikiWord" href="/nlab/show/open+balls">open balls</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math>.</p> <h3 id="in_the_real_numbers">In the real numbers</h3> <h4 id="epsilondelta_definition">Epsilon-delta definition</h4> <p>A function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>ℝ</mi><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">f:\mathbb{R} \to \mathbb{R}</annotation></semantics></math> is <strong>differentiable</strong> if it comes with a function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>d</mi><mi>f</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mo>:</mo><mi>ℝ</mi><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\frac{d f}{d x}:\mathbb{R} \to \mathbb{R}</annotation></semantics></math> and a function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>M</mi> <mi>f</mi></msub><mo>:</mo><msub><mi>ℚ</mi> <mo>+</mo></msub><mo>→</mo><msub><mi>ℚ</mi> <mo>+</mo></msub></mrow><annotation encoding="application/x-tex">M_f:\mathbb{Q}_+ \to \mathbb{Q}_+</annotation></semantics></math> in the positive rational numbers, such that</p> <ul> <li>for every positive rational number <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mo>∈</mo><msub><mi>ℚ</mi> <mo>+</mo></msub></mrow><annotation encoding="application/x-tex">\epsilon \in \mathbb{Q}_+</annotation></semantics></math>, for every real number <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>∈</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">h \in \mathbb{R}</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo><</mo><mo stretchy="false">|</mo><mi>h</mi><mo stretchy="false">|</mo><mo><</mo><msub><mi>M</mi> <mi>f</mi></msub><mo stretchy="false">(</mo><mi>ϵ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">0 \lt | h | \lt M_f(\epsilon)</annotation></semantics></math>, and for every real number <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">x \in \mathbb{R}</annotation></semantics></math>,<div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>|</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>h</mi><mo stretchy="false">)</mo><mo>−</mo><mfrac><mrow><mi>d</mi><mi>f</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>|</mo></mrow><mo><</mo><mi>ϵ</mi><mo stretchy="false">|</mo><mi>h</mi><mo stretchy="false">|</mo></mrow><annotation encoding="application/x-tex">\left|f(x + h) - \frac{d f}{d x}(x)\right| \lt \epsilon |h|</annotation></semantics></math></div></li> </ul> <h4 id="infinitesimal_definition">Infinitesimal definition</h4> <p>Given a predicate <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> on the real numbers <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math>, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> denote the set of all elements in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math> for which <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> holds. A <a class="existingWikiWord" href="/nlab/show/partial+function">partial function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>ℝ</mi><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">f:\mathbb{R} \to \mathbb{R}</annotation></semantics></math> is equivalently a function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>I</mi><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">f:I \to \mathbb{R}</annotation></semantics></math> for any such predicate <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> and set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math>.</p> <p>A function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>I</mi><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">f:I \to \mathbb{R}</annotation></semantics></math> is <strong>differentiable at a subset</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>⊆</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">S \subseteq I</annotation></semantics></math> with injection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi><mo>:</mo><mi>S</mi><mo>↪</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">j:S \hookrightarrow \mathbb{R}</annotation></semantics></math> if it has a function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>d</mi><mi>f</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mo>:</mo><mi>S</mi><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\frac{d f}{d x}:S \to \mathbb{R}</annotation></semantics></math> such that for all Archimedean ordered Artinian local <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>-algebras <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> with ring homomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>:</mo><mi>K</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">h:K \to A</annotation></semantics></math> and <a class="existingWikiWord" href="/nlab/show/nilradical">nilradical</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> such that for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mo>∈</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">\epsilon \in D</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ϵ</mi> <mn>2</mn></msup><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\epsilon^2 = 0</annotation></semantics></math>, for all nilpotent elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mo>∈</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">\epsilon \in D</annotation></semantics></math>,</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>+</mo><mi>ϵ</mi><mo stretchy="false">)</mo><mo>=</mo><mi>h</mi><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>+</mo><mi>h</mi><mrow><mo>(</mo><mfrac><mrow><mi>d</mi><mi>f</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>)</mo></mrow><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">f_A(h(j(a)) + \epsilon) = h(j(a)) + h\left(\frac{d f}{d x}(a)\right) \epsilon</annotation></semantics></math></div> <p>A function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>I</mi><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">f:I \to \mathbb{R}</annotation></semantics></math> is <strong>differentiable at an element</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">a \in I</annotation></semantics></math> if it is differentiable at the <a class="existingWikiWord" href="/nlab/show/singleton+subset">singleton subset</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>a</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{a\}</annotation></semantics></math>, and a function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>I</mi><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">f:I \to \mathbb{R}</annotation></semantics></math> is <strong>differentiable</strong> if it is differentiable at the <a class="existingWikiWord" href="/nlab/show/improper+subset">improper subset</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math>.</p> <h3 id="from_a_cartesian_space_to_the_real_numbers">From a Cartesian space to the real numbers</h3> <div class="num_defn" id="DerivativeOfFunctionFromRnToR"> <h6 id="definition">Definition</h6> <p><strong>(differentiation of real-valued functions on Cartesian space)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math> and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>⊂</mo><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">U \subset \mathbb{R}^n</annotation></semantics></math> be an <a class="existingWikiWord" href="/nlab/show/open+subset">open subset</a>.</p> <p>Then a <a class="existingWikiWord" href="/nlab/show/function">function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>U</mi><mo>⟶</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">f \;\colon\; U \longrightarrow \mathbb{R}</annotation></semantics></math> is called <strong>differentiable</strong> at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">x\in U</annotation></semantics></math> if there exists a <a class="existingWikiWord" href="/nlab/show/linear+map">linear map</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><msub><mi>f</mi> <mi>x</mi></msub><mo>:</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">d f_x : \mathbb{R}^n \to \mathbb{R}</annotation></semantics></math> such that the following <a class="existingWikiWord" href="/nlab/show/limit+of+a+sequence">limit</a> exists as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math> approaches zero “from all directions at once”:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mi>lim</mi> <mrow><mi>h</mi><mo>→</mo><mn>0</mn></mrow></munder><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><mo>−</mo><mi>d</mi><msub><mi>f</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">‖</mo><mi>h</mi><mo stretchy="false">‖</mo></mrow></mfrac><mo>=</mo><mn>0</mn><mo>.</mo></mrow><annotation encoding="application/x-tex"> \lim_{h\to 0} \frac{f(x+h)-f(x) - d f_x(h)}{\Vert h\Vert} = 0. </annotation></semantics></math></div> <p>This means that for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mo>∈</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\epsilon \in (0,\infty)</annotation></semantics></math> there exists an <a class="existingWikiWord" href="/nlab/show/open+subset">open subset</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>⊆</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">V\subseteq U</annotation></semantics></math> containing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> such that whenever <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>+</mo><mi>h</mi><mo>∈</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">x+h\in V</annotation></semantics></math> we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><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><mo>−</mo><mi>d</mi><msub><mi>f</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">‖</mo><mi>h</mi><mo stretchy="false">‖</mo></mrow></mfrac><mo><</mo><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">\frac{f(x+h)-f(x) - d f_x(h)}{\Vert h\Vert} \lt \epsilon</annotation></semantics></math>.</p> <p>We say that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is <strong>differentiable on</strong> a <a class="existingWikiWord" href="/nlab/show/subset">subset</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is differentiable at every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">x\in I</annotation></semantics></math>, and <strong>differentiable</strong> (tout court) if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is differentiable on all of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math>. We say that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is <strong>continuously differentiable</strong> if it is differentiable and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>f</mi></mrow><annotation encoding="application/x-tex">d f</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a>.</p> <p>The map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><msub><mi>f</mi> <mi>x</mi></msub></mrow><annotation encoding="application/x-tex">d f_x</annotation></semantics></math> is called the <strong>derivative</strong> or <strong>differential of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></strong>.</p> </div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p><strong>(Notation and Terminology)</strong></p> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n=1</annotation></semantics></math>, as in classical one-variable calculus, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><msub><mi>f</mi> <mi>x</mi></msub></mrow><annotation encoding="application/x-tex">d f_x</annotation></semantics></math> in def. <a class="maruku-ref" href="#DerivativeOfFunctionFromRnToR"></a> may be identified with a <a class="existingWikiWord" href="/nlab/show/real+number">real number</a>, and that number is also called the derivative of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> and often written <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f'(x)</annotation></semantics></math>. (In that case, the notation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>f</mi></mrow><annotation encoding="application/x-tex">d f</annotation></semantics></math> is generally still reserved for the corresponding linear map, with its input denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">d x</annotation></semantics></math>, so that we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>f</mi><mo>=</mo><mi>f</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>d</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">d f = f'(x) d x</annotation></semantics></math>.)</p> </div> <div class="num_remark"> <h6 id="remark_2">Remark</h6> <p><strong>(equivalent formulation)</strong></p> <p>An equivalent way to state def. <a class="maruku-ref" href="#DerivativeOfFunctionFromRnToR"></a> is to say that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><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><mo>+</mo><mi>d</mi><msub><mi>f</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">)</mo><mo>+</mo><mi>E</mi><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">)</mo><mrow><mo stretchy="false">‖</mo><mi>h</mi><mo stretchy="false">‖</mo></mrow></mrow><annotation encoding="application/x-tex"> f(x+h) = f(x) + d f_x(h) + E(h){\Vert h\Vert} </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> is a function such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>lim</mi> <mrow><mi>h</mi><mo>→</mo><mn>0</mn></mrow></msub><mi>E</mi><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\lim_{h\to 0}E(h) = 0</annotation></semantics></math>. This is easy to see; just let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">)</mo><mo>=</mo><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><mo>−</mo><mi>d</mi><msub><mi>f</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">‖</mo><mi>h</mi><mo stretchy="false">‖</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">E(h) = \frac{f(x+h)-f(x) - d f_x(h)}{\Vert h\Vert}</annotation></semantics></math>.</p> <p>Another equivalent way to say it is that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><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><mo>+</mo><mi>d</mi><msub><mi>f</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mi>E</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">)</mo><msub><mi>h</mi> <mn>1</mn></msub><mo>+</mo><mi>⋯</mi><mo>+</mo><msub><mi>E</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">)</mo><msub><mi>h</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex"> f(x+h) = f(x) + d f_x(h) + E_1(h) h_1 + \cdots + E_n(h) h_n </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">E_i</annotation></semantics></math> are functions such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>lim</mi> <mrow><mi>h</mi><mo>→</mo><mn>0</mn></mrow></msub><msub><mi>E</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\lim_{h\to 0}E_i(h) = 0</annotation></semantics></math>. For if this is true, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mrow><mo stretchy="false">‖</mo><mi>h</mi><mo stretchy="false">‖</mo></mrow></mfrac><mo stretchy="false">(</mo><msub><mi>E</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">)</mo><msub><mi>h</mi> <mn>1</mn></msub><mo>+</mo><mi>⋯</mi><mo>+</mo><msub><mi>E</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">)</mo><msub><mi>h</mi> <mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E(h) = \frac{1}{\Vert h\Vert}(E_1(h) h_1 + \cdots + E_n(h) h_n)</annotation></semantics></math> satisfies the previous definition. Conversely, if the previous definition holds, then defining <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><msub><mi>h</mi> <mi>i</mi></msub></mrow><mrow><mo stretchy="false">‖</mo><mi>h</mi><mo stretchy="false">‖</mo></mrow></mfrac><mi>E</mi><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E_i(h) = \frac{h_i}{\Vert h \Vert} E(h)</annotation></semantics></math> satisfies this definition.</p> </div> <h3 id="partial_and_directional_derivatives">Partial and directional derivatives</h3> <p>A weaker notion of differentiability is the following:</p> <div class="num_defn" id="DirectionalDerivative"> <h6 id="definition_2">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/partial+derivative">directional derivative</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>⊂</mo><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">U \subset \mathbb{R}^n </annotation></semantics></math> an <a class="existingWikiWord" href="/nlab/show/open+subset">open subset</a>.</p> <p>Then a <a class="existingWikiWord" href="/nlab/show/function">function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>U</mi><mo>⟶</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">f \colon U \longrightarrow \mathbb{R}</annotation></semantics></math> is said to have <strong>directional derivative</strong> in the direction of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi><mo>∈</mo><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">v \in \mathbb{R}^n</annotation></semantics></math> at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">x\in U</annotation></semantics></math> if the <a class="existingWikiWord" href="/nlab/show/limit+of+a+sequence">limit</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mi>lim</mi> <mrow><mi>h</mi><mo>→</mo><mn>0</mn></mrow></munder><mfrac><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>h</mi><mi>v</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><annotation encoding="application/x-tex"> \lim_{h\to 0} \frac{f(x+h v)- f(x)}{h} </annotation></semantics></math></div> <p>exists. Here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math> is just a real number.</p> </div> <p>Historically, the term ‘directional derivative’ was reserved for when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi></mrow><annotation encoding="application/x-tex">v</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/unit+vector">unit vector</a> (or divide the derivative above by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">‖</mo><mi>v</mi><mo stretchy="false">‖</mo></mrow><annotation encoding="application/x-tex">\|v\|</annotation></semantics></math>), but the general concept involves less structure and is more important but has no other established name. If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi></mrow><annotation encoding="application/x-tex">v</annotation></semantics></math> is a standard basis vector <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">e_i</annotation></semantics></math>, then the directional derivative is called a <strong><a class="existingWikiWord" href="/nlab/show/partial+derivative">partial derivative</a></strong> with respect to the corresponding coordinate, and often written <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mo>∂</mo><mi>f</mi></mrow><mrow><mo>∂</mo><msub><mi>x</mi> <mi>i</mi></msub></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\partial f}{\partial x_i}</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mrow><msub><mi>x</mi> <mi>i</mi></msub></mrow></msub></mrow><annotation encoding="application/x-tex">f_{x_i}</annotation></semantics></math>.</p> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is differentiable at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> in the sense of def. <a class="maruku-ref" href="#DerivativeOfFunctionFromRnToR"></a>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><msub><mi>f</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">d f_x(v)</annotation></semantics></math> is its directional derivative along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi></mrow><annotation encoding="application/x-tex">v</annotation></semantics></math>. In particular, the coordinates of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><msub><mi>f</mi> <mi>x</mi></msub></mrow><annotation encoding="application/x-tex">d f_x</annotation></semantics></math> are the partial derivatives of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>.</p> <p>In general, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> may have all partial derivatives, and even all directional derivatives, without being differentiable; see the examples <a href="#DifferentiabilityOpposedToPartialDifferentiability">below</a>. However, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> has all partial derivatives and they are <em>continuous</em> as functions of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math>, then in fact <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is differentiable (and indeed continuously differentiable).</p> <h3 id="functions_between_cartesian_spaces">Functions between Cartesian spaces</h3> <div class="num_defn" id="FunctionBetweenCartesianSpacesDifferentiation"> <h6 id="definition_3">Definition</h6> <p><strong>(differentiation of functions between Cartesian spaces)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>n</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>n</mi> <mn>2</mn></msub><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n_1, n_2 \in \mathbb{N}</annotation></semantics></math> and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>⊆</mo><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msup></mrow><annotation encoding="application/x-tex">U\subseteq \mathbb{R}^{n_1}</annotation></semantics></math> be an <a class="existingWikiWord" href="/nlab/show/open+subset">open subset</a>.</p> <p>Then a <a class="existingWikiWord" href="/nlab/show/function">function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>U</mi><mo>⟶</mo><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup></mrow><annotation encoding="application/x-tex">f \;\colon\; U \longrightarrow \mathbb{R}^{n_2}</annotation></semantics></math> is <em>differentiable</em> if for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>∈</mo><mo stretchy="false">{</mo><mn>1</mn><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>n</mi> <mn>2</mn></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">i \in \{1, \cdots, n_2\}</annotation></semantics></math> the component function</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mi>i</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>U</mi><mover><mo>⟶</mo><mi>f</mi></mover><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup><mover><mo>⟶</mo><mrow><msub><mi>pr</mi> <mi>u</mi></msub></mrow></mover><mi>ℝ</mi></mrow><annotation encoding="application/x-tex"> f_i \;\colon\; U \overset{f}{\longrightarrow} \mathbb{R}^{n_2} \overset{pr_u}{\longrightarrow} \mathbb{R} </annotation></semantics></math></div> <p>is differentiable in the sense of def. <a class="maruku-ref" href="#DerivativeOfFunctionFromRnToR"></a>.</p> <p>In this case, the derivatives <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><msub><mi>f</mi> <mi>i</mi></msub><mo lspace="verythinmathspace">:</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">d f_i \colon \mathbb{R}^n \to \mathbb{R}</annotation></semantics></math> of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">f_i</annotation></semantics></math> assemble into a linear map of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>d</mi><msub><mi>f</mi> <mi>x</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msup><mo>→</mo><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> d f_x \;\colon\; \mathbb{R}^{n_1} \to \mathbb{R}^{n_2} \,. </annotation></semantics></math></div></div> <h3 id="functions_between_differentiable_manifolds">Functions between differentiable manifolds</h3> <div class="num_defn"> <h6 id="definition_4">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/differentiable+manifolds">differentiable manifolds</a>, then a <a class="existingWikiWord" href="/nlab/show/function">function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f \colon X \longrightarrow Y</annotation></semantics></math> is called <em>differentiable</em> if for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo>{</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><munderover><mo>→</mo><mo>≃</mo><mrow><msub><mi>ϕ</mi> <mi>i</mi></msub></mrow></munderover><msub><mi>U</mi> <mi>i</mi></msub><mo>⊂</mo><mi>X</mi><mo>}</mo></mrow></mrow><annotation encoding="application/x-tex">\left\{ \mathbb{R}^{n} \underoverset{\simeq}{\phi_i}{\to} U_i \subset X\right\}</annotation></semantics></math> an <a class="existingWikiWord" href="/nlab/show/atlas">atlas</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo>{</mo><msup><mi>ℝ</mi> <mrow><mi>n</mi><mo>′</mo></mrow></msup><munderover><mo>→</mo><mo>≃</mo><mrow><msub><mi>ψ</mi> <mi>j</mi></msub></mrow></munderover><msub><mi>V</mi> <mi>j</mi></msub><mo>⊂</mo><msub><mi>X</mi> <mn>2</mn></msub><mo>}</mo></mrow></mrow><annotation encoding="application/x-tex">\left\{ \mathbb{R}^{n'} \underoverset{\simeq}{\psi_j}{\to} V_j \subset X_2\right\}</annotation></semantics></math> and atlas for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> then for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">i \in I </annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi><mo>∈</mo><mi>J</mi></mrow><annotation encoding="application/x-tex">j \in J</annotation></semantics></math> the function</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>⊃</mo><mphantom><mi>AA</mi></mphantom><mo stretchy="false">(</mo><mi>f</mi><mo>∘</mo><msub><mi>ϕ</mi> <mi>i</mi></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><msub><mi>V</mi> <mi>j</mi></msub><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><msub><mi>ϕ</mi> <mi>i</mi></msub></mrow></mover><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><msub><mi>V</mi> <mi>j</mi></msub><mo stretchy="false">)</mo><mover><mo>⟶</mo><mi>f</mi></mover><msub><mi>V</mi> <mi>j</mi></msub><mover><mo>⟶</mo><mrow><msubsup><mi>ψ</mi> <mi>j</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup></mrow></mover><msup><mi>ℝ</mi> <mrow><mi>n</mi><mo>′</mo></mrow></msup></mrow><annotation encoding="application/x-tex"> \mathbb{R}^n \supset \phantom{AA} (f\circ \phi_i)^{-1}(V_j) \overset{\phi_i}{\longrightarrow} f^{-1}(V_j) \overset{f}{\longrightarrow} V_j \overset{\psi_j^{-1}}{\longrightarrow} \mathbb{R}^{n'} </annotation></semantics></math></div> <p>is differentiable in the sense of def. <a class="maruku-ref" href="#FunctionBetweenCartesianSpacesDifferentiation"></a>.</p> </div> <h2 id="higher_differentiability">Higher differentiability</h2> <h3 id="iterated_differentiability">Iterated differentiability</h3> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>⊂</mo><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">U \subset \mathbb{R}^n</annotation></semantics></math> an <a class="existingWikiWord" href="/nlab/show/open+subset">open subset</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>U</mi><mo>→</mo><msup><mi>ℝ</mi> <mi>m</mi></msup></mrow><annotation encoding="application/x-tex">f \;\colon\; U\to \mathbb{R}^m</annotation></semantics></math> a differentiable function (def. <a class="maruku-ref" href="#DerivativeOfFunctionFromRnToR"></a>), we may regard its differential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>f</mi></mrow><annotation encoding="application/x-tex">d f</annotation></semantics></math> as a function from a subset of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> (the points where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is differentiable) to the space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>,</mo><msup><mi>ℝ</mi> <mi>m</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">L(\mathbb{R}^n,\mathbb{R}^m)</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/linear+maps">linear maps</a>. Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>,</mo><msup><mi>ℝ</mi> <mi>m</mi></msup><mo stretchy="false">)</mo><mo>≅</mo><msup><mi>ℝ</mi> <mrow><mi>n</mi><mi>m</mi></mrow></msup></mrow><annotation encoding="application/x-tex">L(\mathbb{R}^n,\mathbb{R}^m) \cong \mathbb{R}^{n m}</annotation></semantics></math> is again a Cartesian space, we may then ask whether <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>f</mi></mrow><annotation encoding="application/x-tex">d f</annotation></semantics></math> is differentiable.</p> <p>We can then iterate, obtaining the following hierarchy of differentiability. Because iterated differentiability by itself is not very useful, and a differentiable map is necessarily continuous, one generally includes continuity of the last assumed derivatives.</p> <ul> <li> <p>The map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is <strong>continuous</strong> or <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>0</mn></msup></mrow><annotation encoding="application/x-tex">C^0</annotation></semantics></math></strong> if it is a <a class="existingWikiWord" href="/nlab/show/continuous+map">continuous map</a> between underlying <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a>. We begin with this since a differentiable map is necessarily continuous.</p> </li> <li> <p>The map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is <strong>continuously differentiable</strong> or <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">C^1</annotation></semantics></math></strong> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> if it is differentiable at all points of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> and the resulting map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>f</mi></mrow><annotation encoding="application/x-tex">d f</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/continuous+map">continuous</a>. (At this point, if we generalize to infinite-dimensional spaces, we get a variety of notions of when the differential is ‘continuous’; see <a class="existingWikiWord" href="/nlab/show/continuously+differentiable+map">continuously differentiable map</a> for discussion.)</p> </li> <li> <p>The map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is <strong>twice differentiable</strong> if it is differentiable and its derivative <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>f</mi></mrow><annotation encoding="application/x-tex">d f</annotation></semantics></math> is differentiable. A twice differentiable map must be continuously differentiable. Similarly, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is <strong>twice continuously differentiable</strong> or <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">C^2</annotation></semantics></math></strong> if it is twice differentiable and the second derivative <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>d</mi><mi>f</mi></mrow><annotation encoding="application/x-tex">d d f</annotation></semantics></math> is continuous.</p> </li> <li> <p>By <a class="existingWikiWord" href="/nlab/show/recursion">recursion</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n + 1</annotation></semantics></math> times differentiable</strong> if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> times differentiable and the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>th derivative of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is differentiable. Similarly, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> times continuously differentiable</strong> or <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">C^n</annotation></semantics></math></strong> if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> times differentiable and the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>th derivative of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is continuous.</p> </li> <li> <p>The map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is <strong><a class="existingWikiWord" href="/nlab/show/smooth+map">smooth</a></strong> or <strong>infinitely differentiable</strong> or <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup></mrow><annotation encoding="application/x-tex">C^\infty</annotation></semantics></math></strong> if it is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> times differentiable for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>, or equivalently if it is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">C^n</annotation></semantics></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>. (There is no difference between infinite differentiability and infinite continuous differentiability.) One may also define this notion <a class="existingWikiWord" href="/nlab/show/coinductive+definition">coinductively</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is infinitely differentiable if it is differentiable and its derivative <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>f</mi></mrow><annotation encoding="application/x-tex">d f</annotation></semantics></math> is infinitely differentiable.</p> </li> <li> <p>One step higher, we may ask whether <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/analytic+function">analytic</a> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mi>ω</mi></msup></mrow><annotation encoding="application/x-tex">C^\omega</annotation></semantics></math>.</p> </li> </ul> <h3 id="uniform_differentiability">Uniform differentiability</h3> <p>Note that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is differentiable on a set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> iff there is a function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">f'</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> (necessarily unique, assuming that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> has no <a class="existingWikiWord" href="/nlab/show/isolated+points">isolated points</a>) such that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>∀</mo><mspace width="thinmathspace"></mspace><mi>ϵ</mi><mo>></mo><mn>0</mn><mo>,</mo><mspace width="thickmathspace"></mspace><mo>∀</mo><mspace width="thinmathspace"></mspace><mi>x</mi><mo>∈</mo><mi>U</mi><mo>,</mo><mspace width="thickmathspace"></mspace><mo>∃</mo><mspace width="thinmathspace"></mspace><mi>δ</mi><mo>></mo><mn>0</mn><mo>,</mo><mspace width="thickmathspace"></mspace><mo>∀</mo><mspace width="thinmathspace"></mspace><mi>y</mi><mo>∈</mo><mi>U</mi><mo>,</mo><mspace width="thickmathspace"></mspace><mrow><mo stretchy="false">|</mo><mrow><mi>y</mi><mo>−</mo><mi>x</mi></mrow><mo stretchy="false">|</mo></mrow><mo><</mo><mi>δ</mi><mspace width="thickmathspace"></mspace><mo>⇒</mo><mspace width="thickmathspace"></mspace><mrow><mo stretchy="false">|</mo><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>−</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>−</mo><mi>f</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>y</mi><mo>−</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mo stretchy="false">|</mo></mrow><mo><</mo><mi>ϵ</mi><mspace width="thinmathspace"></mspace><mrow><mo stretchy="false">|</mo><mrow><mi>y</mi><mo>−</mo><mi>x</mi></mrow><mo stretchy="false">|</mo></mrow><mo>.</mo></mrow><annotation encoding="application/x-tex"> \forall\, \epsilon \gt 0,\; \forall\, x \in U,\; \exists\, \delta \gt 0,\; \forall\, y \in U,\; {|{y - x}|} \lt \delta \;\Rightarrow\; {|{f(y) - f(x) - f'(x)(y - x)}|} \lt \epsilon \,{|{y - x}|} .</annotation></semantics></math></div> <p>Reverse quantifiers, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is <strong>uniformly differentiable</strong> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> iff there is a function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">f'</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> such that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>∀</mo><mspace width="thinmathspace"></mspace><mi>ϵ</mi><mo>></mo><mn>0</mn><mo>,</mo><mspace width="thickmathspace"></mspace><mo>∃</mo><mspace width="thinmathspace"></mspace><mi>δ</mi><mo>></mo><mn>0</mn><mo>,</mo><mspace width="thickmathspace"></mspace><mo>∀</mo><mspace width="thinmathspace"></mspace><mi>x</mi><mo>∈</mo><mi>U</mi><mo>,</mo><mspace width="thickmathspace"></mspace><mo>∀</mo><mspace width="thinmathspace"></mspace><mi>y</mi><mo>∈</mo><mi>U</mi><mo>,</mo><mspace width="thickmathspace"></mspace><mrow><mo stretchy="false">|</mo><mrow><mi>y</mi><mo>−</mo><mi>x</mi></mrow><mo stretchy="false">|</mo></mrow><mo><</mo><mi>δ</mi><mspace width="thickmathspace"></mspace><mo>⇒</mo><mspace width="thickmathspace"></mspace><mrow><mo stretchy="false">|</mo><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>−</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>−</mo><mi>f</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>y</mi><mo>−</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mo stretchy="false">|</mo></mrow><mo><</mo><mi>ϵ</mi><mspace width="thinmathspace"></mspace><mrow><mo stretchy="false">|</mo><mrow><mi>y</mi><mo>−</mo><mi>x</mi></mrow><mo stretchy="false">|</mo></mrow><mo>.</mo></mrow><annotation encoding="application/x-tex"> \forall\, \epsilon \gt 0,\; \exists\, \delta \gt 0,\; \forall\, x \in U,\; \forall\, y \in U,\; {|{y - x}|} \lt \delta \;\Rightarrow\; {|{f(y) - f(x) - f'(x)(y - x)}|} \lt \epsilon \,{|{y - x}|}.</annotation></semantics></math></div> <p>In <a class="existingWikiWord" href="/nlab/show/classical+mathematics">classical mathematics</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is uniformly differentiable if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is differentiable and its derivative <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">f'</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/uniformly+continuous+map">uniformly continuous</a>; in other words, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is uniformly differentiable iff <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is <em>uniformly-continuously differentiable</em>. The same is true in <a class="existingWikiWord" href="/nlab/show/constructive+mathematics">constructive mathematics</a> as long as one assumes <a class="existingWikiWord" href="/nlab/show/dependent+choice">dependent choice</a>. In the absence of dependent choice, however, the argument only goes one way, and uniform differentiability is stronger. Furthermore, just as pointwise continuity is not as well behaved constructively as uniform continuity, so pointwise differentiability is not as well behaved constructively as uniform differentiability. For this reason, uniform differentiability is particularly important in constructive mathematics.</p> <p>In addition, one could talk about <strong>locally uniform differentiablilty</strong>, which is a continuously differentiable function on a set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> which is uniformly differentiable on every closed and bounded subset <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>⊆</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">V \subseteq U</annotation></semantics></math>.</p> <h3 id="symmetry">Symmetry of higher derivatives</h3> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>U</mi><mo>→</mo><msup><mi>ℝ</mi> <mi>m</mi></msup></mrow><annotation encoding="application/x-tex">f:U\to \mathbb{R}^m</annotation></semantics></math> is twice differentiable with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>⊆</mo><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">U\subseteq \mathbb{R}^n</annotation></semantics></math>, its second derivative</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo stretchy="false">(</mo><mi>d</mi><mi>f</mi><mo stretchy="false">)</mo><mo>:</mo><mi>U</mi><mo>→</mo><mi>L</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>,</mo><mi>L</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>,</mo><msup><mi>ℝ</mi> <mi>m</mi></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≅</mo><mi>Bilin</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>,</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>;</mo><msup><mi>ℝ</mi> <mi>m</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">d(d f) : U \to L(\mathbb{R}^n,L(\mathbb{R}^n,\mathbb{R}^m)) \cong Bilin(\mathbb{R}^n,\mathbb{R}^n;\mathbb{R}^m)</annotation></semantics></math></div> <p>is a function from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> into the space of <a class="existingWikiWord" href="/nlab/show/bilinear+maps">bilinear maps</a> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>×</mo><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n\times \mathbb{R}^n</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>m</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^m</annotation></semantics></math>.</p> <div class="num_theorem" id="symmetrythm"> <h6 id="theorem">Theorem</h6> <p><strong>(Symmetry of higher derivatives)</strong>. If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>U</mi><mo>→</mo><msup><mi>ℝ</mi> <mi>m</mi></msup></mrow><annotation encoding="application/x-tex">f:U\to \mathbb{R}^m</annotation></semantics></math> is twice differentiable, then its second derivative <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo stretchy="false">(</mo><mi>d</mi><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">d(d f)</annotation></semantics></math> lands in the space of <em>symmetric</em> bilinear maps, i.e. for any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">x\in U</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi><mo>,</mo><mi>w</mi><mo>∈</mo><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">v,w\in \mathbb{R}^n</annotation></semantics></math> we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo stretchy="false">(</mo><mi>d</mi><mi>f</mi><msub><mo stretchy="false">)</mo> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mi>d</mi><mo stretchy="false">(</mo><mi>d</mi><mi>f</mi><msub><mo stretchy="false">)</mo> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>w</mi><mo>,</mo><mi>v</mi><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex"> d(d f)_x(v,w) = d(d f)_x(w,v). </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof">Proof</h6> <p>It suffices to assume <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">m=1</annotation></semantics></math>; otherwise we just consider it componentwise. Define a function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>:</mo><mi>ℝ</mi><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">g:\mathbb{R}\to \mathbb{R}</annotation></semantics></math> by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>ξ</mi><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>ξ</mi><mi>v</mi><mo>+</mo><mi>w</mi><mo stretchy="false">)</mo><mo>−</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>ξ</mi><mi>v</mi><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex"> g(\xi) = f(x+\xi v+w) - f(x+\xi v). </annotation></semantics></math></div> <p>Then by the chain rule, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> is differentiable and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mi>g</mi><mo>′</mo><mo stretchy="false">(</mo><mi>ξ</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mi>d</mi><msub><mi>f</mi> <mrow><mi>x</mi><mo>+</mo><mi>ξ</mi><mi>v</mi><mo>+</mo><mi>w</mi></mrow></msub><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo>−</mo><mi>d</mi><msub><mi>f</mi> <mrow><mi>x</mi><mo>+</mo><mi>ξ</mi><mi>v</mi></mrow></msub><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo maxsize="1.8em" minsize="1.8em">(</mo><mi>d</mi><msub><mi>f</mi> <mrow><mi>x</mi><mo>+</mo><mi>ξ</mi><mi>v</mi><mo>+</mo><mi>w</mi></mrow></msub><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo>−</mo><mi>d</mi><msub><mi>f</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo maxsize="1.8em" minsize="1.8em">)</mo><mo>−</mo><mo maxsize="1.8em" minsize="1.8em">(</mo><mi>d</mi><msub><mi>f</mi> <mrow><mi>x</mi><mo>+</mo><mi>ξ</mi><mi>v</mi></mrow></msub><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo>−</mo><mi>d</mi><msub><mi>f</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo maxsize="1.8em" minsize="1.8em">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>d</mi><mo stretchy="false">(</mo><mi>d</mi><mi>f</mi><msub><mo stretchy="false">)</mo> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>ξ</mi><mi>v</mi><mo>+</mo><mi>w</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mi>E</mi> <mn>1</mn></msub><mo stretchy="false">|</mo><mi>v</mi><mo stretchy="false">|</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">|</mo><mi>ξ</mi><mi>v</mi><mo>+</mo><mi>w</mi><mo stretchy="false">|</mo><mo>−</mo><mi>d</mi><mo stretchy="false">(</mo><mi>d</mi><mi>f</mi><msub><mo stretchy="false">)</mo> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>ξ</mi><mi>v</mi><mo stretchy="false">)</mo><mo>−</mo><msub><mi>E</mi> <mn>2</mn></msub><mo stretchy="false">|</mo><mi>v</mi><mo stretchy="false">|</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">|</mo><mi>ξ</mi><mi>v</mi><mo stretchy="false">|</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>d</mi><mo stretchy="false">(</mo><mi>d</mi><mi>f</mi><msub><mo stretchy="false">)</mo> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>+</mo><mi>E</mi><mo stretchy="false">|</mo><mi>v</mi><mo stretchy="false">|</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">(</mo><mo stretchy="false">|</mo><mi>v</mi><mo stretchy="false">|</mo><mo>+</mo><mo stretchy="false">|</mo><mi>w</mi><mo stretchy="false">|</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} g'(\xi) &= d f_{x+\xi v+w}(v) - d f_{x+\xi v}(v)\\ &= \Big(d f_{x+\xi v+w}(v) - d f_{x}(v)\Big) - \Big(d f_{x+\xi v}(v) - d f_x(v)\Big)\\ &= d(d f)_{x}(v,\xi v + w) + E_1 |v|\,|\xi v + w| - d(d f)_x(v,\xi v) - E_2 |v|\,|\xi v|\\ &= d(d f)_x(v,w) + E |v|\, (|v|+|w|) \end{aligned} </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>E</mi> <mn>2</mn></msub><mo>,</mo><mi>E</mi><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">E_1, E_2, E \to 0</annotation></semantics></math> as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">(v,w)\to 0</annotation></semantics></math>. Now the <a class="existingWikiWord" href="/nlab/show/mean+value+theorem">mean value theorem</a> tells us that for some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ξ</mi><mo>∈</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\xi\in(0,1)</annotation></semantics></math> we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>v</mi><mo>+</mo><mi>w</mi><mo stretchy="false">)</mo><mo>−</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>v</mi><mo stretchy="false">)</mo><mo>−</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>w</mi><mo stretchy="false">)</mo><mo>+</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mi>g</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>−</mo><mi>g</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>g</mi><mo>′</mo><mo stretchy="false">(</mo><mi>ξ</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>d</mi><mo stretchy="false">(</mo><mi>d</mi><mi>f</mi><msub><mo stretchy="false">)</mo> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>+</mo><mi>E</mi><mo stretchy="false">(</mo><mo stretchy="false">|</mo><mi>v</mi><mo stretchy="false">|</mo><mo>+</mo><mo stretchy="false">|</mo><mi>w</mi><mo stretchy="false">|</mo><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mo>.</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} f(x+v+w) - f(x+v) - f(x+w) + f(x) &= g(1) - g(0)\\ &= g'(\xi)\\ &= d(d f)_x(v,w) + E (|v|+|w|)^2. \end{aligned} </annotation></semantics></math></div> <p>But the “second-order difference” <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>v</mi><mo>+</mo><mi>w</mi><mo stretchy="false">)</mo><mo>−</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>v</mi><mo stretchy="false">)</mo><mo>−</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>w</mi><mo stretchy="false">)</mo><mo>+</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x+v+w) - f(x+v) - f(x+w) + f(x)</annotation></semantics></math> is manifestly symmetric in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi></mrow><annotation encoding="application/x-tex">v</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>w</mi></mrow><annotation encoding="application/x-tex">w</annotation></semantics></math>, so we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo stretchy="false">(</mo><mi>d</mi><mi>f</mi><msub><mo stretchy="false">)</mo> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>−</mo><mi>d</mi><mo stretchy="false">(</mo><mi>d</mi><mi>f</mi><msub><mo stretchy="false">)</mo> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>w</mi><mo>,</mo><mi>v</mi><mo stretchy="false">)</mo><mo>=</mo><mi>E</mi><mo stretchy="false">(</mo><mo stretchy="false">|</mo><mi>v</mi><mo stretchy="false">|</mo><mo>+</mo><mo stretchy="false">|</mo><mi>w</mi><mo stretchy="false">|</mo><msup><mo stretchy="false">)</mo> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex"> d(d f)_x(v,w) - d(d f)_x(w,v) = E (|v|+|w|)^2 </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>lim</mi> <mrow><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>→</mo><mn>0</mn></mrow></msub><mi>E</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\lim_{(v,w)\to 0} E = 0</annotation></semantics></math>. But bilinearity of the LHS then implies that it is identically zero.</p> </div> <p>The components of the bilinear map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo stretchy="false">(</mo><mi>d</mi><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">d(d f)</annotation></semantics></math> are the second-order partial derivatives <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mrow><msup><mo>∂</mo> <mn>2</mn></msup><mi>f</mi></mrow><mrow><mo>∂</mo><msub><mi>x</mi> <mi>i</mi></msub><mo>∂</mo><msub><mi>x</mi> <mi>j</mi></msub></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\partial^2 f}{\partial x_i \partial x_j}</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>. Thus, this theorem says that if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is twice differentiable, then the <em>mixed partials</em> are equal,</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><msup><mo>∂</mo> <mn>2</mn></msup><mi>f</mi></mrow><mrow><mo>∂</mo><msub><mi>x</mi> <mi>i</mi></msub><mo>∂</mo><msub><mi>x</mi> <mi>j</mi></msub></mrow></mfrac><mo>=</mo><mfrac><mrow><msup><mo>∂</mo> <mn>2</mn></msup><mi>f</mi></mrow><mrow><mo>∂</mo><msub><mi>x</mi> <mi>j</mi></msub><mo>∂</mo><msub><mi>x</mi> <mi>i</mi></msub></mrow></mfrac><mo>.</mo></mrow><annotation encoding="application/x-tex">\frac{\partial^2 f}{\partial x_i \partial x_j} = \frac{\partial^2 f}{\partial x_j \partial x_i}.</annotation></semantics></math></div> <p>The second-order partial derivatives may <em>exist</em> without the mixed partials being equal; see below for a counterexample. However, the theorem shows that if we require <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> to actually be twice differentiable rather than merely having second-order partial derivatives, then this cannot happen.</p> <p>In particular, we have the following corollary, which is more commonly found in textbooks.</p> <div class="num_cor" id="symmetrycontcor"> <h6 id="corollary">Corollary</h6> <p><strong>(Symmetry of continuous higher derivatives)</strong>. If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> has first and second-order partial derivatives, and the latter are continuous in a neighborhood of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math>, then the mixed partial derivatives are equal,</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><msup><mo>∂</mo> <mn>2</mn></msup><mi>f</mi></mrow><mrow><mo>∂</mo><msub><mi>x</mi> <mi>i</mi></msub><mo>∂</mo><msub><mi>x</mi> <mi>j</mi></msub></mrow></mfrac><mo>=</mo><mfrac><mrow><msup><mo>∂</mo> <mn>2</mn></msup><mi>f</mi></mrow><mrow><mo>∂</mo><msub><mi>x</mi> <mi>j</mi></msub><mo>∂</mo><msub><mi>x</mi> <mi>i</mi></msub></mrow></mfrac><mo>.</mo></mrow><annotation encoding="application/x-tex">\frac{\partial^2 f}{\partial x_i \partial x_j} = \frac{\partial^2 f}{\partial x_j \partial x_i}.</annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>Continuity of the second-order partials implies that the first-order partials are differentiable, and hence so is the differential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>f</mi></mrow><annotation encoding="application/x-tex">d f</annotation></semantics></math>.</p> </div> <h3 id="a_direct_definition_of_the_second_derivative">A direct definition of the second derivative</h3> <p>Note that the proof of the theorem implies that if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is twice differentiable at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math>, then there exists a bilinear map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mo>∂</mo> <mn>2</mn></msup><msub><mi>f</mi> <mi>x</mi></msub><mo>:</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>×</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>→</mo><msup><mi>ℝ</mi> <mi>m</mi></msup></mrow><annotation encoding="application/x-tex">\partial^2 f_x : \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}^m</annotation></semantics></math> such that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>v</mi><mo>+</mo><mi>w</mi><mo stretchy="false">)</mo><mo>−</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>v</mi><mo stretchy="false">)</mo><mo>−</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>w</mi><mo stretchy="false">)</mo><mo>+</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mo>∂</mo> <mn>2</mn></msup><msub><mi>f</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>+</mo><mi>E</mi><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mrow><mo stretchy="false">|</mo><mi>v</mi><mo stretchy="false">|</mo><mo>+</mo><mo stretchy="false">|</mo><mi>w</mi><mo stretchy="false">|</mo></mrow><msup><mo stretchy="false">)</mo> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">f(x+v+w) - f(x+v) - f(x+w) + f(x) = \partial^2 f_x(v,w) + E(v,w) ({|v|+|w|})^2 </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>lim</mi> <mrow><mi>v</mi><mo>,</mo><mi>w</mi><mo>→</mo><mn>0</mn></mrow></msub><mi>E</mi><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\lim_{v,w\to 0} E(v,w) = 0</annotation></semantics></math>. This is a condition that makes sense as a condition on an arbitrary <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> without assuming differentiability. and if it holds, then the bilinear map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mo>∂</mo> <mn>2</mn></msup><msub><mi>f</mi> <mi>x</mi></msub></mrow><annotation encoding="application/x-tex">\partial^2 f_x</annotation></semantics></math> must be symmetric.</p> <p>Moreover, as explained <a href="http://mathoverflow.net/a/165733/49">here</a>, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is differentiable in a neighborhood of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> and satisfies this condition at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math>, then it is in fact twice differentiable at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math>. To see this, it suffices to show that each coordinate of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>f</mi></mrow><annotation encoding="application/x-tex">d f</annotation></semantics></math> is differentiable, so let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>w</mi><mo>=</mo><mi>δ</mi><msub><mi>e</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">w=\delta e_i</annotation></semantics></math>, with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">e_i</annotation></semantics></math> a unit basis vector and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>δ</mi></mrow><annotation encoding="application/x-tex">\delta</annotation></semantics></math> a real number <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≠</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\neq 0</annotation></semantics></math>. Then we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>δ</mi><msub><mi>e</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>v</mi><mo>+</mo><mi>δ</mi><msub><mi>e</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>−</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>v</mi><mo stretchy="false">)</mo><mo>−</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>δ</mi><msub><mi>e</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>+</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>−</mo><msup><mo>∂</mo> <mn>2</mn></msup><msub><mi>f</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>δ</mi><mi>w</mi><mo stretchy="false">)</mo></mrow><mrow><mrow><mo stretchy="false">|</mo><mi>v</mi><mo stretchy="false">|</mo></mrow><mi>δ</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex"> E(v,\delta e_i) = \frac{f(x+v+\delta e_i) - f(x+v) - f(x+\delta e_i) + f(x) - \partial^2 f_x(v,\delta w)}{{|v|}{\delta}}</annotation></semantics></math></div> <p>Taking the limit as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>δ</mi><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\delta\to 0</annotation></semantics></math> and using differentiability of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>+</mo><mi>v</mi></mrow><annotation encoding="application/x-tex">x+v</annotation></semantics></math> (for sufficiently small <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi></mrow><annotation encoding="application/x-tex">v</annotation></semantics></math>), we get</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mi>lim</mi> <mrow><mi>δ</mi><mo>→</mo><mn>0</mn></mrow></munder><mi>E</mi><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>δ</mi><msub><mi>e</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi>d</mi><msub><mi>f</mi> <mrow><mi>x</mi><mo>+</mo><mi>v</mi></mrow></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>−</mo><mi>d</mi><msub><mi>f</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>−</mo><msup><mo>∂</mo> <mn>2</mn></msup><msub><mi>f</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><msub><mi>e</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">|</mo><mi>v</mi><mo stretchy="false">|</mo></mrow></mfrac><mo>.</mo></mrow><annotation encoding="application/x-tex"> \lim_{\delta \to 0} E(v,\delta e_i) = \frac{d f_{x+v}(e_i) - d f_x(e_i) - \partial^2 f_x(v,e_i)}{{|v|}}. </annotation></semantics></math></div> <p>Now take the limit as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">v\to 0</annotation></semantics></math>; on the left we get <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math> by assumption, so the function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>↦</mo><mi>d</mi><msub><mi>f</mi> <mi>y</mi></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">y\mapsto d f_y(e_i)</annotation></semantics></math> is differentiable at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> with derivative <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mo>∂</mo> <mn>2</mn></msup><msub><mi>f</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><msub><mi>e</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\partial^2 f_x(-,e_i)</annotation></semantics></math>. Thus, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>f</mi></mrow><annotation encoding="application/x-tex">d f</annotation></semantics></math> is differentiable at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math>.</p> <p>On the other hand, this condition by itself does not even imply that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is continuous. For instance, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Q}</annotation></semantics></math>-linear map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}\to \mathbb{R}</annotation></semantics></math>, then the second-order difference <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>v</mi><mo>+</mo><mi>w</mi><mo stretchy="false">)</mo><mo>−</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>v</mi><mo stretchy="false">)</mo><mo>−</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>w</mi><mo stretchy="false">)</mo><mo>+</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x+v+w) - f(x+v) - f(x+w) + f(x)</annotation></semantics></math> is identically zero.</p> <h3 id="higher_differentiability_on_tangent_spaces_and_the_chain_rule">Higher differentiability on tangent spaces and the chain rule</h3> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">f:\mathbb{R}^n\to \mathbb{R}</annotation></semantics></math> be differentiable. Instead of asking whether <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>f</mi><mo>:</mo><mi>U</mi><mo>→</mo><mi>L</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>,</mo><mi>ℝ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">d f : U \to L(\mathbb{R}^n,\mathbb{R})</annotation></semantics></math> is differentiable, we can ask whether its exponential transpose <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>f</mi><mo>:</mo><mi>U</mi><mo>×</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">d f : U\times \mathbb{R}^n \to \mathbb{R}</annotation></semantics></math> is differentiable. (Note that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>×</mo><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">U\times \mathbb{R}^n</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>⊆</mo><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">U\subseteq \mathbb{R}^n</annotation></semantics></math>.) This amounts to asking that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">x\in U</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi><mo>∈</mo><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">v\in \mathbb{R}^n</annotation></semantics></math>, we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>d</mi><msub><mi>f</mi> <mrow><mi>x</mi><mo>+</mo><mi>w</mi></mrow></msub><mo stretchy="false">(</mo><mi>v</mi><mo>+</mo><mi>h</mi><mo stretchy="false">)</mo><mo>−</mo><mi>d</mi><msub><mi>f</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>d</mi> <mn>2</mn></msup><msub><mi>f</mi> <mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo>,</mo><mi>h</mi><mo stretchy="false">)</mo><mo>+</mo><mi>E</mi><mo stretchy="false">(</mo><mo stretchy="false">|</mo><mi>w</mi><mo stretchy="false">|</mo><mo>+</mo><mo stretchy="false">|</mo><mi>h</mi><mo stretchy="false">|</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> d f_{x+w}(v+h) - d f_x(v) = d^2 f_{(x,v)}(w,h) + E(|w|+|h|) </annotation></semantics></math></div> <p>for a linear map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>d</mi> <mn>2</mn></msup><msub><mi>f</mi> <mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></msub><mo>:</mo><msup><mi>ℝ</mi> <mrow><mn>2</mn><mi>n</mi></mrow></msup><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">d^2 f_{(x,v)}:\mathbb{R}^{2n} \to \mathbb{R}</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>lim</mi> <mrow><mo stretchy="false">(</mo><mi>w</mi><mo>,</mo><mi>h</mi><mo stretchy="false">)</mo><mo>→</mo><mn>0</mn></mrow></msub><mi>E</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\lim_{(w,h)\to 0} E = 0</annotation></semantics></math>. Setting <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">h=0</annotation></semantics></math>, we see that this implies that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>f</mi><mo>:</mo><mi>U</mi><mo>→</mo><mi>L</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>,</mo><mi>ℝ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">d f : U \to L(\mathbb{R}^n,\mathbb{R})</annotation></semantics></math> is differentiable, with differential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo stretchy="false">(</mo><mi>d</mi><mi>f</mi><msub><mo stretchy="false">)</mo> <mi>x</mi></msub><mo>=</mo><msup><mi>d</mi> <mn>2</mn></msup><msub><mi>f</mi> <mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">d(d f)_x = d^2 f_{(x,v)}(w,0)</annotation></semantics></math> for any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi></mrow><annotation encoding="application/x-tex">v</annotation></semantics></math>. And setting <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>w</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">w=0</annotation></semantics></math>, we obtain <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><msub><mi>f</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>d</mi> <mn>2</mn></msup><msub><mi>f</mi> <mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mi>h</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">d f_x(h) = d^2 f_{(x,v)}(0,h)</annotation></semantics></math> for any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi></mrow><annotation encoding="application/x-tex">v</annotation></semantics></math>. Thus, we can write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>d</mi> <mn>2</mn></msup><msub><mi>f</mi> <mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>w</mi><mo>,</mo><mi>h</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mo>∂</mo> <mn>2</mn></msup><msub><mi>f</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>+</mo><mi>d</mi><msub><mi>f</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> d^2 f_{(x,v)}(w,h) = \partial^2 f_x(v,w) + d f_x(h) </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mo>∂</mo> <mn>2</mn></msup><msub><mi>f</mi> <mi>x</mi></msub></mrow><annotation encoding="application/x-tex">\partial^2 f_x</annotation></semantics></math> is the symmetric bilinear map from the previous section. Conversely, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is twice differentiable, then using linearity and continuity of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>f</mi></mrow><annotation encoding="application/x-tex">d f</annotation></semantics></math> it is easy to see that the above condition holds.</p> <p>Therefore, these two kinds of twice-differentiability of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> are equivalent as <em>conditions</em> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>, but the resulting <em>second differential</em> is different. In the second case, we get</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>d</mi> <mn>2</mn></msup><mi>f</mi><mo>=</mo><msup><mo>∂</mo> <mn>2</mn></msup><mi>f</mi><mo>+</mo><mi>d</mi><mi>f</mi><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></munder><mfrac><mrow><msup><mo>∂</mo> <mn>2</mn></msup><mi>f</mi></mrow><mrow><mo>∂</mo><msub><mi>x</mi> <mi>i</mi></msub><mo>∂</mo><msub><mi>x</mi> <mi>j</mi></msub></mrow></mfrac><mi>d</mi><msub><mi>x</mi> <mi>i</mi></msub><mspace width="thinmathspace"></mspace><mi>d</mi><msub><mi>x</mi> <mi>j</mi></msub><mo>+</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mi>i</mi></munder><mfrac><mrow><mo>∂</mo><mi>f</mi></mrow><mrow><mo>∂</mo><msub><mi>x</mi> <mi>i</mi></msub></mrow></mfrac><msup><mi>d</mi> <mn>2</mn></msup><msub><mi>x</mi> <mi>i</mi></msub><mo>.</mo></mrow><annotation encoding="application/x-tex">d^2f = \partial^2 f + d f = \sum_{i,j} \frac{\partial^2f}{\partial x_i \partial x_j} d x_i \, d x_j + \sum_i \frac{\partial f}{\partial x_i} d^2 x_i. </annotation></semantics></math></div> <p>rather than merely the first term <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub><mfrac><mrow><msup><mo>∂</mo> <mn>2</mn></msup><mi>f</mi></mrow><mrow><mo>∂</mo><msub><mi>x</mi> <mi>i</mi></msub><mo>∂</mo><msub><mi>x</mi> <mi>j</mi></msub></mrow></mfrac><mi>d</mi><msub><mi>x</mi> <mi>i</mi></msub><mspace width="thinmathspace"></mspace><mi>d</mi><msub><mi>x</mi> <mi>j</mi></msub></mrow><annotation encoding="application/x-tex">\sum_{i,j} \frac{\partial^2f}{\partial x_i \partial x_j} d x_i \, d x_j</annotation></semantics></math>. There are two advantages to the second approach (asking that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>f</mi><mo>:</mo><mi>U</mi><mo>×</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">d f : U\times \mathbb{R}^n \to \mathbb{R}</annotation></semantics></math> be differentiable).</p> <p>Firstly, we can reformulate it in terms of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> by asking that there exists a linear map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><msub><mi>f</mi> <mi>x</mi></msub><mo>:</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>→</mo><msup><mi>ℝ</mi> <mi>m</mi></msup></mrow><annotation encoding="application/x-tex">d f_x : \mathbb{R}^n \to \mathbb{R}^m</annotation></semantics></math> and a bilinear map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mo>∂</mo> <mn>2</mn></msup><msub><mi>f</mi> <mi>x</mi></msub><mo>:</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>×</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>→</mo><msup><mi>ℝ</mi> <mi>m</mi></msup></mrow><annotation encoding="application/x-tex">\partial^2 f_x : \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}^m</annotation></semantics></math> such that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>v</mi><mo>+</mo><mi>w</mi><mo>+</mo><mi>h</mi><mo stretchy="false">)</mo><mo>−</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>v</mi><mo stretchy="false">)</mo><mo>−</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>w</mi><mo stretchy="false">)</mo><mo>+</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mo>∂</mo> <mn>2</mn></msup><msub><mi>f</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>+</mo><mi>d</mi><msub><mi>f</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">)</mo><mo>+</mo><mi>E</mi><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>w</mi><mo>,</mo><mi>h</mi><mo stretchy="false">)</mo><msqrt><mrow><msup><mrow><mo stretchy="false">|</mo><mi>v</mi><mo stretchy="false">|</mo></mrow> <mn>2</mn></msup><msup><mrow><mo stretchy="false">|</mo><mi>w</mi><mo stretchy="false">|</mo></mrow> <mn>2</mn></msup><mo>+</mo><msup><mrow><mo stretchy="false">|</mo><mi>h</mi><mo stretchy="false">|</mo></mrow> <mn>2</mn></msup></mrow></msqrt><mo>.</mo></mrow><annotation encoding="application/x-tex">f(x+v+w+h) - f(x+v) - f(x+w) + f(x) = \partial^2 f_x(v,w) + d f_x(h) + E(v,w,h) \sqrt{{|v|}^2{|w|}^2 + {|h|}^2}.</annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>lim</mi> <mrow><mi>v</mi><mo>,</mo><mi>w</mi><mo>,</mo><mi>h</mi><mo>→</mo><mn>0</mn></mrow></msub><mi>E</mi><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>w</mi><mo>,</mo><mi>h</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\lim_{v,w,h \to 0} E(v,w,h) = 0</annotation></semantics></math>. This holds if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is twice differentiable, for then we can write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>v</mi><mo>+</mo><mi>w</mi><mo>+</mo><mi>h</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>v</mi><mo>+</mo><mi>w</mi><mo stretchy="false">)</mo><mo>+</mo><mi>d</mi><msub><mi>f</mi> <mrow><mi>x</mi><mo>+</mo><mi>v</mi><mo>+</mo><mi>w</mi></mrow></msub><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">)</mo><mo>+</mo><mi>E</mi><mrow><mo stretchy="false">|</mo><mi>h</mi><mo stretchy="false">|</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>v</mi><mo>+</mo><mi>w</mi><mo stretchy="false">)</mo><mo>+</mo><mi>d</mi><msub><mi>f</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">)</mo><mo>+</mo><mi>d</mi><mo stretchy="false">(</mo><mi>d</mi><mi>f</mi><msub><mo stretchy="false">)</mo> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>v</mi><mo>+</mo><mi>w</mi><mo>,</mo><mi>h</mi><mo stretchy="false">)</mo><mo>+</mo><mi>E</mi><mrow><mo stretchy="false">|</mo><mi>v</mi><mo>+</mo><mi>w</mi><mo stretchy="false">|</mo></mrow><mrow><mo stretchy="false">|</mo><mi>h</mi><mo stretchy="false">|</mo></mrow><mo>+</mo><mi>E</mi><mrow><mo stretchy="false">|</mo><mi>h</mi><mo stretchy="false">|</mo></mrow></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} f(x+v+w+h) &= f(x+v+w) + d f_{x+v+w}(h) + E {|h|}\\ &= f(x+v+w) + d f_x(h) + d(d f)_x(v+w,h) + E {|v+w|}{|h|} + E{|h|} \end{aligned} </annotation></semantics></math></div> <p>and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo stretchy="false">(</mo><mi>d</mi><mi>f</mi><msub><mo stretchy="false">)</mo> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>v</mi><mo>+</mo><mi>w</mi><mo>,</mo><mi>h</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">d(d f)_x(v+w,h)</annotation></semantics></math> can also be incorporated into the error term. Of course, setting <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">h=0</annotation></semantics></math> we obtain the characterization of twice-differentiability from the previous section. But setting <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi><mo>=</mo><mi>w</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">v=w=0</annotation></semantics></math>, we find that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is differentiable at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> with derivative <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><msub><mi>f</mi> <mi>x</mi></msub></mrow><annotation encoding="application/x-tex">d f_x</annotation></semantics></math>. So here we have a direct characterization of the second derivative which also implies that the first derivative exists (although it seems that it doesn’t imply differentiability in a neighborhood of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math>, so that the resulting “second derivative” may not actually be the derivative of the first derivative).</p> <p>Secondly, the virtue of a second differential incorporating the first derivatives is that like the first differential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>f</mi></mrow><annotation encoding="application/x-tex">d f</annotation></semantics></math>, but unlike the bilinear map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mo>∂</mo> <mn>2</mn></msup><mi>f</mi></mrow><annotation encoding="application/x-tex">\partial^2 f</annotation></semantics></math>, it satisfies <a class="existingWikiWord" href="/nlab/show/Cauchy%27s+invariant+rule">Cauchy's invariant rule</a>. This means that we can express the <a class="existingWikiWord" href="/nlab/show/chain+rule">chain rule</a> for second differentials of composite maps simply by substitution: if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">y = f(u)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mo>=</mo><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">u = g(x)</annotation></semantics></math>, then finding <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>d</mi> <mn>2</mn></msup><mi>y</mi></mrow><annotation encoding="application/x-tex">d^2 y</annotation></semantics></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>u</mi></mrow><annotation encoding="application/x-tex">d u</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>d</mi> <mn>2</mn></msup><mi>u</mi></mrow><annotation encoding="application/x-tex">d^2 u</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>u</mi></mrow><annotation encoding="application/x-tex">d u</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>d</mi> <mn>2</mn></msup><mi>u</mi></mrow><annotation encoding="application/x-tex">d^2 u</annotation></semantics></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">d x</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>d</mi> <mn>2</mn></msup><mi>x</mi></mrow><annotation encoding="application/x-tex">d^2 x</annotation></semantics></math>, and substituting, gives the correct expression for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>d</mi> <mn>2</mn></msup><mi>y</mi></mrow><annotation encoding="application/x-tex">d^2 y</annotation></semantics></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">d x</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>d</mi> <mn>2</mn></msup><mi>x</mi></mrow><annotation encoding="application/x-tex">d^2 x</annotation></semantics></math>.</p> <p>In fact, this can be proven using the above direct characterization of the second differential, in essentially exactly the same way that we prove the ordinary chain rule for first derivatives. We sketch the proof, omitting the explicit error terms. We can write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>v</mi><mo>+</mo><mi>w</mi><mo>+</mo><mi>h</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>−</mo><mi>g</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>v</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>−</mo><mi>g</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>w</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>+</mo><mi>g</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g(f(x+v+w+h)) - g(f(x+v)) - g(f(x+w)) + g(f(x)) </annotation></semantics></math></div> <p>as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><mi>v</mi><mo>′</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mi>w</mi><mo>′</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mi>h</mi><mo>′</mo><mo stretchy="false">)</mo><mo>−</mo><mi>g</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><mi>v</mi><mo>′</mo><mo stretchy="false">)</mo><mo>−</mo><mi>g</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><mi>w</mi><mo>′</mo><mo stretchy="false">)</mo><mo>+</mo><mi>g</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> g(f(x) + v' + w' + h') - g(f(x) + v') - g(f(x) + w') + g(f(x)) </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi><mo>′</mo><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>v</mi><mo stretchy="false">)</mo><mo>−</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">v' = f(x+v) - f(x)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>w</mi><mo>′</mo><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>w</mi><mo stretchy="false">)</mo><mo>−</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">w' = f(x+w) - f(x)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>′</mo><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>v</mi><mo>+</mo><mi>w</mi><mo>+</mo><mi>h</mi><mo stretchy="false">)</mo><mo>−</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>v</mi><mo stretchy="false">)</mo><mo>−</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>w</mi><mo stretchy="false">)</mo><mo>+</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">h' = f(x+v+w+h) - f(x+v) - f(x+w) + f(x)</annotation></semantics></math>. Now by extra-strong twice differentiability of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math>, this is approximately equal to</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mo>∂</mo> <mn>2</mn></msup><msub><mi>g</mi> <mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>v</mi><mo>′</mo><mo>,</mo><mi>w</mi><mo>′</mo><mo stretchy="false">)</mo><mo>+</mo><mi>d</mi><msub><mi>g</mi> <mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>h</mi><mo>′</mo><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex"> \partial^2 g_{f(x)}(v',w') + d g_{f(x)}(h'). </annotation></semantics></math></div> <p>But by differentiability of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>, we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi><mo>′</mo><mo>≈</mo><mi>d</mi><msub><mi>f</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">v' \approx d f_x(v)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>w</mi><mo>′</mo><mo>≈</mo><mi>d</mi><msub><mi>f</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">w' \approx d f_x(w)</annotation></semantics></math>, while by extra-strong twice differentiability of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>′</mo><mo>≈</mo><msup><mo>∂</mo> <mn>2</mn></msup><msub><mi>f</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>+</mo><mi>d</mi><msub><mi>f</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">h' \approx \partial^2 f_x(v,w) + d f_x(h)</annotation></semantics></math>. Thus, we obtain approximately</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mo>∂</mo> <mn>2</mn></msup><msub><mi>g</mi> <mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>d</mi><msub><mi>f</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo>,</mo><mi>d</mi><msub><mi>f</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>+</mo><mi>d</mi><msub><mi>g</mi> <mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msup><mo>∂</mo> <mn>2</mn></msup><msub><mi>f</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>+</mo><mi>d</mi><msub><mi>f</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \partial^2 g_{f(x)}(d f_x(v), d f_x(w)) + d g_{f(x)}(\partial^2 f_x(v,w) + d f_x(h)) </annotation></semantics></math></div> <p>which is exactly what we would get by substitution.</p> <h3 id="for_maps_between_manifolds">For maps between manifolds</h3> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mi>k</mi></msup></mrow><annotation encoding="application/x-tex">C^k</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/differentiable+manifolds">differentiable manifolds</a>, then we may define what it means for a map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f:X\to Y</annotation></semantics></math> to be <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> times differentiable, or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">C^n</annotation></semantics></math>, for any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≤</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">n\le k</annotation></semantics></math>, by asking that it yield such a map when restricted to any charts. The most common case is when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> are smooth (infinitely differentiable) manifolds, so that we can define <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">C^n</annotation></semantics></math> functions between them for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≤</mo><mn>∞</mn></mrow><annotation encoding="application/x-tex">n\le \infty</annotation></semantics></math>. (Analytic manifolds, which are necessary in order to define analyticity of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>, are somewhat rarer.)</p> <p>A differentiable map between manifolds induces a map between their <a class="existingWikiWord" href="/nlab/show/tangent+bundles">tangent bundles</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>f</mi><mo>:</mo><mi>T</mi><mi>X</mi><mo>→</mo><mi>T</mi><mi>Y</mi></mrow><annotation encoding="application/x-tex">d f : T X \to T Y</annotation></semantics></math>; this operation extends to a <a class="existingWikiWord" href="/nlab/show/functor">functor</a> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">C^{k+1}</annotation></semantics></math> manifolds and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">C^{k+1}</annotation></semantics></math> maps to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mi>k</mi></msup></mrow><annotation encoding="application/x-tex">C^k</annotation></semantics></math> manifolds and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mi>k</mi></msup></mrow><annotation encoding="application/x-tex">C^k</annotation></semantics></math> maps. See <a class="existingWikiWord" href="/nlab/show/differentiation">differentiation</a> for more.</p> <h2 id="examples_and_nonexamples">Examples and non-examples</h2> <h3 id="DifferentiabilityOpposedToPartialDifferentiability">Differentiability versus partial differentiability</h3> <p>The function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><msup><mi>ℝ</mi> <mn>2</mn></msup><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">f:\mathbb{R}^2 \to \mathbb{R}</annotation></semantics></math> defined by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo><mrow><mtable displaystyle="false" columnalign="left left"><mtr><mtd><mfrac><mrow><msup><mi>y</mi> <mn>3</mn></msup></mrow><mrow><msup><mi>x</mi> <mn>2</mn></msup><mo>+</mo><msup><mi>y</mi> <mn>2</mn></msup></mrow></mfrac></mtd> <mtd><mspace width="1em"></mspace><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>≠</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><mspace width="1em"></mspace><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow></mrow><annotation encoding="application/x-tex"> f(x,y) = \begin{cases} \frac{y^3}{x^2+y^2} &\quad (x,y) \neq (0,0)\\ 0 &\quad (x,y) = (0,0) \end{cases} </annotation></semantics></math></div> <p>is continuous everywhere, and has directional derivatives (def. <a class="maruku-ref" href="#DirectionalDerivative"></a>) at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(0,0)</annotation></semantics></math> in all directions, but is not differentiable at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(0,0)</annotation></semantics></math> in the sense of def. <a class="maruku-ref" href="#DerivativeOfFunctionFromRnToR"></a>. Note that the (unnormalized) directional derivative along the vector <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(a,b)</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mrow><msup><mi>b</mi> <mn>3</mn></msup></mrow><mrow><msup><mi>a</mi> <mn>2</mn></msup><mo>+</mo><msup><mi>b</mi> <mn>2</mn></msup></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{b^3}{a^2+b^2}</annotation></semantics></math>, which is not linear.</p> <p>One may wonder whether existence of a linear map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><msub><mi>f</mi> <mi>x</mi></msub></mrow><annotation encoding="application/x-tex">d f_x</annotation></semantics></math> is enough, but this is also not the case. The function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><msup><mi>ℝ</mi> <mn>2</mn></msup><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">f:\mathbb{R}^2 \to \mathbb{R}</annotation></semantics></math> defined by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo><mrow><mtable displaystyle="false" columnalign="left left"><mtr><mtd><mfrac><mrow><msup><mi>y</mi> <mn>3</mn></msup></mrow><mi>x</mi></mfrac></mtd> <mtd><mspace width="1em"></mspace><mi>x</mi><mo>≠</mo><mn>0</mn></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><mspace width="1em"></mspace><mi>x</mi><mo>=</mo><mn>0</mn></mtd></mtr></mtable></mrow></mrow></mrow><annotation encoding="application/x-tex"> f(x,y) = \begin{cases} \frac{y^3}{x} &\quad x\neq 0\\ 0 &\quad x=0 \end{cases} </annotation></semantics></math></div> <p>has all directional derivatives at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(0,0)</annotation></semantics></math> equaling <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math>, so that in particular there is a linear map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><msub><mi>f</mi> <mrow><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">d f_{(0,0)}</annotation></semantics></math> whose values are the directional derivatives. But it is not differentiable at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(0,0)</annotation></semantics></math>; in fact, it is not even continuous at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(0,0)</annotation></semantics></math>. (Thus it also provides an example of a discontinuous function which has all directional derivatives.)</p> <p>The discontinuity kind of gives this last one away; maybe a continuous function with all directional derivatives must be differentiable? But no, because if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math> is a <span class="newWikiWord">Weierstrass function<a href="/nlab/new/Weierstrass+function">?</a></span> (continuous everywhere but differentiable nowhere) from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math> with <a class="existingWikiWord" href="/nlab/show/periodic+function">period</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><mi>π</mi></mrow><annotation encoding="application/x-tex">2\pi</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="0em" rspace="thinmathspace">atan2</mo></mrow><annotation encoding="application/x-tex">\operatorname{atan2}</annotation></semantics></math> is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-argument <a class="existingWikiWord" href="/nlab/show/arctangent">arctangent</a> function from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^2</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math> (so that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msqrt><mrow><msup><mi>x</mi> <mn>2</mn></msup><mo>+</mo><msup><mi>y</mi> <mn>2</mn></msup></mrow></msqrt><mspace width="thinmathspace"></mspace><mi>sin</mi><mspace width="thinmathspace"></mspace><mo lspace="0em" rspace="thinmathspace">atan2</mo><mo stretchy="false">(</mo><mi>y</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">\sqrt{x^2+y^2} \,\sin\,\operatorname{atan2}(y,x) = y</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msqrt><mrow><msup><mi>x</mi> <mn>2</mn></msup><mo>+</mo><msup><mi>y</mi> <mn>2</mn></msup></mrow></msqrt><mspace width="thinmathspace"></mspace><mi>cos</mi><mspace width="thinmathspace"></mspace><mo lspace="0em" rspace="thinmathspace">atan2</mo><mo stretchy="false">(</mo><mi>y</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">\sqrt{x^2+y^2} \,\cos\,\operatorname{atan2}(y,x) = x</annotation></semantics></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math>), then set</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>2</mn></msup><mo>+</mo><msup><mi>y</mi> <mn>2</mn></msup><mo stretchy="false">)</mo><mi>W</mi><mo stretchy="false">(</mo><mo lspace="0em" rspace="thinmathspace">atan2</mo><mo stretchy="false">(</mo><mi>y</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex"> f(x,y) = (x^2 + y^2) W(\operatorname{atan2}(y,x)) .</annotation></semantics></math></div> <p>Then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is, like <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math>, continuous everywhere and differentiable nowhere, yet the directional derivatives through the origin are all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math>. There are further examples (including one that is analytic except at the origin) at <a href="https://math.stackexchange.com/questions/1497043/is-there-a-function-thats-continuous-and-has-all-directional-derivatives-as-a-l">Math StackExchange question 1497043</a>.</p> <h3 id="differentiability_versus_continuous_and_higher_differentiability">Differentiability versus continuous and higher differentiability</h3> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/natural+number">natural number</a>, and consider the function</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mi>k</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≔</mo><msup><mi>x</mi> <mi>k</mi></msup><mi>sin</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">/</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> f_k(x) \coloneqq x^k \sin(1/x) </annotation></semantics></math></div> <p>from the <a class="existingWikiWord" href="/nlab/show/real+line">real line</a> to itself, with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>≔</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">f(0) \coloneqq 0</annotation></semantics></math>. Away from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">f_k</annotation></semantics></math> is smooth (even analytic); but at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">f_0</annotation></semantics></math> is not continuous, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">f_1</annotation></semantics></math> is continuous but not differentiable, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">f_2</annotation></semantics></math> is differentiable but not continuously differentiable, and so on:</p> <ul> <li>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>=</mo><mn>2</mn><mi>n</mi></mrow><annotation encoding="application/x-tex">k = 2 n</annotation></semantics></math> is even, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">f_k</annotation></semantics></math> is differentiable <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> times but not continuously differentiable <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> times;</li> <li>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>=</mo><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">k = 2 n + 1</annotation></semantics></math> is odd, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">f_k</annotation></semantics></math> is continuously differentiable <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> times but not differentiable <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n + 1</annotation></semantics></math> times.</li> </ul> <p>(For a fractional value of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">f_k</annotation></semantics></math> has the same behaviour as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mrow><mo stretchy="false">⌈</mo><mi>k</mi><mo stretchy="false">⌉</mo></mrow></msub></mrow><annotation encoding="application/x-tex">f_{\lceil{k}\rceil}</annotation></semantics></math>.)</p> <p>Similarly, in two dimensions we can consider functions such as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>2</mn></msup><mo>+</mo><msup><mi>y</mi> <mn>2</mn></msup><mo stretchy="false">)</mo><mi>sin</mi><mo stretchy="false">(</mo><mfrac><mn>1</mn><msqrt><mrow><msup><mi>x</mi> <mn>2</mn></msup><mo>+</mo><msup><mi>y</mi> <mn>2</mn></msup></mrow></msqrt></mfrac><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex"> f(x,y) = (x^2+y^2)\sin(\frac{1}{\sqrt{x^2+y^2}}).</annotation></semantics></math></div> <p>together with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">f(0,0) = 0</annotation></semantics></math>. This is smooth away from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math>, and once differentiable at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math>, even in the strong sense that it is well-approximated by a linear function near <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math>. However, its derivative is not continuous at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math>. In particular, this shows that the converse of the theorem “if the partial derivatives exist and are continuous at a point, then the function is differentiable there” fails, even in higher dimensions.</p> <p>Uniform differentiability is stronger than continuous differentiability but independent of twice differentiability. For an example that is uniformly differentiable but not twice differentiable, use <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex">f_3</annotation></semantics></math> again. For an example that that is twice differentiable (and hence continuously differentiable) but not uniformly differentiable, use <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>↦</mo><msup><mi>x</mi> <mn>3</mn></msup></mrow><annotation encoding="application/x-tex">x \mapsto x^3</annotation></semantics></math>. However, on a <a class="existingWikiWord" href="/nlab/show/compact+set">compact</a> domain, any continuously differentiable function (and a fortiori any twice differentiable function) must be uniformly continuous (at least in <a class="existingWikiWord" href="/nlab/show/classical+mathematics">classical</a> and <a class="existingWikiWord" href="/nlab/show/intuitionistic+mathematics">intuitionistic mathematics</a>).</p> <h3 id="symmetry_of_the_second_partial_derivatives">Symmetry of the second partial derivatives</h3> <p>The function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><msup><mi>ℝ</mi> <mn>2</mn></msup><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">f:\mathbb{R}^2 \to \mathbb{R}</annotation></semantics></math> defined by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi>x</mi><mi>y</mi><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>2</mn></msup><mo>−</mo><msup><mi>y</mi> <mn>2</mn></msup><mo stretchy="false">)</mo></mrow><mrow><msup><mi>x</mi> <mn>2</mn></msup><mo>+</mo><msup><mi>y</mi> <mn>2</mn></msup></mrow></mfrac></mrow><annotation encoding="application/x-tex"> f(x,y) = \frac{x y (x^2-y^2)}{x^2+y^2}</annotation></semantics></math></div> <p>plus <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">f(0,0) = 0</annotation></semantics></math>, has partial derivatives <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mrow><msup><mo>∂</mo> <mn>2</mn></msup><mi>f</mi></mrow><mrow><mo>∂</mo><mi>x</mi><mo>∂</mo><mi>y</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\partial^2f}{\partial x \partial y}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mrow><msup><mo>∂</mo> <mn>2</mn></msup><mi>f</mi></mrow><mrow><mo>∂</mo><mi>y</mi><mo>∂</mo><mi>x</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\partial^2f}{\partial y \partial x}</annotation></semantics></math> that both exist but are not equal at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(0,0)</annotation></semantics></math> (nor are they continuous at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(0,0)</annotation></semantics></math>). Therefore, by the theorem proven above, it is not twice differentiable at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(0,0)</annotation></semantics></math>.</p> <h3 id="twice_differentiability_versus_quadratic_approximation">Twice differentiability versus quadratic approximation</h3> <p>While differentiability means approximability by a linear function, twice differentiability does <em>not</em> mean approximability by a quadratic function. For example, the function</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>x</mi> <mn>3</mn></msup><mi>sin</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">/</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> f(x) = x^3 \sin(1/x) </annotation></semantics></math></div> <p>(with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">f(0) =0</annotation></semantics></math>, to make it continuous there) is not twice differentiable at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">x=0</annotation></semantics></math>, but it is well-approximated by the quadratic polynomial <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">p(x) = 0</annotation></semantics></math> in the sense that their difference is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>o</mi><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">o(x^2)</annotation></semantics></math> as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">x\to 0</annotation></semantics></math>. Even worse, the function</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>e</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><mo stretchy="false">/</mo><msup><mi>x</mi> <mn>2</mn></msup></mrow></msup><mi>sin</mi><mo stretchy="false">(</mo><msup><mi>e</mi> <mrow><mn>1</mn><mo stretchy="false">/</mo><msup><mi>x</mi> <mn>2</mn></msup></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> f(x) = e^{-1/x^2} \sin(e^{1/x^2}) </annotation></semantics></math></div> <p>is not twice differentiable at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">x=0</annotation></semantics></math>, but it is well-approximated by a polynomial of <em>any</em> finite degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> (namely, the zero polynomial), in the sense that their difference is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>o</mi><mo stretchy="false">(</mo><msup><mi>x</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">o(x^n)</annotation></semantics></math> as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">x\to 0</annotation></semantics></math>.</p> <p>In some contexts, it is useful to say that functions such as these have “pointwise second derivatives”. More precisely, we say that a function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> has a <strong>pointwise <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>k</mi> <mi>th</mi></msup></mrow><annotation encoding="application/x-tex">k^{th}</annotation></semantics></math> derivative</strong> at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math> if it is continuous at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math> and there exists a polynomial <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> of degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> such that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mi>lim</mi> <mrow><mi>x</mi><mo>→</mo><mi>a</mi></mrow></munder><mfrac><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>−</mo><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mi>a</mi><msup><mo stretchy="false">)</mo> <mi>k</mi></msup></mrow></mfrac><mo>=</mo><mn>0</mn><mo>.</mo></mrow><annotation encoding="application/x-tex"> \lim_{x\to a} \frac{f(x) - p(x)}{(x-a)^k} = 0. </annotation></semantics></math></div> <p>(We have to state continuity explicitly in order to match the usual notion of differentiability when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">k = 1</annotation></semantics></math>, since nothing in this limit constrains the value of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(a)</annotation></semantics></math>.) In this case, the pointwise <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>k</mi> <mi>th</mi></msup></mrow><annotation encoding="application/x-tex">k^{th}</annotation></semantics></math> derivative is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>f</mi> <mi>pt</mi> <mrow><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msubsup><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>p</mi> <mrow><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f^{(k)}_{pt}(a) = p^{(k)}(a)</annotation></semantics></math> (and it follows that all lower-order pointwise derivatives match as well). Thus we would say that while <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>x</mi> <mn>3</mn></msup><mi>sin</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">/</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x) = x^3 \sin(1/x)</annotation></semantics></math> does not have a second derivative at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math>, it does have a <em>pointwise</em> second derivative at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mi>pt</mi></msub><mo>″</mo><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">f_{pt}''(0) = 0</annotation></semantics></math>. Similarly, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>e</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><mo stretchy="false">/</mo><msup><mi>x</mi> <mn>2</mn></msup></mrow></msup><mi>sin</mi><mo stretchy="false">(</mo><msup><mi>e</mi> <mrow><mn>1</mn><mo stretchy="false">/</mo><msup><mi>x</mi> <mn>2</mn></msup></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">e^{-1/x^2} \sin(e^{1/x^2})</annotation></semantics></math> is pointwise smooth. See e.g. <a href="http://math.stackexchange.com/a/160383/91608">this MSE answer</a>.</p> <p>The polynomial <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> may be thought of as the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>th-order <a class="existingWikiWord" href="/nlab/show/Taylor+polynomial">Taylor polynomial</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math>, and the limit above is Taylor’s theorem with the Peano remainder. (Thus the subscript <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>pt</mi></mrow><annotation encoding="application/x-tex">pt</annotation></semantics></math> can be thought of as standing for ‘Peano–Taylor’ as well as for ‘pointwise’.) Note that while <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-times pointwise differentiability is weaker than <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-times differentiability for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>></mo><mn>1</mn></mrow><annotation encoding="application/x-tex">k \gt 1</annotation></semantics></math> and equivalent for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">k = 1</annotation></semantics></math>, it is stronger for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">k = 0</annotation></semantics></math> (since even <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math>-times pointwise differentiability requires continuity).</p> <h3 id="the_second_derivative_as_a_quadratic_form">The second derivative as a quadratic form</h3> <p>In the definition of strong twice-differentiability, we cannot replace the symmetric <a class="existingWikiWord" href="/nlab/show/bilinear+map">bilinear map</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mo>∂</mo> <mn>2</mn></msup><msub><mi>f</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\partial^2 f_x(v,w)</annotation></semantics></math> by the corresponding <a class="existingWikiWord" href="/nlab/show/quadratic+form">quadratic form</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Q</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mo>∂</mo> <mn>2</mn></msup><msub><mi>f</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>v</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Q_x(v) = \partial^2f_x(v,v)</annotation></semantics></math>. In other words, if we suppose that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>2</mn><mi>v</mi><mo stretchy="false">)</mo><mo>−</mo><mn>2</mn><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>v</mi><mo stretchy="false">)</mo><mo>+</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>Q</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo>+</mo><mi>E</mi><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><msup><mrow><mo stretchy="false">|</mo><mi>v</mi><mo stretchy="false">|</mo></mrow> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">f(x+2v) - 2 f(x+v) + f(x) = Q_x(v) + E(v) {|v|}^2 </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>lim</mi> <mrow><mi>v</mi><mo>→</mo><mn>0</mn></mrow></msub><mi>E</mi><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\lim_{v\to 0} E(v) = 0</annotation></semantics></math>, for some quadratic form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Q</mi> <mi>x</mi></msub></mrow><annotation encoding="application/x-tex">Q_x</annotation></semantics></math>, it does not follow that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is twice differentiable at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math>, even in one dimension. The same old counterexample</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>x</mi> <mn>3</mn></msup><mi>sin</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">/</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> f(x) = x^3 \sin(1/x) </annotation></semantics></math></div> <p>(with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">f(0)=0</annotation></semantics></math>) works: we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mn>0</mn><mo>+</mo><mn>2</mn><mi>v</mi><mo stretchy="false">)</mo><mo>−</mo><mn>2</mn><mi>f</mi><mo stretchy="false">(</mo><mn>0</mn><mo>+</mo><mi>v</mi><mo stretchy="false">)</mo><mo>+</mo><mi>f</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mn>2</mn><msup><mi>v</mi> <mn>3</mn></msup><mo stretchy="false">(</mo><mn>4</mn><mi>sin</mi><mo stretchy="false">(</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>v</mi></mrow></mfrac><mo stretchy="false">)</mo><mo>−</mo><mi>sin</mi><mo stretchy="false">(</mo><mfrac><mn>1</mn><mi>v</mi></mfrac><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(0+2v) - 2 f(0+v) + f(0)= 2v^3(4\sin(\frac{1}{2v}) - \sin(\frac{1}{v}))</annotation></semantics></math></div> <p>so <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo>=</mo><mi>v</mi><mo stretchy="false">(</mo><mn>4</mn><mi>sin</mi><mo stretchy="false">(</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>v</mi></mrow></mfrac><mo stretchy="false">)</mo><mo>−</mo><mi>sin</mi><mo stretchy="false">(</mo><mfrac><mn>1</mn><mi>v</mi></mfrac><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">E(v) = v (4\sin(\frac{1}{2v}) - \sin(\frac{1}{v})) \to 0</annotation></semantics></math>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extremum">extremum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+rule">chain rule</a>, <a class="existingWikiWord" href="/nlab/show/total+derivative">total derivative</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a>, <strong>differentiable function</strong>, <a class="existingWikiWord" href="/nlab/show/continuously+differentiable+function">continuously differentiable function</a>, <a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a>, <a class="existingWikiWord" href="/nlab/show/analytic+function">analytic function</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derivative+of+distributions">derivative of distributions</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hasse-Schmidt+derivative">Hasse-Schmidt derivative</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integrable+function">integrable function</a>, <a class="existingWikiWord" href="/nlab/show/square-integrable+function">square-integrable function</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bounded+function">bounded function</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compactly+supported+function">compactly supported function</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/measurable+function">measurable function</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/rapidly+decreasing+function">rapidly decreasing function</a></p> </li> </ul> <h2 id="references">References</h2> <p>Early account, in the context of <a class="existingWikiWord" href="/nlab/show/Cohomotopy">Cohomotopy</a>, <a class="existingWikiWord" href="/nlab/show/cobordism+theory">cobordism theory</a> and the <a class="existingWikiWord" href="/nlab/show/Pontryagin-Thom+construction">Pontryagin-Thom construction</a>:</p> <ul> <li id="Pontrjagin55"><a class="existingWikiWord" href="/nlab/show/Lev+Pontrjagin">Lev Pontrjagin</a>, Chapter I of: <em>Smooth manifolds and their applications in Homotopy theory</em>, Trudy Mat. Inst. im Steklov, No 45, Izdat. Akad. Nauk. USSR, Moscow, 1955 (AMS Translation Series 2, Vol. 11, 1959) (<a href="https://www.worldscientific.com/doi/abs/10.1142/9789812772107_0001">doi:10.1142/9789812772107_0001</a>, <a href="https://www.maths.ed.ac.uk/~v1ranick/papers/pont001.pdf">pdf</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on December 1, 2023 at 20:14:06. 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