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id="searchField" name="query" value="Search" 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>Contents</title></head> <body> <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="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#statements'>Statements</a></li> <li><a href='#related_theorems'>Related theorems</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>On a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> <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> the smooth <a class="existingWikiWord" href="/nlab/show/tangent+vector+fields">tangent vector fields</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi><mo>∈</mo><msub><mi>Γ</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mi>T</mi><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">v \in \Gamma_X(T X)</annotation></semantics></math> are geometrically defined as smoothly varying collections of smooth <a class="existingWikiWord" href="/nlab/show/paths">paths</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>γ</mi> <mi>x</mi></msub><mo lspace="verythinmathspace">:</mo><msup><mi>ℝ</mi> <mn>1</mn></msup><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\gamma_x \colon \mathbb{R}^1 \to X</annotation></semantics></math> through every point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x \in X</annotation></semantics></math> (i.e. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>γ</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">\gamma(0) = x</annotation></semantics></math>), with two such paths regarded as equivalent if their first <a class="existingWikiWord" href="/nlab/show/derivative">derivative</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mi>x</mi></msub><mo>∈</mo><msub><mi>T</mi> <mi>x</mi></msub><mi>X</mi></mrow><annotation encoding="application/x-tex">v_x \in T_x X</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>, seen in any <a class="existingWikiWord" href="/nlab/show/chart">chart</a>, coincides.</p> <p>By pre-composing a <a class="existingWikiWord" href="/nlab/show/smooth+function">smooth 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><msup><mi>ℝ</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">f \colon X \longrightarrow \mathbb{R}^1</annotation></semantics></math> with such a path, we obtain a function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∘</mo><msub><mi>γ</mi> <mi>x</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>ℝ</mi> <mn>1</mn></msup><mo>→</mo><msup><mi>ℝ</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">f\circ \gamma_x \;\colon\; \mathbb{R}^1 \to \mathbb{R}^1</annotation></semantics></math> from the <a class="existingWikiWord" href="/nlab/show/real+line">real line</a> to itself. Therefore its <a class="existingWikiWord" href="/nlab/show/derivative">derivative</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>D</mi> <mi>v</mi></msub><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≔</mo><mi>d</mi><mo stretchy="false">(</mo><mi>f</mi><mo>∘</mo><mi>γ</mi><msub><mo stretchy="false">)</mo> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex"> (D_v f)(x) \coloneqq d (f \circ \gamma)_0 </annotation></semantics></math></div> <p>may be identified with a <a class="existingWikiWord" href="/nlab/show/real+number">real number</a> (measuring the rate of change 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> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>γ</mi></mrow><annotation encoding="application/x-tex">\gamma</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> to first order). Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>γ</mi> <mi>x</mi></msub></mrow><annotation encoding="application/x-tex">\gamma_x</annotation></semantics></math>, or rather its derivative <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mi>x</mi></msub><mo>∈</mo><msub><mi>T</mi> <mi>x</mi></msub><mi>X</mi></mrow><annotation encoding="application/x-tex">v_x \in T_x X</annotation></semantics></math>, is assumed to depend smoothly on <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>, this defines a new smooth function</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>D</mi> <mi>v</mi></msub><mi>f</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>⟶</mo><msup><mi>ℝ</mi> <mn>1</mn></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> D_v f \;\colon\; X \longrightarrow \mathbb{R}^1 \,. </annotation></semantics></math></div> <p>Due to the <a class="existingWikiWord" href="/nlab/show/product+rule">product rule</a> of differentiation, this assignment <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>↦</mo><msub><mi>D</mi> <mi>v</mi></msub><mi>f</mi></mrow><annotation encoding="application/x-tex">f \mapsto D_v f</annotation></semantics></math> is such that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>,</mo><mi>g</mi><mo>∈</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f,g \in C^\infty(X)</annotation></semantics></math> two smooth functions, with pointwise product function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>⋅</mo><mi>g</mi></mrow><annotation encoding="application/x-tex">f \cdot g</annotation></semantics></math> then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>D</mi> <mi>v</mi></msub><mo stretchy="false">(</mo><mi>f</mi><mo>⋅</mo><mi>g</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><msub><mi>D</mi> <mi>v</mi></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>g</mi><mo>+</mo><mi>f</mi><mo>⋅</mo><msub><mi>D</mi> <mi>v</mi></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> D_v(f \cdot g) \;=\; D_v(f) \cdot g + f \cdot D_v(g) \,. </annotation></semantics></math></div> <p>This means that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>D</mi> <mi>v</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">D_v \;\colon\; C^\infty(X) \to C^\infty(X)</annotation></semantics></math> is a <em><a class="existingWikiWord" href="/nlab/show/derivation">derivation</a></em> on the algebra of smooth functions.</p> <p>Remarkably, <em>all</em> derivations on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^\infty(X)</annotation></semantics></math> arise this way, and for a unique vector field <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi><mo>∈</mo><msub><mi>Γ</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mi>T</mi><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">v \in \Gamma_X(T X)</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mtext>derivations on</mtext></mtd></mtr> <mtr><mtd><mtext>algebra of smooth functions</mtext></mtd></mtr> <mtr><mtd><mtext>on</mtext><mspace width="thinmathspace"></mspace><mi>X</mi></mtd></mtr></mtable></mrow><mo>}</mo></mrow><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mtext>smooth</mtext></mtd></mtr> <mtr><mtd><mtext>tangent vector fields</mtext></mtd></mtr> <mtr><mtd><mtext>on</mtext><mspace width="thinmathspace"></mspace><mi>X</mi></mtd></mtr></mtable></mrow><mo>}</mo></mrow></mrow><annotation encoding="application/x-tex"> \left\{ \array{ \text{derivations on} \\ \text{algebra of smooth functions} \\ \text{on}\, X } \right\} \;\simeq\; \left\{ \array{ \text{smooth} \\ \text{tangent vector fields} \\ \text{on}\, X } \right\} </annotation></semantics></math></div> <p>This is prop. <a class="maruku-ref" href="#Statement"></a> below. What makes this work is the <em><a class="existingWikiWord" href="/nlab/show/Hadamard+lemma">Hadamard lemma</a></em>, see the <a href="#Proof">proof</a> below for details.</p> <p>Notice that the concept of derivations is purely a concept of <a class="existingWikiWord" href="/nlab/show/algebra">algebra</a>, with no input from the <a class="existingWikiWord" href="/nlab/show/topology">topology</a> and <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a> that goes into the definition and construction of smooth manifolds and their tangent bundles. Therefore the identification of smooth vector fields with derivations is an algebraic incarnation of an aspect of differential geometry, an identification comparable two two other such phenomena:</p> <ul> <li> <p>the <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> – which identifies smooth manifolds themselves with the <a class="existingWikiWord" href="/nlab/show/formal+dual">formal dual</a> of their real algebras of smooth functions;</p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/smooth+Serre-Swan+theorem">smooth Serre-Swan theorem</a> which identifies <a class="existingWikiWord" href="/nlab/show/smooth+vector+bundles">smooth vector bundles</a> over a smooth manifold with the <a class="existingWikiWord" href="/nlab/show/projective+modules">projective modules</a> of the algebra of smooth functions.</p> </li> </ul> <p>These statements mean that <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a> has more in common with <a class="existingWikiWord" href="/nlab/show/algebraic+geometry">algebraic geometry</a> than is manifest from the traditional definitions. This serves as the bases for the definition of <a class="existingWikiWord" href="/nlab/show/formal+smooth+manifolds">formal smooth manifolds</a> and the theory of <a class="existingWikiWord" href="/nlab/show/synthetic+differential+geometry">synthetic differential geometry</a>. For more exposition of this relation see at <em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+supergeometry">geometry of physics – supergeometry</a></em>.</p> <p>Beware however that related algebraic properties familiar from <a class="existingWikiWord" href="/nlab/show/affine+schemes">affine schemes</a> may break in differential geometry: For the <a class="existingWikiWord" href="/nlab/show/K%C3%A4hler+differential+forms">Kähler differential forms</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^\infty(X)</annotation></semantics></math> to come out as the expected smooth <a class="existingWikiWord" href="/nlab/show/differential+forms">differential forms</a> one needs to refine the plain <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>-<a class="existingWikiWord" href="/nlab/show/commutative+algebra">commutative 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><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^\infty(X)</annotation></semantics></math> to the structure of a <em><a class="existingWikiWord" href="/nlab/show/smooth+algebra">smooth algebra</a></em>. See at <em><a class="existingWikiWord" href="/nlab/show/K%C3%A4hler+differential+forms">Kähler differential forms</a></em> for discussion of this issue.</p> <h2 id="statements">Statements</h2> <div class="num_theorem" id="Statement"> <h6 id="theorem">Theorem</h6> <p>Let <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> be a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a>.</p> <p>Write</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^\infty(X)</annotation></semantics></math> for its real <a class="existingWikiWord" href="/nlab/show/algebra+of+functions">algebra</a> of <a class="existingWikiWord" href="/nlab/show/smooth+functions">smooth functions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><msup><mi>ℝ</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">X \to \mathbb{R}^1</annotation></semantics></math> (under pointwise multiplication);</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Γ</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mi>T</mi><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Gamma_X(T X)</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/real+vector+space">real vector space</a> of smooth <a class="existingWikiWord" href="/nlab/show/tangent+vector+fields">tangent vector fields</a> on <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>, hence the space of smooth <a class="existingWikiWord" href="/nlab/show/sections">sections</a> of the <a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">T X</annotation></semantics></math> 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>, regarded as a <a class="existingWikiWord" href="/nlab/show/smooth+vector+bundle">smooth vector bundle</a>.</p> </li> </ol> <p>1, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Der</mi><mo stretchy="false">(</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Der(C^\infty(X))</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/real+vector+space">real vector space</a> of <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>-linear <a class="existingWikiWord" href="/nlab/show/derivations">derivations</a> of the <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>-algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^\infty(X)</annotation></semantics></math></p> <p>Then:</p> <ol> <li> <p>The operation of <a class="existingWikiWord" href="/nlab/show/differentiation">differentiation</a> with respect to a <a class="existingWikiWord" href="/nlab/show/vector+field">vector field</a> is a <a class="existingWikiWord" href="/nlab/show/derivation">derivation</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^\infty(X)</annotation></semantics></math>.</p> </li> <li> <p>Every <a class="existingWikiWord" href="/nlab/show/derivation">derivation</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^\infty(X)</annotation></semantics></math> arises this way, for a unique vector field.</p> </li> </ol> <p>Hence there is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>D</mi> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>Γ</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mi>T</mi><mi>X</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mo>≃</mo></mover><mi>Der</mi><mo stretchy="false">(</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> D_{(-)} \;\colon\; \Gamma_X(T X) \stackrel{\simeq}{\longrightarrow} Der(C^\infty(X)) \,. </annotation></semantics></math></div></div> <div class="proof" id="Proof"> <h6 id="proof">Proof</h6> <p>First to discuss that vector fields induce derivations. Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi><mo>∈</mo><msub><mi>Γ</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mi>T</mi><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">v \in \Gamma_X(T X)</annotation></semantics></math> be a smooth tangent vector field.</p> <p>This means first of all that for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x \in X</annotation></semantics></math> there is a <a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>γ</mi> <mi>x</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>ℝ</mi> <mn>1</mn></msup><mo>⟶</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\gamma_x \;\colon\; \mathbb{R}^1 \longrightarrow X</annotation></semantics></math> such that</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>γ</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">\gamma_x(0) = x</annotation></semantics></math></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><msub><mi>γ</mi> <mi>x</mi></msub><msub><mo></mo><mn>0</mn></msub><mo>=</mo><msub><mi>v</mi> <mi>x</mi></msub><mo>∈</mo><msub><mi>T</mi> <mi>x</mi></msub><mi>X</mi></mrow><annotation encoding="application/x-tex">d \gamma_x_0 = v_x \in T_x X</annotation></semantics></math>.</p> </li> </ol> <p>Then for <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><msup><mi>ℝ</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">f \colon X \to \mathbb{R}^1</annotation></semantics></math> a smooth function, define a new function</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>D</mi> <mi>v</mi></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>⟶</mo><msup><mi>ℝ</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex"> D_v(f) \;\colon\; X \longrightarrow \mathbb{R}^1 </annotation></semantics></math></div> <p>by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>D</mi> <mi>v</mi></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≔</mo><mi>d</mi><mo stretchy="false">(</mo><mi>f</mi><mo>∘</mo><msub><mi>γ</mi> <mi>x</mi></msub><msub><mo stretchy="false">)</mo> <mn>0</mn></msub><mo>=</mo><mfrac><mi>d</mi><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>γ</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mo stretchy="false">|</mo> <mrow><mi>t</mi><mo>=</mo><mn>0</mn></mrow></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (D_v(f))(x) \coloneqq d (f \circ \gamma_x)_0 = \frac{d}{d t} (f(\gamma_x(t)))\vert_{t = 0} \,. </annotation></semantics></math></div> <p>By the linearity of differentation we have for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">c \in \mathbb{R}</annotation></semantics></math> that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>D</mi> <mi>v</mi></msub><mo stretchy="false">(</mo><mi>c</mi><mo>⋅</mo><mi>f</mi><mo stretchy="false">)</mo><mo>=</mo><mi>c</mi><mo>⋅</mo><msub><mi>D</mi> <mi>v</mi></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">D_v(c \cdot f) = c \cdot D_v(f)</annotation></semantics></math>. Moreover, by the <a class="existingWikiWord" href="/nlab/show/product+rule">product rule</a> for differentiation we have for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>,</mo><mi>g</mi><mo>∈</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f,g \in C^\infty(X)</annotation></semantics></math> two smooth functions, that</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><msub><mi>D</mi> <mi>v</mi></msub><mo stretchy="false">(</mo><mi>f</mi><mo>⋅</mo><mi>g</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mfrac><mi>d</mi><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>f</mi><mo>⋅</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>γ</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mo stretchy="false">|</mo> <mrow><mi>t</mi><mo>=</mo><mn>0</mn></mrow></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mfrac><mi>d</mi><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>γ</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⋅</mo><mi>g</mi><mo stretchy="false">(</mo><msub><mi>γ</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mo stretchy="false">|</mo> <mrow><mi>t</mi><mo>=</mo><mn>0</mn></mrow></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mrow><mo>(</mo><mfrac><mi>d</mi><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mi>f</mi><mo stretchy="false">(</mo><msub><mi>γ</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mo stretchy="false">|</mo> <mrow><mi>t</mi><mo>=</mo><mn>0</mn></mrow></msub><mo>)</mo></mrow><mo>⋅</mo><mi>g</mi><mo stretchy="false">(</mo><msub><mi>γ</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>+</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>γ</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⋅</mo><mrow><mo>(</mo><mfrac><mi>d</mi><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mi>g</mi><mo stretchy="false">(</mo><mi>γ</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mo stretchy="false">|</mo> <mrow><mi>t</mi><mo>=</mo><mn>0</mn></mrow></msub><mo>)</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo stretchy="false">(</mo><msub><mi>D</mi> <mi>v</mi></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>g</mi><mo>+</mo><mi>f</mi><mo>⋅</mo><msub><mi>D</mi> <mi>v</mi></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} D_v( f \cdot g) & = \frac{d}{d t}( (f \cdot g)(\gamma(t)) )\vert_{t = 0} \\ & = \frac{d}{ d t}( f(\gamma_x(t)) \cdot g(\gamma_x(t)) )\vert_{t = 0} \\ & = \left( \frac{d}{d t} f(\gamma_x(t))\vert_{t = 0} \right) \cdot g(\gamma_x(0)) + f(\gamma_x(0)) \cdot \left( \frac{d}{d t} g(\gamma(t))\vert_{t = 0} \right) \\ & = (D_v(f) \cdot g + f \cdot D_v(g))(x) \end{aligned} \,. </annotation></semantics></math></div> <p>Therefore it only remains to check that the function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>D</mi> <mi>v</mi></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">D_v(f)</annotation></semantics></math> is a smooth function in the first place.</p> <p>Let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" 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></div> <p>be an <a class="existingWikiWord" href="/nlab/show/atlas">atlas</a> that exhibits the <a class="existingWikiWord" href="/nlab/show/smooth+structure">smooth structure</a> 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>, hence an <a class="existingWikiWord" href="/nlab/show/open+cover">open cover</a> by subsets equipped with a <a class="existingWikiWord" href="/nlab/show/diffeomorphism">diffeomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\phi_i</annotation></semantics></math> from a <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a> <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> with its standard smooth structure.</p> <p>By definition of the <a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">T X</annotation></semantics></math>, its restriction to any <a class="existingWikiWord" href="/nlab/show/chart">chart</a> of this atlas is given by a diffeomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ψ</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\psi_i</annotation></semantics></math> as in the following diagram</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><mo>×</mo><msup><mi>ℝ</mi> <mi>n</mi></msup></mtd> <mtd></mtd> <mtd><munderover><mo>⟶</mo><mo>≃</mo><mrow><msub><mi>ψ</mi> <mi>i</mi></msub></mrow></munderover></mtd> <mtd></mtd> <mtd><mi>T</mi><mi>X</mi><msub><mo stretchy="false">|</mo> <mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mrow><msub><mi>ϕ</mi> <mi>i</mi></msub><mo>∘</mo><mpadded width="0" lspace="-100%width"><mrow><msub><mi>pr</mi> <mn>1</mn></msub></mrow></mpadded></mrow></msub><mo>↘</mo></mtd> <mtd></mtd> <mtd><msub><mo>↙</mo> <mrow><mi>p</mi><msub><mo stretchy="false">|</mo> <mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow></msub></mrow></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msub><mi>U</mi> <mi>i</mi></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \mathbb{R}^n \times \mathbb{R}^n && \underoverset{\simeq}{\psi_i}{\longrightarrow} && T X\vert_{U_i} \\ & {}_{\phi_i \circ \mathllap{pr_1}}\searrow && \swarrow_{p\vert_{U_i}} \\ && U_i } \,. </annotation></semantics></math></div> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>D</mi> <mi>v</mi></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><msup><mi>ℝ</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">D_v(f) \colon X \to \mathbb{R}^1</annotation></semantics></math> to be continuous and smooth, it is sufficient to see that its restrictions</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>D</mi> <mi>v</mi></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>∘</mo><msub><mi>ψ</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex"> D_v(f) \circ \psi_i </annotation></semantics></math></div> <p>are continuous and smooth, 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>. (For continuity this follows by <a href="Top#ClosedSubspacesGluing">this example</a> and for differentiability this is clear from the fact that derivatives at any point may be computed in an arbitrary open neighbourhood).</p> <p>With respect to this chart the tangent vector field <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi><mo>∈</mo><msub><mi>Γ</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mi>T</mi><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">v \in \Gamma_X(T X)</annotation></semantics></math> is given by a smooth function</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mi>i</mi></msub><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><msub><mi>pr</mi> <mn>2</mn></msub><mo>∘</mo><msubsup><mi>ψ</mi> <mi>i</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>v</mi><msub><mo stretchy="false">|</mo> <mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>U</mi> <mi>i</mi></msub><mo>⟶</mo><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex"> v_i \;\coloneqq\; pr_2 \circ \psi_i^{-1}(v \vert_{U_i}) \;\colon\; U_i \longrightarrow \mathbb{R}^n </annotation></semantics></math></div> <p>By the identification of tangent vectors on <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> with elements of <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>, these tangent vectors are represented by straight smooth curves (<a href="Introduction+to+Topology+--+1EuclideanSpaceTangentBundle">this remark</a>):</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>γ</mi> <mrow><mi>x</mi><mo>,</mo><mi>i</mi></mrow></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>t</mi><mo>↦</mo><msubsup><mi>ϕ</mi> <mi>i</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><mi>t</mi><mo>⋅</mo><msub><mi>v</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \gamma_{x, i} \;\colon\; t \mapsto \phi_i^{-1}(x) + t \cdot v_i(x) \,. </annotation></semantics></math></div> <p>and for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mi>i</mi></msub><mo>≔</mo><mi>f</mi><mo>∘</mo><msub><mi>ϕ</mi> <mi>i</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>→</mo><msup><mi>ℝ</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">f_i \coloneqq f \circ \phi_i \;\colon\; \mathbb{R}^n \to \mathbb{R}^1</annotation></semantics></math> the restriction of the given smooth function, we have, by the <a class="existingWikiWord" href="/nlab/show/chain+rule">chain rule</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mi>d</mi><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo stretchy="false">(</mo><msub><mi>f</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><msub><mi>γ</mi> <mrow><mi>x</mi><mo>,</mo><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mo stretchy="false">|</mo> <mrow><mi>t</mi><mo>=</mo><mn>0</mn></mrow></msub><mo>=</mo><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow> <mi>n</mi></munderover><mo stretchy="false">(</mo><msub><mi>v</mi> <mi>i</mi></msub><msup><mo stretchy="false">)</mo> <mi>k</mi></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>⋅</mo><mfrac><mrow><mo>∂</mo><msub><mi>f</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><mrow><mo>∂</mo><msup><mi>y</mi> <mi>k</mi></msup></mrow></mfrac><msub><mo stretchy="false">|</mo> <mrow><mi>y</mi><mo>=</mo><mi>x</mi></mrow></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \frac{d}{d t} ( f_i(\gamma_{x,i}(t)) )\vert_{t = 0} = \sum_{k = 1}^n (v_i)^k(x) \cdot \frac{\partial f_i(y)}{\partial y^k}\vert_{y = x} \,. </annotation></semantics></math></div> <p>(Here and in the following we are writing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mo stretchy="false">)</mo> <mi>k</mi></msup></mrow><annotation encoding="application/x-tex">(-)^k</annotation></semantics></math> for 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 component of an element of <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> <p>This is the value of a function 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 is the sum over the product of two smooth functions, namely the component functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>v</mi> <mi>i</mi></msub><msup><mo stretchy="false">)</mo> <mi>k</mi></msup></mrow><annotation encoding="application/x-tex">(v_i)^k</annotation></semantics></math> of the smooth section in the given chart, and the derivatives of <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> (which are functions by the assumption 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 smooth).</p> <p>This shows that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>D</mi> <mi>v</mi></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">D_v(f)</annotation></semantics></math> is a smooth function.</p> <p>Now for the converse. Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>∈</mo><mi>Der</mi><mo stretchy="false">(</mo><msup><mo lspace="0em" rspace="thinmathspace">C</mo> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">D \in Der(\C^\infty(X))</annotation></semantics></math> be a derivation, we need to show that it arises on each chart <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\phi_i</annotation></semantics></math> from smooth vector field <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">v_i</annotation></semantics></math> as above.</p> <p>The <a class="existingWikiWord" href="/nlab/show/Hadamard+lemma">Hadamard lemma</a> gives that every <a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mi>i</mi></msub><mo lspace="verythinmathspace">:</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>→</mo><msup><mi>ℝ</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">f_i \colon \mathbb{R}^n \to \mathbb{R}^1</annotation></semantics></math> may be written for each point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">x \in \mathbb{R}^n</annotation></semantics></math> 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>y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo>−</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow> <mi>n</mi></munderover><mo stretchy="false">(</mo><msup><mi>y</mi> <mi>k</mi></msup><mo>−</mo><msup><mi>x</mi> <mi>k</mi></msup><mo stretchy="false">)</mo><mo>⋅</mo><msub><mi>g</mi> <mrow><mi>x</mi><mo>,</mo><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> f(y) = f(y-x) + \sum_{k = 0}^n (y^k - x^k) \cdot g_{x,k}(y) </annotation></semantics></math></div> <p>for smooth functions</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>g</mi> <mi>k</mi></msub><mo>∈</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><msub><mo stretchy="false">}</mo> <mrow><mi>k</mi><mo>∈</mo><mo stretchy="false">{</mo><mn>1</mn><mo>,</mo><mi>⋯</mi><mo>,</mo><mi>n</mi><mo stretchy="false">}</mo></mrow></msub></mrow><annotation encoding="application/x-tex"> \{g_k \in C^\infty(\mathbb{R}^n)\}_{k \in \{1,\cdots, n\}} </annotation></semantics></math></div> <p>satisfying</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mrow><mi>x</mi><mo>,</mo><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mo>∂</mo><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><mrow><mo>∂</mo><msup><mi>y</mi> <mi>k</mi></msup></mrow></mfrac><msub><mo stretchy="false">|</mo> <mrow><mi>y</mi><mo>=</mo><mi>x</mi></mrow></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> g_{x,k}(x) = \frac{\partial f(y)}{\partial y^k}\vert_{y = x} \,. </annotation></semantics></math></div> <p>Now by the derivation property 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>D</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mi>D</mi><mrow><mo>(</mo><mi>f</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>+</mo><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow> <mi>n</mi></munderover><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mo stretchy="false">)</mo> <mi>k</mi></msup><mo>−</mo><msup><mi>x</mi> <mi>k</mi></msup><mo stretchy="false">)</mo><mo>⋅</mo><msub><mi>g</mi> <mrow><mi>x</mi><mo>,</mo><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>)</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow> <mi>n</mi></munderover><mo maxsize="1.2em" minsize="1.2em">(</mo><mrow><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mo stretchy="false">)</mo> <mi>k</mi></msup><mo>−</mo><msup><mi>x</mi> <mi>k</mi></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mo>⋅</mo><msub><mi>g</mi> <mrow><mi>x</mi><mo>,</mo><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mo stretchy="false">)</mo> <mi>k</mi></msup><mo>−</mo><msup><mi>x</mi> <mi>k</mi></msup><mo stretchy="false">)</mo><mo>⋅</mo><mi>D</mi><msub><mi>g</mi> <mrow><mi>x</mi><mo>,</mo><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow> <mi>n</mi></munderover><mo stretchy="false">(</mo><msub><mi>v</mi> <mi>i</mi></msub><msup><mo stretchy="false">)</mo> <mi>k</mi></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>⋅</mo><mfrac><mrow><mo>∂</mo><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><mrow><mo>∂</mo><msup><mi>y</mi> <mi>k</mi></msup></mrow></mfrac><msub><mo stretchy="false">|</mo> <mrow><mi>y</mi><mo>=</mo><mi>x</mi></mrow></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} D(f)(x) & = D \left( f(0) + \sum_{k = 1}^n ((-)^k - x^k) \cdot g_{x,k}(-) \right) \\ & = \sum_{k = 1}^n \big( {(D( (-)^k - x^k ))} \cdot g_{x,k}(-) + ((-)^k - x^k) \cdot D g_{x,k}(-) \big)(x) \\ & = \sum_{k = 1}^n (v_i)^k(x) \cdot \frac{\partial f(y)}{\partial y^k}\vert_{y = x} \end{aligned} \,. </annotation></semantics></math></div> <p>Here in the last step we <em>defined</em></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>v</mi> <mi>i</mi></msub><msup><mo stretchy="false">)</mo> <mi>k</mi></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≔</mo><mi>D</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mo stretchy="false">)</mo> <mi>k</mi></msup><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (v_i)^k(x) \coloneqq D( (-)^k )(x) </annotation></semantics></math></div> <p>to be the value 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> of the function which the derivation produces when applied to 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 coordinate function (which is a smooth function 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> by definition of <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>), and we used the above property of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mrow><mi>x</mi><mo>,</mo><mi>k</mi></mrow></msub></mrow><annotation encoding="application/x-tex">g_{x,k}</annotation></semantics></math> as well as that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mo stretchy="false">)</mo> <mi>k</mi></msup><mo>−</mo><msup><mi>x</mi> <mi>k</mi></msup></mrow><annotation encoding="application/x-tex">(-)^k - x^k</annotation></semantics></math> vanishes 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>This shows that the derivation <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> comes, in the chart <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\phi_i</annotation></semantics></math>, from the vector field <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">v_i</annotation></semantics></math> as above.</p> </div> <h2 id="related_theorems">Related theorems</h2> <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/Borel%27s+theorem">Borel's theorem</a></p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/Tietze+extension+theorem">Tietze extension theorem</a></p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/Whitney+extension+theorem">Whitney extension theorem</a></p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/Steenrod-Wockel+approximation+theorem">Steenrod-Wockel approximation theorem</a></p> </li> </ul> <h2 id="references">References</h2> <p>The case of analytic manifolds is discussed in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Claude+Chevalley">Claude Chevalley</a>, Theory of Lie Groups I (Princeton, 1946).</li> </ul> <p>The case of <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>-manifolds is discussed in Section 4 of</p> <ul> <li>Harley Flanders. Development of an extended exterior differential calculus. Transactions of the American Mathematical Society 75:2 (1953), 311-326. <a href="https://doi.org/10.1090/s0002-9947-1953-0057005-8">doi</a>.</li> </ul> <p>The case of <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 is discussed in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/W.+F.+Newns">W. F. Newns</a>, <a class="existingWikiWord" href="/nlab/show/A.+G.+Walker">A. G. Walker</a>, <em>Tangent Planes To a Differentiable Manifold</em>, Journal of the London Mathematical Society s1-31:4 (1956) 400–407 [<a href="https://doi.org/10.1112/jlms/s1-31.4.400">doi:10.1112/jlms/s1-31.4.400</a>]</li> </ul> <p>In Section 2, formula (2), Newns–Walker anticipate the notion of a <a class="existingWikiWord" href="/nlab/show/C%5E%E2%88%9E-derivation">C^∞-derivation</a>.</p> <p>See also</p> <ul> <li> <p>Thm. 3.7 in <a href="http://www.math.toronto.edu/mgualt/MAT1300/week9.pdf">this pdf</a></p> </li> <li> <p>Thm. 4.15 in <a href="https://www.math.toronto.edu/mgualt/courses/17-1300/docs/17-1300-notes-12.pdf">this pdf</a></p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on February 6, 2025 at 17:55:38. 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