CINXE.COM

<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg-flat.dtd" > <html xmlns="http://www.w3.org/1999/xhtml"> <head> <title> vector field (changes) in nLab </title> <meta http-equiv="Content-Type" content="text/html; charset=UTF-8" /> <meta name="robots" content="noindex,nofollow" /> <meta name="viewport" content="width=device-width, initial-scale=1" /> <link href="/stylesheets/instiki.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/mathematics.css?1660229990" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/syntax.css?1660229990" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/nlab.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <link rel="stylesheet" type="text/css" href="https://cdn.jsdelivr.net/gh/dreampulse/computer-modern-web-font@master/fonts.css"/> <style type="text/css"> h1#pageName, div.info, .newWikiWord a, a.existingWikiWord, .newWikiWord a:hover, [actiontype="toggle"]:hover, #TextileHelp h3 { color: #226622; } a:visited.existingWikiWord { color: #164416; } </style> <style type="text/css"><![CDATA[/*></style> <script src="/javascripts/prototype.js?1660229990" type="text/javascript"></script> <script src="/javascripts/effects.js?1660229990" type="text/javascript"></script> <script src="/javascripts/dragdrop.js?1660229990" type="text/javascript"></script> <script src="/javascripts/controls.js?1660229990" type="text/javascript"></script> <script src="/javascripts/application.js?1660229990" type="text/javascript"></script> <script src="/javascripts/page_helper.js?1660229990" type="text/javascript"></script> <script src="/javascripts/thm_numbering.js?1660229990" type="text/javascript"></script> <script type="text/x-mathjax-config"> <![CDATA[//><!]]> </script> <script type="text/javascript"> <![CDATA[//><!]]> </script> <link href="https://ncatlab.org/nlab/atom_with_headlines" rel="alternate" title="Atom with headlines" type="application/atom+xml" /> <link href="https://ncatlab.org/nlab/atom_with_content" rel="alternate" title="Atom with full content" type="application/atom+xml" /> <script type="text/javascript"> document.observe("dom:loaded", function() { generateThmNumbers(); }); </script> </head> <body> <div id="Container"> <div id="Content"> <h1 id="pageName"> <span style="float: left; margin: 0.5em 0.25em -0.25em 0"> <svg xmlns="http://www.w3.org/2000/svg" width="1.872em" height="1.8em" viewBox="0 0 190 181"> <path fill="#226622" d="M72.8 145c-1.6 17.3-15.7 10-23.6 20.2-5.6 7.3 4.8 15 11.4 15 11.5-.2 19-13.4 26.4-20.3 3.3-3 8.2-4 11.2-7.2a14 14 0 0 0 2.9-11.1c-1.4-9.6-12.4-18.6-16.9-27.2-5-9.6-10.7-27.4-24.1-27.7-17.4-.3-.4 26 4.7 30.7 2.4 2.3 5.4 4.1 7.3 6.9 1.6 2.3 2.1 5.8-1 7.2-5.9 2.6-12.4-6.3-15.5-10-8.8-10.6-15.5-23-26.2-31.8-5.2-4.3-11.8-8-18-3.7-7.3 4.9-4.2 12.9.2 18.5a81 81 0 0 0 30.7 23c3.3 1.5 12.8 5.6 10 10.7-2.5 5.2-11.7 3-15.6 1.1-8.4-3.8-24.3-21.3-34.4-13.7-3.5 2.6-2.3 7.6-1.2 11.1 2.8 9 12.2 17.2 20.9 20.5 17.3 6.7 34.3-8 50.8-12.1z"/> <path fill="#a41e32" d="M145.9 121.3c-.2-7.5 0-19.6-4.5-26-5.4-7.5-12.9-1-14.1 5.8-1.4 7.8 2.7 14.1 4.8 21.3 3.4 12 5.8 29-.8 40.1-3.6-6.7-5.2-13-7-20.4-2.1-8.2-12.8-13.2-15.1-1.9-2 9.7 9 21.2 12 30.1 1.2 4 2 8.8 6.4 10.3 6.9 2.3 13.3-4.7 17.7-8.8 12.2-11.5 36.6-20.7 43.4-36.4 6.7-15.7-13.7-14-21.3-7.2-9.1 8-11.9 20.5-23.6 25.1 7.5-23.7 31.8-37.6 38.4-61.4 2-7.3-.8-29.6-13-19.8-14.5 11.6-6.6 37.6-23.3 49.2z"/> <path fill="#193c78" d="M86.3 47.5c0-13-10.2-27.6-5.8-40.4 2.8-8.4 14.1-10.1 17-1 3.8 11.6-.3 26.3-1.8 38 11.7-.7 10.5-16 14.8-24.3 2.1-4.2 5.7-9.1 11-6.7 6 2.7 7.4 9.2 6.6 15.1-2.2 14-12.2 18.8-22.4 27-3.4 2.7-8 6.6-5.9 11.6 2 4.4 7 4.5 10.7 2.8 7.4-3.3 13.4-16.5 21.7-16 14.6.7 12 21.9.9 26.2-5 1.9-10.2 2.3-15.2 3.9-5.8 1.8-9.4 8.7-15.7 8.9-6.1.1-9-6.9-14.3-9-14.4-6-33.3-2-44.7-14.7-3.7-4.2-9.6-12-4.9-17.4 9.3-10.7 28 7.2 35.7 12 2 1.1 11 6.9 11.4 1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> vector field (changes) </h1> <div class="navigation"> <span class="skipNav"><a href='#navEnd'>Skip the Navigation Links</a> | </span> <span style="display:inline-block; width: 0.3em;"></span> <a href="/nlab/show/diff/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/12573/#Item_1" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" 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"> <p class="show_diff"> Showing changes from revision #13 to #14: <ins class="diffins">Added</ins> | <del class="diffdel">Removed</del> | <del class="diffmod">Chan</del><ins class="diffmod">ged</ins> </p> <div class='rightHandSide'> <div class='toc clickDown' tabindex='0'> <h3 id='context'>Context</h3> <h4 id='differential_geometry'>Differential geometry</h4> <div class='hide'> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/synthetic+differential+geometry'>synthetic</a> <a class='existingWikiWord' href='/nlab/show/diff/differential+geometry'>differential geometry</a></strong></p> <p><strong>Introductions</strong></p> <p><a class='existingWikiWord' href='/nlab/show/diff/Introduction+to+Topology+--+1'>from point-set topology to differentiable manifolds</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/geometry+of+physics'>geometry of physics</a>: <a class='existingWikiWord' href='/nlab/show/diff/geometry+of+physics+--+coordinate+systems'>coordinate systems</a>, <a class='existingWikiWord' href='/nlab/show/diff/geometry+of+physics+--+smooth+sets'>smooth spaces</a>, <a class='existingWikiWord' href='/nlab/show/diff/geometry+of+physics+--+manifolds+and+orbifolds'>manifolds</a>, <a class='existingWikiWord' href='/nlab/show/diff/geometry+of+physics+--+smooth+homotopy+types'>smooth homotopy types</a>, <a class='existingWikiWord' href='/nlab/show/diff/geometry+of+physics+--+supergeometry'>supergeometry</a></p> <p><strong>Differentials</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/differentiation'>differentiation</a>, <a class='existingWikiWord' href='/nlab/show/diff/chain+rule'>chain rule</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/differentiable+map'>differentiable function</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/infinitesimal+object'>infinitesimal space</a>, <a class='existingWikiWord' href='/nlab/show/diff/infinitesimally+thickened+point'>infinitesimally thickened point</a>, <a class='existingWikiWord' href='/nlab/show/diff/amazing+right+adjoint'>amazing right adjoint</a></p> </li> </ul> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/V-manifold'>V-manifolds</a></strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/differentiable+manifold'>differentiable manifold</a>, <a class='existingWikiWord' href='/nlab/show/diff/coordinate+system'>coordinate chart</a>, <a class='existingWikiWord' href='/nlab/show/diff/atlas'>atlas</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/smooth+manifold'>smooth manifold</a>, <a class='existingWikiWord' href='/nlab/show/diff/smooth+structure'>smooth structure</a>, <a class='existingWikiWord' href='/nlab/show/diff/exotic+smooth+structure'>exotic smooth structure</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/analytic+manifold'>analytic manifold</a>, <a class='existingWikiWord' href='/nlab/show/diff/complex+manifold'>complex manifold</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/formal+smooth+manifold'>formal smooth manifold</a>, <a class='existingWikiWord' href='/nlab/show/diff/derived+smooth+manifold'>derived smooth manifold</a></p> </li> </ul> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/smooth+set'>smooth space</a></strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/diffeological+space'>diffeological space</a>, <a class='existingWikiWord' href='/nlab/show/diff/Fr%C3%B6licher+space'>Frölicher space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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/diff/tangent+bundle'>tangent bundle</a>, <a class='existingWikiWord' href='/nlab/show/diff/frame+bundle'>frame bundle</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/vector+field'>vector field</a>, <a class='existingWikiWord' href='/nlab/show/diff/multivector+field'>multivector field</a>, <a class='existingWikiWord' href='/nlab/show/diff/tangent+Lie+algebroid'>tangent Lie algebroid</a>;</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/differential+forms+in+synthetic+differential+geometry'>differential forms</a>, <a class='existingWikiWord' href='/nlab/show/diff/de+Rham+complex'>de Rham complex</a>, <a class='existingWikiWord' href='/nlab/show/diff/Dolbeault+complex'>Dolbeault complex</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/pullback+of+a+differential+form'>pullback of differential forms</a>, <a class='existingWikiWord' href='/nlab/show/diff/invariant+differential+form'>invariant differential form</a>, <a class='existingWikiWord' href='/nlab/show/diff/Maurer-Cartan+form'>Maurer-Cartan form</a>, <a class='existingWikiWord' href='/nlab/show/diff/horizontal+differential+form'>horizontal differential form</a>,</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cogerm+differential+form'>cogerm differential form</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/integration+of+differential+forms'>integration of differential forms</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/local+diffeomorphism'>local diffeomorphism</a>, <a class='existingWikiWord' href='/nlab/show/diff/formally+%C3%A9tale+morphism'>formally étale morphism</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/submersion'>submersion</a>, <a class='existingWikiWord' href='/nlab/show/diff/formally+smooth+morphism'>formally smooth morphism</a>,</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/immersion'>immersion</a>, <a class='existingWikiWord' href='/nlab/show/diff/formally+unramified+morphism'>formally unramified morphism</a>,</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/de+Rham+space'>de Rham space</a>, <a class='existingWikiWord' href='/nlab/show/diff/crystal'>crystal</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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/diff/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/diff/smooth+Serre-Swan+theorem'>smooth Serre-Swan theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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/diff/Hadamard+lemma'>Hadamard lemma</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Borel%27s+theorem'>Borel's theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Boman%27s+theorem'>Boman's theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Whitney+extension+theorem'>Whitney extension theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Steenrod+approximation+theorem'>Steenrod-Wockel approximation theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Whitney+embedding+theorem'>Whitney embedding theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Poincar%C3%A9+lemma'>Poincare lemma</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Stokes+theorem'>Stokes theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/de+Rham+theorem'>de Rham theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Hochschild-Kostant-Rosenberg+theorem'>Hochschild-Kostant-Rosenberg theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/differential+cohomology+diagram'>differential cohomology hexagon</a></p> </li> </ul> <p><strong>Axiomatics</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Kock-Lawvere+axiom'>Kock-Lawvere axiom</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/smooth+topos'>smooth topos</a>, <a class='existingWikiWord' href='/nlab/show/diff/super+smooth+topos'>super smooth topos</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/microlinear+space'>microlinear space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/integration+axiom'>integration axiom</a></p> </li> </ul> </li> </ul> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/cohesive'>cohesion</a></strong></p> <ul> <li> <p>(<a class='existingWikiWord' href='/nlab/show/diff/shape+modality'>shape modality</a> <math class='maruku-mathml' display='inline' id='mathml_ff8f054063f166fbf6a72fba0743f8c9a0634729_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>⊣</mo></mrow><annotation encoding='application/x-tex'>\dashv</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/flat+modality'>flat modality</a> <math class='maruku-mathml' display='inline' id='mathml_ff8f054063f166fbf6a72fba0743f8c9a0634729_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>⊣</mo></mrow><annotation encoding='application/x-tex'>\dashv</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/sharp+modality'>sharp modality</a>)</p> <p><math class='maruku-mathml' display='inline' id='mathml_ff8f054063f166fbf6a72fba0743f8c9a0634729_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mo lspace='0em' rspace='thinmathspace'>esh</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/diff/discrete+object'>discrete object</a>, <a class='existingWikiWord' href='/nlab/show/diff/codiscrete+object'>codiscrete object</a>, <a class='existingWikiWord' href='/nlab/show/diff/concrete+object'>concrete object</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/points-to-pieces+transform'>points-to-pieces transform</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cohesive+%28infinity%2C1%29-topos+--+structures'>structures in cohesion</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/dR-shape+modality'>dR-shape modality</a> <math class='maruku-mathml' display='inline' id='mathml_ff8f054063f166fbf6a72fba0743f8c9a0634729_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>⊣</mo></mrow><annotation encoding='application/x-tex'>\dashv</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/dR-flat+modality'>dR-flat modality</a></p> <p><math class='maruku-mathml' display='inline' id='mathml_ff8f054063f166fbf6a72fba0743f8c9a0634729_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mo lspace='0em' rspace='thinmathspace'>esh</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/diff/infinitesimal+cohesive+%28infinity%2C1%29-topos'>infinitesimal cohesion</a></strong></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/classical+modality'>classical modality</a></li> </ul> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/tangent+cohesive+%28%E2%88%9E%2C1%29-topos'>tangent cohesion</a></strong></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/differential+cohomology+diagram'>differential cohomology diagram</a></li> </ul> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/differential+cohesive+%28infinity%2C1%29-topos'>differential cohesion</a></strong></p> <ul> <li> <p>(<a class='existingWikiWord' href='/nlab/show/diff/reduction+modality'>reduction modality</a> <math class='maruku-mathml' display='inline' id='mathml_ff8f054063f166fbf6a72fba0743f8c9a0634729_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>⊣</mo></mrow><annotation encoding='application/x-tex'>\dashv</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/infinitesimal+shape+modality'>infinitesimal shape modality</a> <math class='maruku-mathml' display='inline' id='mathml_ff8f054063f166fbf6a72fba0743f8c9a0634729_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>⊣</mo></mrow><annotation encoding='application/x-tex'>\dashv</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/infinitesimal+flat+modality'>infinitesimal flat modality</a>)</p> <p><math class='maruku-mathml' display='inline' id='mathml_ff8f054063f166fbf6a72fba0743f8c9a0634729_8' xmlns='http://www.w3.org/1998/Math/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/diff/reduced+object'>reduced object</a>, <a class='existingWikiWord' href='/nlab/show/diff/coreduced+object'>coreduced object</a>, <a class='existingWikiWord' href='/nlab/show/diff/formally+smooth+object'>formally smooth object</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/formally+%C3%A9tale+morphism'>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/diff/super+formal+smooth+infinity-groupoid'>graded differential cohesion</a></strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/bosonic+modality'>fermionic modality</a> <math class='maruku-mathml' display='inline' id='mathml_ff8f054063f166fbf6a72fba0743f8c9a0634729_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>⊣</mo></mrow><annotation encoding='application/x-tex'>\dashv</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/bosonic+modality'>bosonic modality</a> <math class='maruku-mathml' display='inline' id='mathml_ff8f054063f166fbf6a72fba0743f8c9a0634729_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>⊣</mo></mrow><annotation encoding='application/x-tex'>\dashv</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/rheonomy+modality'>rheonomy modality</a></p> <p><math class='maruku-mathml' display='inline' id='mathml_ff8f054063f166fbf6a72fba0743f8c9a0634729_11' xmlns='http://www.w3.org/1998/Math/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/diff/orbifold+cohomology'>singular cohesion</a></strong></p> <div class='maruku-equation' id='Diagram'><math class='maruku-mathml' display='block' id='mathml_ff8f054063f166fbf6a72fba0743f8c9a0634729_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd /> <mtd /> <mtd><mi>id</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi>id</mi></mtd></mtr> <mtr><mtd /> <mtd /> <mtd><mo>∨</mo></mtd> <mtd /> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd /> <mtd><mover><mrow /><mi>fermionic</mi></mover></mtd> <mtd><mo>⇉</mo></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>⇝</mo></mtd> <mtd><mover><mrow /><mi>bosonic</mi></mover></mtd></mtr> <mtr><mtd /> <mtd /> <mtd><mo>⊥</mo></mtd> <mtd /> <mtd><mo>⊥</mo></mtd></mtr> <mtr><mtd /> <mtd><mover><mrow /><mi>bosonic</mi></mover></mtd> <mtd><mo>⇝</mo></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi mathvariant='normal'>R</mi><mspace width='negativethinmathspace' /><mspace width='negativethinmathspace' /><mi mathvariant='normal'>h</mi></mtd> <mtd><mover><mrow /><mi>rheonomic</mi></mover></mtd></mtr> <mtr><mtd /> <mtd /> <mtd><mo>∨</mo></mtd> <mtd /> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd /> <mtd><mover><mrow /><mi>reduced</mi></mover></mtd> <mtd><mi>ℜ</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi>ℑ</mi></mtd> <mtd><mover><mrow /><mi>infinitesimal</mi></mover></mtd></mtr> <mtr><mtd /> <mtd /> <mtd><mo>⊥</mo></mtd> <mtd /> <mtd><mo>⊥</mo></mtd></mtr> <mtr><mtd /> <mtd><mover><mrow /><mi>infinitesimal</mi></mover></mtd> <mtd><mi>ℑ</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi>&</mi></mtd> <mtd><mover><mrow /><mtext>étale</mtext></mover></mtd></mtr> <mtr><mtd /> <mtd /> <mtd><mo>∨</mo></mtd> <mtd /> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd /> <mtd><mover><mrow /><mi>cohesive</mi></mover></mtd> <mtd><mo lspace='0em' rspace='thinmathspace'>esh</mo></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>♭</mo></mtd> <mtd><mover><mrow /><mi>discrete</mi></mover></mtd></mtr> <mtr><mtd /> <mtd /> <mtd><mo>⊥</mo></mtd> <mtd /> <mtd><mo>⊥</mo></mtd></mtr> <mtr><mtd /> <mtd><mover><mrow /><mi>discrete</mi></mover></mtd> <mtd><mo>♭</mo></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>♯</mo></mtd> <mtd><mover><mrow /><mi>continuous</mi></mover></mtd></mtr> <mtr><mtd /> <mtd /> <mtd><mo>∨</mo></mtd> <mtd /> <mtd><mo>∨</mo></mtd></mtr> <mtr><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> <p id='models_2'><strong>Models</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Models+for+Smooth+Infinitesimal+Analysis'>Models for Smooth Infinitesimal Analysis</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/C%5E%E2%88%9E-ring'>smooth algebra</a> (<math class='maruku-mathml' display='inline' id='mathml_ff8f054063f166fbf6a72fba0743f8c9a0634729_13' xmlns='http://www.w3.org/1998/Math/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/diff/smooth+locus'>smooth locus</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Fermat+theory'>Fermat theory</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Cahiers+topos'>Cahiers topos</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/smooth+infinity-groupoid'>smooth ∞-groupoid</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/formal+smooth+infinity-groupoid'>formal smooth ∞-groupoid</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/super+formal+smooth+infinity-groupoid'>super formal smooth ∞-groupoid</a></p> </li> </ul> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/Lie+theory'>Lie theory</a>, <a class='existingWikiWord' href='/nlab/show/diff/infinity-Lie+theory'>∞-Lie theory</a></strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Lie+algebra'>Lie algebra</a>, <a class='existingWikiWord' href='/nlab/show/diff/L-infinity-algebra'>Lie n-algebra</a>, <a class='existingWikiWord' href='/nlab/show/diff/L-infinity-algebra'>L-∞ algebra</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Lie+group'>Lie group</a>, <a class='existingWikiWord' href='/nlab/show/diff/Lie+2-group'>Lie 2-group</a>, <a class='existingWikiWord' href='/nlab/show/diff/smooth+infinity-groupoid'>smooth ∞-group</a></p> </li> </ul> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/differential+equation'>differential equations</a>, <a class='existingWikiWord' href='/nlab/show/diff/variational+calculus'>variational calculus</a></strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/D-geometry'>D-geometry</a>, <a class='existingWikiWord' href='/nlab/show/diff/D-module'>D-module</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/jet+bundle'>jet bundle</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/variational+bicomplex'>variational bicomplex</a>, <a class='existingWikiWord' href='/nlab/show/diff/Euler-Lagrange+complex'>Euler-Lagrange complex</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Euler-Lagrange+equation'>Euler-Lagrange equation</a>, <a class='existingWikiWord' href='/nlab/show/diff/De+Donder-Weyl-Hamilton+equation'>de Donder-Weyl formalism</a>,</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/phase+space'>phase space</a></p> </li> </ul> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/Chern-Weil+theory'>Chern-Weil theory</a>, <a class='existingWikiWord' href='/nlab/show/diff/Chern-Weil+theory+in+Smooth%E2%88%9EGrpd'>∞-Chern-Weil theory</a></strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/connection+on+a+bundle'>connection on a bundle</a>, <a class='existingWikiWord' href='/nlab/show/diff/connection+on+a+smooth+principal+infinity-bundle'>connection on an ∞-bundle</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/differential+cohomology'>differential cohomology</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/ordinary+differential+cohomology'>ordinary differential cohomology</a>, <a class='existingWikiWord' href='/nlab/show/diff/Deligne+cohomology'>Deligne complex</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/differential+K-theory'>differential K-theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/differential+cobordism+cohomology'>differential cobordism cohomology</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/parallel+transport'>parallel transport</a>, <a class='existingWikiWord' href='/nlab/show/diff/higher+parallel+transport'>higher parallel transport</a>, <a class='existingWikiWord' href='/nlab/show/diff/fiber+integration+in+differential+cohomology'>fiber integration in differential cohomology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/holonomy'>holonomy</a>, <a class='existingWikiWord' href='/nlab/show/diff/higher+parallel+transport'>higher holonomy</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/gauge+theory'>gauge theory</a>, <a class='existingWikiWord' href='/nlab/show/diff/higher+gauge+field'>higher gauge theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Wilson+loop'>Wilson line</a>, <a class='existingWikiWord' href='/nlab/show/diff/Wilson+surface'>Wilson surface</a></p> </li> </ul> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/Cartan+geometry'>Cartan geometry</a> (<a class='existingWikiWord' href='/nlab/show/diff/super-Cartan+geometry'>super</a>, <a class='existingWikiWord' href='/nlab/show/diff/higher+Cartan+geometry'>higher</a>)</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Klein+geometry'>Klein geometry</a>, (<a class='existingWikiWord' href='/nlab/show/diff/higher+Klein+geometry'>higher</a>)</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/G-structure'>G-structure</a>, <a class='existingWikiWord' href='/nlab/show/diff/torsion+of+a+G-structure'>torsion of a G-structure</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Euclidean+geometry'>Euclidean geometry</a>, <a class='existingWikiWord' href='/nlab/show/diff/hyperbolic+geometry'>hyperbolic geometry</a>, <a class='existingWikiWord' href='/nlab/show/diff/elliptic+geometry'>elliptic geometry</a></p> </li> <li> <p>(<a class='existingWikiWord' href='/nlab/show/diff/pseudo-Riemannian+metric'>pseudo</a>-)<a class='existingWikiWord' href='/nlab/show/diff/Riemannian+geometry'>Riemannian geometry</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/orthogonal+structure'>orthogonal structure</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/isometry'>isometry</a>, <a class='existingWikiWord' href='/nlab/show/diff/Killing+vector+field'>Killing vector field</a>, <a class='existingWikiWord' href='/nlab/show/diff/Killing+spinor'>Killing spinor</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/spacetime'>spacetime</a>, <a class='existingWikiWord' href='/nlab/show/diff/super+spacetime'>super-spacetime</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/complex+geometry'>complex geometry</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/symplectic+geometry'>symplectic geometry</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/conformal+geometry'>conformal geometry</a></p> </li> </ul> </div> </div> </div> <h1 id='contents'>Contents</h1> <div class='maruku_toc'><ul><li><a href='#idea'>Idea</a></li><li><a href='#properties'>Properties</a><ul><li><a href='#as_derivations_of_smooth_functions'>As derivations of smooth functions</a></li></ul></li><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 <strong>vector field</strong> is a <a class='existingWikiWord' href='/nlab/show/diff/section'>section</a> of a <a class='existingWikiWord' href='/nlab/show/diff/vector+bundle'>vector bundle</a>.</p> <p>More specifically, a vector field is a <em><a class='existingWikiWord' href='/nlab/show/diff/tangent+bundle'>tangent vector field</a></em> which is a <a class='existingWikiWord' href='/nlab/show/diff/section'>section</a> of a <a class='existingWikiWord' href='/nlab/show/diff/tangent+bundle'>tangent bundle</a>. Hence this is a function which picks a <a class='existingWikiWord' href='/nlab/show/diff/tangent+bundle'>tangent vector</a> at each point of a manifold, such that this assignment is suitably <a class='existingWikiWord' href='/nlab/show/diff/differentiable+map'>differentiable</a>.</p> <p>Equivalently this is a rank <math class='maruku-mathml' display='inline' id='mathml_ff8f054063f166fbf6a72fba0743f8c9a0634729_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(1,0)</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/tensor'>tensor field</a> on <math class='maruku-mathml' display='inline' id='mathml_ff8f054063f166fbf6a72fba0743f8c9a0634729_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math></p> <h2 id='properties'>Properties</h2> <h3 id='as_derivations_of_smooth_functions'>As derivations of smooth functions</h3> <p>Vector fields may be identified with <a class='existingWikiWord' href='/nlab/show/diff/derivation'>derivations</a> on the <a class='existingWikiWord' href='/nlab/show/diff/algebra+of+functions'>algebra of</a> <a class='existingWikiWord' href='/nlab/show/diff/smooth+map'>smooth functions</a>. See the article <em><a class='existingWikiWord' href='/nlab/show/diff/derivations+of+smooth+functions+are+vector+fields'>derivations of smooth functions are vector fields</a></em>.</p> <h2 id='related_concepts'>Related concepts</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/flow+of+a+vector+field'>flow of a vector field</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/pushforward+of+vector+fields'>pushforward of vector fields</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/derivations+of+smooth+functions+are+vector+fields'>derivations of smooth functions are vector fields</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/differential+form'>differential form</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/frame+field'>frame field</a></p> </li> <ins class='diffins'><li> <p><a class='existingWikiWord' href='/nlab/show/diff/Euler+vector+field'>Euler vector field</a></p> </li></ins><ins class='diffins'> </ins><li> <p><a class='existingWikiWord' href='/nlab/show/diff/Hamiltonian+vector+field'>Hamiltonian vector field</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Reeb+vector+field'>Reeb vector field</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/evolutionary+vector+field'>evolutionary vector field</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/field+%28physics%29'>field (physics)</a></p> </li> </ul> <h2 id='references'>References</h2> <p>For instance section 1.3 and 1.4 of</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Theodore+Frankel'>Theodore Frankel</a>, <em><a class='existingWikiWord' href='/nlab/show/diff/The+Geometry+of+Physics+-+An+Introduction'>The Geometry of Physics - An Introduction</a></em></li> </ul> <p> </p> </div> <div class="revisedby"> <p> Last revised on August 28, 2024 at 16:37:48. See the <a href="/nlab/history/vector+field" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/vector+field" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/12573/#Item_1">Discuss</a><span class="backintime"><a href="/nlab/revision/diff/vector+field/13" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/vector+field" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Hide changes</a><a href="/nlab/history/vector+field" accesskey="S" class="navlink" id="history" rel="nofollow">History (13 revisions)</a> <a href="/nlab/show/vector+field/cite" style="color: black">Cite</a> <a href="/nlab/print/vector+field" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/vector+field" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>