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Jet bundle

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</style><div role="note" class="hatnote navigation-not-searchable">"Jet space" redirects here; not to be confused with <a href="Space_jet_(disambiguation)" class="mw-redirect mw-disambig" title="Space jet (disambiguation)">space jet</a>.</div>

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</style>In <a href="Differential_topology" title="Differential topology">differential topology</a>, the jet bundle is a certain construction that makes a new <a href="Smooth_manifold" class="mw-redirect" title="Smooth manifold">smooth</a> <a href="Fiber_bundle" title="Fiber bundle">fiber bundle</a> out of a given smooth fiber bundle. It makes it possible to write <a href="Differential_equation" title="Differential equation">differential equations</a> on <a href="Fiber_bundle#Sections" title="Fiber bundle">sections</a> of a fiber bundle in an invariant form. <a href="Jet_(mathematics)" title="Jet (mathematics)">Jets</a> may also be seen as the coordinate free versions of <a href="Taylor_expansions" class="mw-redirect" title="Taylor expansions">Taylor expansions</a>.

Historically, jet bundles are attributed to <a href="Charles_Ehresmann" title="Charles Ehresmann">Charles Ehresmann</a>, and were an advance on the method (<a href="Cartan's_equivalence_method" title="Cartan's equivalence method">prolongation</a>) of <a href="%C3%89lie_Cartan" title="Élie Cartan">Élie Cartan</a>, of dealing geometrically with <a href="Derivative" title="Derivative">higher derivatives</a>, by imposing <a href="Differential_form" title="Differential form">differential form</a> conditions on newly introduced formal variables. Jet bundles are sometimes called sprays, although <a href="Spray_(mathematics)" title="Spray (mathematics)">sprays</a> usually refer more specifically to the associated <a href="Vector_field" title="Vector field">vector field</a> induced on the corresponding bundle (e.g., the <a href="Geodesic_spray" class="mw-redirect" title="Geodesic spray">geodesic spray</a> on <a href="Finsler_manifold" title="Finsler manifold">Finsler manifolds</a>.)

Since the early 1980s, jet bundles have appeared as a concise way to describe phenomena associated with the derivatives of maps, particularly those associated with the <a href="Calculus_of_variations" title="Calculus of variations">calculus of variations</a>.<a href="#cite_note-1">[1]</a> Consequently, the jet bundle is now recognized as the correct domain for a <a href="Covariant_classical_field_theory" title="Covariant classical field theory">geometrical covariant field theory</a> and much work is done in <a href="General_relativity" title="General relativity">general relativistic</a> formulations of fields using this approach.

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Jet_(mathematics)" title="Jet (mathematics)">Jet (mathematics)</a></div>

Suppose M is an m-dimensional <a href="Manifold" title="Manifold">manifold</a> and that (E, π, M) is a <a href="Fiber_bundle" title="Fiber bundle">fiber bundle</a>. For p ∈ M, let Γ(p) denote the set of all local sections whose domain contains p. Let ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I=(I(1),I(2),...,I(m))}">

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<annotation encoding="application/x-tex">{\displaystyle I=(I(1),I(2),...,I(m))}</annotation>

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</math><img src="./06359bf782029772e5a4b722bd11790adf76f33f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.592ex; height:2.843ex;" alt="{\displaystyle I=(I(1),I(2),...,I(m))}" loading="lazy">⁠ be a <a href="Multi-index" class="mw-redirect" title="Multi-index">multi-index</a> (an m-tuple of non-negative integers, not necessarily in ascending order), then define:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}|I|&:=\sum _{i=1}^{m}I(i)\\{\frac {\partial ^{|I|}}{\partial x^{I}}}&:=\prod _{i=1}^{m}\left({\frac {\partial }{\partial x^{i}}}\right)^{I(i)}.\end{aligned}}}</annotation>

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</math><img src="./6d502f8c534450c41b6ec203d754044baa148190.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.671ex; width:23.657ex; height:14.509ex;" alt="{\displaystyle {\begin{aligned}|I|&:=\sum _{i=1}^{m}I(i)\\{\frac {\partial ^{|I|}}{\partial x^{I}}}&:=\prod _{i=1}^{m}\left({\frac {\partial }{\partial x^{i}}}\right)^{I(i)}.\end{aligned}}}" loading="lazy"></dd></dl>

Define the local sections σ, η ∈ Γ(p) to have the same r-jet at p if

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<annotation encoding="application/x-tex">{\displaystyle \left.{\frac {\partial ^{|I|}\sigma ^{\alpha }}{\partial x^{I}}}\right|_{p}=\left.{\frac {\partial ^{|I|}\eta ^{\alpha }}{\partial x^{I}}}\right|_{p},\quad 0\leq |I|\leq r.}</annotation>

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</math><img src="./d47e1008f011d7aef85066c725839fd14f953b27.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:34.77ex; height:7.343ex;" alt="{\displaystyle \left.{\frac {\partial ^{|I|}\sigma ^{\alpha }}{\partial x^{I}}}\right|_{p}=\left.{\frac {\partial ^{|I|}\eta ^{\alpha }}{\partial x^{I}}}\right|_{p},\quad 0\leq |I|\leq r.}" loading="lazy"></dd></dl>

The relation that two maps have the same r-jet is an <a href="Equivalence_relation" title="Equivalence relation">equivalence relation</a>. An r-jet is an <a href="Equivalence_class" title="Equivalence class">equivalence class</a> under this relation, and the r-jet with representative σ is denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j_{p}^{r}\sigma }">

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<annotation encoding="application/x-tex">{\displaystyle j_{p}^{r}\sigma }</annotation>

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</math><img src="./98367c763409135ece4a409dd841a057affa4e7e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.027ex; width:3.374ex; height:2.843ex;" alt="{\displaystyle j_{p}^{r}\sigma }" loading="lazy">. The integer r is also called the order of the jet, p is its source and σ(p) is its target.

<div class="mw-heading mw-heading2"><h2 id="Jet_manifolds">Jet manifolds</h2></div>

The r-th jet manifold of π is the set

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<annotation encoding="application/x-tex">{\displaystyle J^{r}(\pi )=\left\{j_{p}^{r}\sigma :p\in M,\sigma \in \Gamma (p)\right\}.}</annotation>

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</math><img src="./2079b8ee8125c2986c7262e8c89368c08a3f71f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:33.47ex; height:3.009ex;" alt="{\displaystyle J^{r}(\pi )=\left\{j_{p}^{r}\sigma :p\in M,\sigma \in \Gamma (p)\right\}.}" loading="lazy"></dd></dl>

We may define projections πr and πr,0 called the source and target projections respectively, by

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<annotation encoding="application/x-tex">{\displaystyle {\begin{cases}\pi _{r}:J^{r}(\pi )\to M\\j_{p}^{r}\sigma \mapsto p\end{cases}},\qquad {\begin{cases}\pi _{r,0}:J^{r}(\pi )\to E\\j_{p}^{r}\sigma \mapsto \sigma (p)\end{cases}}}</annotation>

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</math><img src="./76d16f693ff341a1edd87559aa0b43efa76edfd4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:43.147ex; height:6.176ex;" alt="{\displaystyle {\begin{cases}\pi _{r}:J^{r}(\pi )\to M\\j_{p}^{r}\sigma \mapsto p\end{cases}},\qquad {\begin{cases}\pi _{r,0}:J^{r}(\pi )\to E\\j_{p}^{r}\sigma \mapsto \sigma (p)\end{cases}}}" loading="lazy"></dd></dl>

If 1 ≤ k ≤ r, then the k-jet projection is the function πr,k defined by

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<annotation encoding="application/x-tex">{\displaystyle {\begin{cases}\pi _{r,k}:J^{r}(\pi )\to J^{k}(\pi )\\j_{p}^{r}\sigma \mapsto j_{p}^{k}\sigma \end{cases}}}</annotation>

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From this definition, it is clear that πr = π o πr,0 and that if 0 ≤ m ≤ k, then πr,m = πk,m o πr,k. It is conventional to regard πr,r as the <a href="Identity_function" title="Identity function">identity map</a> on J r(π) and to identify J 0(π) with E.

The functions πr,k, πr,0 and πr are <a href="Smooth_function" class="mw-redirect" title="Smooth function">smooth</a> <a href="Surjective" class="mw-redirect" title="Surjective">surjective</a> <a href="Submersion_(mathematics)" title="Submersion (mathematics)">submersions</a>.

A <a href="Coordinate_system" title="Coordinate system">coordinate system</a> on E will generate a coordinate system on J r(π). Let (U, u) be an adapted <a href="Coordinate_chart" class="mw-redirect" title="Coordinate chart">coordinate chart</a> on E, where u = (xi, uα). The induced coordinate chart (Ur, ur) on J r(π) is defined by

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}U^{r}&=\left\{j_{p}^{r}\sigma :p\in M,\sigma (p)\in U\right\}\\u^{r}&=\left(x^{i},u^{\alpha },u_{I}^{\alpha }\right)\end{aligned}}}</annotation>

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where

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x^{i}\left(j_{p}^{r}\sigma \right)&=x^{i}(p)\\u^{\alpha }\left(j_{p}^{r}\sigma \right)&=u^{\alpha }(\sigma (p))\end{aligned}}}</annotation>

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and the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\left({\binom {m+r}{r}}-1\right)}">

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<annotation encoding="application/x-tex">{\displaystyle n\left({\binom {m+r}{r}}-1\right)}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{cases}u_{I}^{\alpha }:U^{k}\to \mathbf {R} \\u_{I}^{\alpha }\left(j_{p}^{r}\sigma \right)=\left.{\frac {\partial ^{|I|}\sigma ^{\alpha }}{\partial x^{I}}}\right|_{p}\end{cases}}}</annotation>

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Given an atlas of adapted charts (U, u) on E, the corresponding collection of charts (U r, u r) is a <a href="Finite-dimensional" class="mw-redirect" title="Finite-dimensional">finite-dimensional</a> C∞ atlas on J r(π).

<div class="mw-heading mw-heading2"><h2 id="Jet_bundles">Jet bundles</h2></div>

Since the atlas on each <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J^{r}(\pi )}">

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<annotation encoding="application/x-tex">{\displaystyle J^{r}(\pi )}</annotation>

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</math><img src="./9649cbfa3f6636b19fbbf7deb7f2446a7f74fd6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.641ex; height:2.843ex;" alt="{\displaystyle J^{r}(\pi )}" loading="lazy"> defines a manifold, the triples <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (J^{r}(\pi ),\pi _{r,k},J^{k}(\pi ))}">

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<annotation encoding="application/x-tex">{\displaystyle (J^{r}(\pi ),\pi _{r,k},J^{k}(\pi ))}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle (J^{r}(\pi ),\pi _{r,0},E)}</annotation>

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</math><img src="./5949a975ca16a143d5b93888c8b9a79652994eaa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.872ex; height:3.009ex;" alt="{\displaystyle (J^{r}(\pi ),\pi _{r,0},E)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (J^{r}(\pi ),\pi _{r},M)}">

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<annotation encoding="application/x-tex">{\displaystyle (J^{r}(\pi ),\pi _{r},M)}</annotation>

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</math><img src="./dc71851cb18da5d7ad5e7197d71052327936df92.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.259ex; height:2.843ex;" alt="{\displaystyle (J^{r}(\pi ),\pi _{r},M)}" loading="lazy"> all define fibered manifolds. In particular, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (E,\pi ,M)}">

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<annotation encoding="application/x-tex">{\displaystyle (E,\pi ,M)}</annotation>

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</math><img src="./d071b29bc6684043ec325007c5fdb45477c22210.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.427ex; height:2.843ex;" alt="{\displaystyle (E,\pi ,M)}" loading="lazy">is a fiber bundle, the triple <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (J^{r}(\pi ),\pi _{r},M)}">

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<annotation encoding="application/x-tex">{\displaystyle (J^{r}(\pi ),\pi _{r},M)}</annotation>

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If W ⊂ M is an open submanifold, then

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<annotation encoding="application/x-tex">{\displaystyle J^{r}\left(\pi |_{\pi ^{-1}(W)}\right)\cong \pi _{r}^{-1}(W).\,}</annotation>

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</math><img src="./72772559b5c37c310ed32ac5b82aaf550fa1750c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:25.732ex; height:4.843ex;" alt="{\displaystyle J^{r}\left(\pi |_{\pi ^{-1}(W)}\right)\cong \pi _{r}^{-1}(W).\,}" loading="lazy"></dd></dl>

If p ∈ M, then the fiber <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{r}^{-1}(p)\,}">

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<annotation encoding="application/x-tex">{\displaystyle \pi _{r}^{-1}(p)\,}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle J_{p}^{r}(\pi )}</annotation>

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</math><img src="./46ebd501acd1676eb03eefadca06bb7c0fba4000.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.641ex; height:3.009ex;" alt="{\displaystyle J_{p}^{r}(\pi )}" loading="lazy">.

Let σ be a local section of π with domain W ⊂ M. The r-th jet prolongation of σ is the map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j^{r}\sigma :W\rightarrow J^{r}(\pi )}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle j^{r}\sigma :W\rightarrow J^{r}(\pi )}</annotation>

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</math><img src="./0ef3a3191c9394682e21d135d50bd4f2520b84f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.027ex; width:16.916ex; height:2.843ex;" alt="{\displaystyle j^{r}\sigma :W\rightarrow J^{r}(\pi )}" loading="lazy"> defined by

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (j^{r}\sigma )(p)=j_{p}^{r}\sigma .\,}</annotation>

</semantics>

</math><img src="./b2420a54a5e9e03c15e7df1192d73d42506f8c7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.529ex; height:3.009ex;" alt="{\displaystyle (j^{r}\sigma )(p)=j_{p}^{r}\sigma .\,}" loading="lazy"></dd></dl>

Note that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{r}\circ j^{r}\sigma =\mathbb {id} _{W}}">

<msub>

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<msup>

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</msup>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \pi _{r}\circ j^{r}\sigma =\mathbb {id} _{W}}</annotation>

</semantics>

</math><img src="./a77260bb8a24bda1e319e2d9cd2e0de03d6ad753.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.747ex; height:2.676ex;" alt="{\displaystyle \pi _{r}\circ j^{r}\sigma =\mathbb {id} _{W}}" loading="lazy">, so <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j^{r}\sigma }">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle j^{r}\sigma }</annotation>

</semantics>

</math><img src="./57b59e7273b9b37cc2f8c4ec9f9678e88343b5af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:3.288ex; height:2.676ex;" alt="{\displaystyle j^{r}\sigma }" loading="lazy"> really is a section. In local coordinates, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j^{r}\sigma }">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle j^{r}\sigma }</annotation>

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<mrow>

<mrow>

<msup>

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</msup>

<mfrac>

<mrow>

<msup>

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</mfrac>

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<annotation encoding="application/x-tex">{\displaystyle \left(\sigma ^{\alpha },{\frac {\partial ^{|I|}\sigma ^{\alpha }}{\partial x^{I}}}\right)\qquad 1\leq |I|\leq r.\,}</annotation>

</semantics>

</math><img src="./1ead5cd0b09054bf8a0bd952cf7e9880c5d61d87.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.779ex; height:6.343ex;" alt="{\displaystyle \left(\sigma ^{\alpha },{\frac {\partial ^{|I|}\sigma ^{\alpha }}{\partial x^{I}}}\right)\qquad 1\leq |I|\leq r.\,}" loading="lazy"></dd></dl>

We identify <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j^{0}\sigma }">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle j^{0}\sigma }</annotation>

</semantics>

</math><img src="./f5e0f70e4e81874f7bb7fbb9936aa57e6dbea44e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:3.369ex; height:3.009ex;" alt="{\displaystyle j^{0}\sigma }" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation>

</semantics>

</math><img src="./59f59b7c3e6fdb1d0365a494b81fb9a696138c36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" loading="lazy"> .

<div class="mw-heading mw-heading3"><h3 id="Algebro-geometric_perspective">Algebro-geometric perspective</h3></div>

An independently motivated construction of the sheaf of sections <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma J^{k}\left(\pi _{TM}\right)}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \Gamma J^{k}\left(\pi _{TM}\right)}</annotation>

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</math><img src="./58ffed706956287399ae53667d15739e02fcec67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.705ex; height:3.176ex;" alt="{\displaystyle \Gamma J^{k}\left(\pi _{TM}\right)}" loading="lazy"> is given.

Consider a diagonal map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \Delta _{n}:M\to \prod _{i=1}^{n+1}M}">

<msub>

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</munderover>

</mstyle>

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<annotation encoding="application/x-tex">{\textstyle \Delta _{n}:M\to \prod _{i=1}^{n+1}M}</annotation>

</semantics>

</math><img src="./0a151859d8457595e9c19294b5448bd4e8040329.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.49ex; height:3.509ex;" alt="{\textstyle \Delta _{n}:M\to \prod _{i=1}^{n+1}M}" loading="lazy">, where the smooth manifold <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> is a <a href="Locally_ringed_space" class="mw-redirect" title="Locally ringed space">locally ringed space</a> by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{k}(U)}">

<msup>

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</msup>

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<annotation encoding="application/x-tex">{\displaystyle C^{k}(U)}</annotation>

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</math><img src="./073d4c8080827cd63cadcb5f0521b856f702b4e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.478ex; height:3.176ex;" alt="{\displaystyle C^{k}(U)}" loading="lazy"> for each open <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U}</annotation>

</semantics>

</math><img src="./458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy">. Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {I}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {I}}}</annotation>

</semantics>

</math><img src="./0e9730a0ada0426927ff64141eb9f505eca132d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-left: -0.069ex; width:1.561ex; height:2.176ex;" alt="{\displaystyle {\mathcal {I}}}" loading="lazy"> be the <a href="Ideal_sheaf" title="Ideal sheaf">ideal sheaf</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{n}(M)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta _{n}(M)}</annotation>

</semantics>

</math><img src="./fdfe236d2038f7010dab50bf5d780f899ab1b653.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.406ex; height:2.843ex;" alt="{\displaystyle \Delta _{n}(M)}" loading="lazy">, equivalently let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {I}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {I}}}</annotation>

</semantics>

</math><img src="./0e9730a0ada0426927ff64141eb9f505eca132d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-left: -0.069ex; width:1.561ex; height:2.176ex;" alt="{\displaystyle {\mathcal {I}}}" loading="lazy"> be the <a href="Sheaf_(mathematics)" title="Sheaf (mathematics)">sheaf</a> of smooth <a href="Germ_(mathematics)" title="Germ (mathematics)">germs</a> which vanish on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{n}(M)}">

<msub>

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</msub>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \Delta _{n}(M)}</annotation>

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</math><img src="./fdfe236d2038f7010dab50bf5d780f899ab1b653.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.406ex; height:2.843ex;" alt="{\displaystyle \Delta _{n}(M)}" loading="lazy"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0<n\leq k}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle 0<n\leq k}</annotation>

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</math><img src="./1e3a62288b780861486a82dbbe04ab616acd8166.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.965ex; height:2.343ex;" alt="{\displaystyle 0<n\leq k}" loading="lazy">. The <a href="Inverse_image_functor" title="Inverse image functor">pullback</a> of the <a href="Quotient_sheaf" class="mw-redirect" title="Quotient sheaf">quotient sheaf</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\Delta _{n}}^{*}\left({\mathcal {I}}/{\mathcal {I}}^{n+1}\right)}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle {\Delta _{n}}^{*}\left({\mathcal {I}}/{\mathcal {I}}^{n+1}\right)}</annotation>

</semantics>

</math><img src="./92e22c78b5e0125b2680387920c3d07cbc3d7236.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.259ex; height:3.343ex;" alt="{\displaystyle {\Delta _{n}}^{*}\left({\mathcal {I}}/{\mathcal {I}}^{n+1}\right)}" loading="lazy"> from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \prod _{i=1}^{n+1}M}">

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<annotation encoding="application/x-tex">{\textstyle \prod _{i=1}^{n+1}M}</annotation>

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</math><img src="./a46bb6fa126c64f2446c3a44fd9ebd3dde0e6032.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.342ex; height:3.509ex;" alt="{\textstyle \prod _{i=1}^{n+1}M}" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

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</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta _{n}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \Delta _{n}}</annotation>

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</math><img src="./caca5839d47294236c4bb71f41a578e36e0604fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.154ex; height:2.509ex;" alt="{\displaystyle \Delta _{n}}" loading="lazy"> is the sheaf of k-jets.<a href="#cite_note-2">[2]</a>

The <a href="Direct_limit" title="Direct limit">direct limit</a> of the sequence of injections given by the canonical inclusions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {I}}^{n+1}\hookrightarrow {\mathcal {I}}^{n}}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {I}}^{n+1}\hookrightarrow {\mathcal {I}}^{n}}</annotation>

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</math><img src="./626d1fc88053f1f293d581f4b07ba7c7f790b3fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-left: -0.069ex; width:11.633ex; height:2.676ex;" alt="{\displaystyle {\mathcal {I}}^{n+1}\hookrightarrow {\mathcal {I}}^{n}}" loading="lazy"> of sheaves, gives rise to the infinite jet sheaf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {J}}^{\infty }(TM)}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {J}}^{\infty }(TM)}</annotation>

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</math><img src="./824478470a4e7da20917379fc44711d47a7269af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.825ex; height:2.843ex;" alt="{\displaystyle {\mathcal {J}}^{\infty }(TM)}" loading="lazy">. Observe that by the direct limit construction it is a filtered ring.

<div class="mw-heading mw-heading3"><h3 id="Example">Example</h3></div>

If π is the <a href="Trivial_bundle" class="mw-redirect" title="Trivial bundle">trivial bundle</a> (M × R, pr1, M), then there is a canonical <a href="Diffeomorphism" title="Diffeomorphism">diffeomorphism</a> between the first jet bundle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J^{1}(\pi )}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle J^{1}(\pi )}</annotation>

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</math><img src="./853ef765ac56e27fff16d54f41405345d6ecc230.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.721ex; height:3.176ex;" alt="{\displaystyle J^{1}(\pi )}" loading="lazy"> and T*M × R. To construct this diffeomorphism, for each σ in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma _{M}(\pi )}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle \Gamma _{M}(\pi )}</annotation>

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</math><img src="./c6a130da0d700ae8fc471186d20d531c9a914d86.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.553ex; height:2.843ex;" alt="{\displaystyle \Gamma _{M}(\pi )}" loading="lazy"> write <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {\sigma }}=pr_{2}\circ \sigma \in C^{\infty }(M)\,}">

<mover>

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle {\bar {\sigma }}=pr_{2}\circ \sigma \in C^{\infty }(M)\,}</annotation>

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</math><img src="./8e77d39c0d0986dc413eab19c184fe8f3bae7878.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.377ex; height:2.843ex;" alt="{\displaystyle {\bar {\sigma }}=pr_{2}\circ \sigma \in C^{\infty }(M)\,}" loading="lazy">.

Then, whenever p ∈ M

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<annotation encoding="application/x-tex">{\displaystyle j_{p}^{1}\sigma =\left\{\psi :\psi \in \Gamma _{p}(\pi );{\bar {\psi }}(p)={\bar {\sigma }}(p);d{\bar {\psi }}_{p}=d{\bar {\sigma }}_{p}\right\}.\,}</annotation>

</semantics>

</math><img src="./ddebc1bd1348edf2da04dafbd60271ba4ec400b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; margin-left: -0.027ex; width:48.684ex; height:3.343ex;" alt="{\displaystyle j_{p}^{1}\sigma =\left\{\psi :\psi \in \Gamma _{p}(\pi );{\bar {\psi }}(p)={\bar {\sigma }}(p);d{\bar {\psi }}_{p}=d{\bar {\sigma }}_{p}\right\}.\,}" loading="lazy"></dd></dl>

Consequently, the mapping

<mrow>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{cases}J^{1}(\pi )\to T^{*}M\times \mathbf {R} \\j_{p}^{1}\sigma \mapsto \left(d{\bar {\sigma }}_{p},{\bar {\sigma }}(p)\right)\end{cases}}}</annotation>

</semantics>

</math><img src="./8340619fe74944b74cc6ec8acc6ae1c548525156.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.89ex; height:6.176ex;" alt="{\displaystyle {\begin{cases}J^{1}(\pi )\to T^{*}M\times \mathbf {R} \\j_{p}^{1}\sigma \mapsto \left(d{\bar {\sigma }}_{p},{\bar {\sigma }}(p)\right)\end{cases}}}" loading="lazy"></dd></dl>

is well-defined and is clearly <a href="Injective" class="mw-redirect" title="Injective">injective</a>. Writing it out in coordinates shows that it is a diffeomorphism, because if (xi, u) are coordinates on M × R, where u = idR is the identity coordinate, then the derivative coordinates ui on J1(π) correspond to the coordinates ∂i on T*M.

Likewise, if π is the trivial bundle (R × M, pr1, R), then there exists a canonical diffeomorphism between <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J^{1}(\pi )}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle J^{1}(\pi )}</annotation>

</semantics>

<div class="mw-heading mw-heading2"><h2 id="Contact_structure">Contact structure</h2></div>

The space Jr(π) carries a natural <a href="Distribution_(differential_geometry)" title="Distribution (differential geometry)">distribution</a>, that is, a sub-bundle of the <a href="Tangent_bundle" title="Tangent bundle">tangent bundle</a> TJr(π)), called the Cartan distribution. The Cartan distribution is spanned by all tangent planes to graphs of holonomic sections; that is, sections of the form jrφ for φ a section of π.

The annihilator of the Cartan distribution is a space of <a href="One-form" class="mw-redirect" title="One-form">differential one-forms</a> called <a href="Contact_form" class="mw-redirect" title="Contact form">contact forms</a>, on Jr(π). The space of differential one-forms on Jr(π) is denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda ^{1}J^{r}(\pi )}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \Lambda ^{1}J^{r}(\pi )}</annotation>

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</math><img src="./9b601757c46f393365efc60578b7557fe2189791.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.308ex; height:3.176ex;" alt="{\displaystyle \Lambda ^{1}J^{r}(\pi )}" loading="lazy"> and the space of contact forms is denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda _{C}^{r}\pi }">

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</msubsup>

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<annotation encoding="application/x-tex">{\displaystyle \Lambda _{C}^{r}\pi }</annotation>

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</math><img src="./a3814be6f1ca391302add414be94fc7f2bad0e42.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.426ex; height:2.843ex;" alt="{\displaystyle \Lambda _{C}^{r}\pi }" loading="lazy">. A one form is a contact form provided its <a href="Pullback_(differential_geometry)" title="Pullback (differential geometry)">pullback</a> along every prolongation is zero. In other words, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta \in \Lambda ^{1}J^{r}\pi }">

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<annotation encoding="application/x-tex">{\displaystyle \theta \in \Lambda ^{1}J^{r}\pi }</annotation>

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</math><img src="./320d8e9d62fc4d73664b09d6463dd46f86466c0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.43ex; height:2.676ex;" alt="{\displaystyle \theta \in \Lambda ^{1}J^{r}\pi }" loading="lazy"> is a contact form if and only if

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<annotation encoding="application/x-tex">{\displaystyle \left(j^{r+1}\sigma \right)^{*}\theta =0}</annotation>

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</math><img src="./92aaf298403227f5b7b9dd0fc0c9219b453bb99c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.897ex; height:3.509ex;" alt="{\displaystyle \left(j^{r+1}\sigma \right)^{*}\theta =0}" loading="lazy"></dd></dl>

for all local sections σ of π over M.

The Cartan distribution is the main geometrical structure on jet spaces and plays an important role in the geometric theory of <a href="Partial_differential_equation" title="Partial differential equation">partial differential equations</a>. The Cartan distributions are completely non-integrable. In particular, they are not <a href="Distribution_(differential_geometry)" title="Distribution (differential geometry)">involutive</a>. The dimension of the Cartan distribution grows with the order of the jet space. However, on the space of infinite jets J∞ the Cartan distribution becomes involutive and finite-dimensional: its dimension coincides with the dimension of the base manifold M.

<div class="mw-heading mw-heading3"><h3 id="Example_2">Example</h3></div>

Consider the case (E, π, M), where E ≃ R2 and M ≃ R. Then, (J1(π), π, M) defines the first jet bundle, and may be coordinated by (x, u, u1), where

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x\left(j_{p}^{1}\sigma \right)&=x(p)=x\\u\left(j_{p}^{1}\sigma \right)&=u(\sigma (p))=u(\sigma (x))=\sigma (x)\\u_{1}\left(j_{p}^{1}\sigma \right)&=\left.{\frac {\partial \sigma }{\partial x}}\right|_{p}=\sigma '(x)\end{aligned}}}</annotation>

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</math><img src="./bea275f562c0cab986befdd74b416cd129cbd935.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.76ex; margin-bottom: -0.244ex; width:37.818ex; height:13.176ex;" alt="{\displaystyle {\begin{aligned}x\left(j_{p}^{1}\sigma \right)&=x(p)=x\\u\left(j_{p}^{1}\sigma \right)&=u(\sigma (p))=u(\sigma (x))=\sigma (x)\\u_{1}\left(j_{p}^{1}\sigma \right)&=\left.{\frac {\partial \sigma }{\partial x}}\right|_{p}=\sigma '(x)\end{aligned}}}" loading="lazy"></dd></dl>

for all p ∈ M and σ in Γp(π). A general 1-form on J1(π) takes the form

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<annotation encoding="application/x-tex">{\displaystyle \theta =a(x,u,u_{1})dx+b(x,u,u_{1})du+c(x,u,u_{1})du_{1}\,}</annotation>

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</math><img src="./7c3864834d40978996def5d6e1179b4b399c42bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:48.943ex; height:2.843ex;" alt="{\displaystyle \theta =a(x,u,u_{1})dx+b(x,u,u_{1})du+c(x,u,u_{1})du_{1}\,}" loading="lazy"></dd></dl>

A section σ in Γp(π) has first prolongation

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<annotation encoding="application/x-tex">{\displaystyle j^{1}\sigma =(u,u_{1})=\left(\sigma (p),\left.{\frac {\partial \sigma }{\partial x}}\right|_{p}\right).}</annotation>

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</math><img src="./38248dbe553e825feb9afc705dd1bb59f80ee514.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; margin-left: -0.027ex; width:31.11ex; height:6.343ex;" alt="{\displaystyle j^{1}\sigma =(u,u_{1})=\left(\sigma (p),\left.{\frac {\partial \sigma }{\partial x}}\right|_{p}\right).}" loading="lazy"></dd></dl>

Hence, (j1σ)*θ can be calculated as

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\left(j_{p}^{1}\sigma \right)^{*}\theta &=\theta \circ j_{p}^{1}\sigma \\&=a(x,\sigma (x),\sigma '(x))dx+b(x,\sigma (x),\sigma '(x))d(\sigma (x))+c(x,\sigma (x),\sigma '(x))d(\sigma '(x))\\&=a(x,\sigma (x),\sigma '(x))dx+b(x,\sigma (x),\sigma '(x))\sigma '(x)dx+c(x,\sigma (x),\sigma '(x))\sigma ''(x)dx\\&=[a(x,\sigma (x),\sigma '(x))+b(x,\sigma (x),\sigma '(x))\sigma '(x)+c(x,\sigma (x),\sigma '(x))\sigma ''(x)]dx\end{aligned}}}</annotation>

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</math><img src="./30a959d2fa3a6248866a919b98d607dea8e76829.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:83.273ex; height:12.843ex;" alt="{\displaystyle {\begin{aligned}\left(j_{p}^{1}\sigma \right)^{*}\theta &=\theta \circ j_{p}^{1}\sigma \\&=a(x,\sigma (x),\sigma '(x))dx+b(x,\sigma (x),\sigma '(x))d(\sigma (x))+c(x,\sigma (x),\sigma '(x))d(\sigma '(x))\\&=a(x,\sigma (x),\sigma '(x))dx+b(x,\sigma (x),\sigma '(x))\sigma '(x)dx+c(x,\sigma (x),\sigma '(x))\sigma ''(x)dx\\&=[a(x,\sigma (x),\sigma '(x))+b(x,\sigma (x),\sigma '(x))\sigma '(x)+c(x,\sigma (x),\sigma '(x))\sigma ''(x)]dx\end{aligned}}}" loading="lazy"></dd></dl>

This will vanish for all sections σ if and only if c = 0 and a = −bσ′(x). Hence, θ = b(x, u, u1)θ0 must necessarily be a multiple of the basic contact form θ0 = du − u1dx. Proceeding to the second jet space J2(π) with additional coordinate u2, such that

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<annotation encoding="application/x-tex">{\displaystyle u_{2}(j_{p}^{2}\sigma )=\left.{\frac {\partial ^{2}\sigma }{\partial x^{2}}}\right|_{p}=\sigma ''(x)\,}</annotation>

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</math><img src="./ec2c97efbdac2a290990c0dc3b997642a04ccb8b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:26ex; height:6.843ex;" alt="{\displaystyle u_{2}(j_{p}^{2}\sigma )=\left.{\frac {\partial ^{2}\sigma }{\partial x^{2}}}\right|_{p}=\sigma ''(x)\,}" loading="lazy"></dd></dl>

a general 1-form has the construction

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<annotation encoding="application/x-tex">{\displaystyle \theta =a(x,u,u_{1},u_{2})dx+b(x,u,u_{1},u_{2})du+c(x,u,u_{1},u_{2})du_{1}+e(x,u,u_{1},u_{2})du_{2}\,}</annotation>

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</math><img src="./af24ae9c1f9859ad19c48de3bb64ddac54f41dfc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:79.059ex; height:2.843ex;" alt="{\displaystyle \theta =a(x,u,u_{1},u_{2})dx+b(x,u,u_{1},u_{2})du+c(x,u,u_{1},u_{2})du_{1}+e(x,u,u_{1},u_{2})du_{2}\,}" loading="lazy"></dd></dl>

This is a contact form if and only if

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\left(j_{p}^{2}\sigma \right)^{*}\theta &=\theta \circ j_{p}^{2}\sigma \\&=a(x,\sigma (x),\sigma '(x),\sigma ''(x))dx+b(x,\sigma (x),\sigma '(x),\sigma ''(x))d(\sigma (x))+{}\\&\qquad \qquad c(x,\sigma (x),\sigma '(x),\sigma ''(x))d(\sigma '(x))+e(x,\sigma (x),\sigma '(x),\sigma ''(x))d(\sigma ''(x))\\&=adx+b\sigma '(x)dx+c\sigma ''(x)dx+e\sigma '''(x)dx\\&=[a+b\sigma '(x)+c\sigma ''(x)+e\sigma '''(x)]dx\\&=0\end{aligned}}}</annotation>

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</math><img src="./11dfc307319b49dd700c8bf6df06a16de4993719.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.005ex; width:82.346ex; height:19.176ex;" alt="{\displaystyle {\begin{aligned}\left(j_{p}^{2}\sigma \right)^{*}\theta &=\theta \circ j_{p}^{2}\sigma \\&=a(x,\sigma (x),\sigma '(x),\sigma ''(x))dx+b(x,\sigma (x),\sigma '(x),\sigma ''(x))d(\sigma (x))+{}\\&\qquad \qquad c(x,\sigma (x),\sigma '(x),\sigma ''(x))d(\sigma '(x))+e(x,\sigma (x),\sigma '(x),\sigma ''(x))d(\sigma ''(x))\\&=adx+b\sigma '(x)dx+c\sigma ''(x)dx+e\sigma '''(x)dx\\&=[a+b\sigma '(x)+c\sigma ''(x)+e\sigma '''(x)]dx\\&=0\end{aligned}}}" loading="lazy"></dd></dl>

which implies that e = 0 and a = −bσ′(x) − cσ′′(x). Therefore, θ is a contact form if and only if

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<annotation encoding="application/x-tex">{\displaystyle \theta =b(x,\sigma (x),\sigma '(x))\theta _{0}+c(x,\sigma (x),\sigma '(x))\theta _{1},}</annotation>

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</math><img src="./4309aa5bf40ee7cfe2b7f2e796d8465e4e6ccb55.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.629ex; height:3.009ex;" alt="{\displaystyle \theta =b(x,\sigma (x),\sigma '(x))\theta _{0}+c(x,\sigma (x),\sigma '(x))\theta _{1},}" loading="lazy"></dd></dl>

where θ1 = du1 − u2dx is the next basic contact form (Note that here we are identifying the form θ0 with its pull-back <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\pi _{2,1}\right)^{*}\theta _{0}}">

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<annotation encoding="application/x-tex">{\displaystyle \left(\pi _{2,1}\right)^{*}\theta _{0}}</annotation>

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</math><img src="./346219f1d3142544149de300d7f00bf47996cbbb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.667ex; height:3.176ex;" alt="{\displaystyle \left(\pi _{2,1}\right)^{*}\theta _{0}}" loading="lazy"> to J2(π)).

In general, providing x, u ∈ R, a contact form on Jr+1(π) can be written as a <a href="Linear_combination" title="Linear combination">linear combination</a> of the basic contact forms

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<annotation encoding="application/x-tex">{\displaystyle \theta _{k}=du_{k}-u_{k+1}dx\qquad k=0,\ldots ,r-1\,}</annotation>

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</math><img src="./c8f43640780c3d267f361a50f6b7803e47c656a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:39.551ex; height:2.509ex;" alt="{\displaystyle \theta _{k}=du_{k}-u_{k+1}dx\qquad k=0,\ldots ,r-1\,}" loading="lazy"></dd></dl>

where

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<annotation encoding="application/x-tex">{\displaystyle u_{k}\left(j^{k}\sigma \right)=\left.{\frac {\partial ^{k}\sigma }{\partial x^{k}}}\right|_{p}.}</annotation>

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</math><img src="./aa3fd2114a7c6f32264bb75f8563c1983f871fe8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:18.361ex; height:6.843ex;" alt="{\displaystyle u_{k}\left(j^{k}\sigma \right)=\left.{\frac {\partial ^{k}\sigma }{\partial x^{k}}}\right|_{p}.}" loading="lazy"></dd></dl>

Similar arguments lead to a complete characterization of all contact forms.

In local coordinates, every contact one-form on Jr+1(π) can be written as a linear combination

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<annotation encoding="application/x-tex">{\displaystyle \theta =\sum _{|I|=0}^{r}P_{\alpha }^{I}\theta _{I}^{\alpha }}</annotation>

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</math><img src="./0bd7ddb855d6b8cb94d0cefa96a5d0670cddbce0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:13.677ex; height:7.343ex;" alt="{\displaystyle \theta =\sum _{|I|=0}^{r}P_{\alpha }^{I}\theta _{I}^{\alpha }}" loading="lazy"></dd></dl>

with smooth coefficients <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{i}^{\alpha }(x^{i},u^{\alpha },u_{I}^{\alpha })}">

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<annotation encoding="application/x-tex">{\displaystyle \theta _{I}^{\alpha }=du_{I}^{\alpha }-u_{I,i}^{\alpha }dx^{i}\,}</annotation>

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</math><img src="./ef597c7049ccc59a3601b2cc106cef7eb45d708a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:19.291ex; height:3.676ex;" alt="{\displaystyle \theta _{I}^{\alpha }=du_{I}^{\alpha }-u_{I,i}^{\alpha }dx^{i}\,}" loading="lazy"></dd></dl>

|I| is known as the order of the contact form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{i}^{\alpha }}">

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</math><img src="./0a3465c262bd336ecfbcb836b78e65f93af98fe2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.375ex; height:2.843ex;" alt="{\displaystyle \theta _{i}^{\alpha }}" loading="lazy">. Note that contact forms on Jr+1(π) have orders at most r. Contact forms provide a characterization of those local sections of πr+1 which are prolongations of sections of π.

Let ψ ∈ ΓW(πr+1), then ψ = jr+1σ where σ ∈ ΓW(π) if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi ^{*}(\theta |_{W})=0,\forall \theta \in \Lambda _{C}^{1}\pi _{r+1,r}.\,}">

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<annotation encoding="application/x-tex">{\displaystyle \psi ^{*}(\theta |_{W})=0,\forall \theta \in \Lambda _{C}^{1}\pi _{r+1,r}.\,}</annotation>

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</math><img src="./338a7600f5e0eb91bba4df969facdd601c988e6a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:28.313ex; height:3.176ex;" alt="{\displaystyle \psi ^{*}(\theta |_{W})=0,\forall \theta \in \Lambda _{C}^{1}\pi _{r+1,r}.\,}" loading="lazy">

<div class="mw-heading mw-heading2"><h2 id="Vector_fields">Vector fields</h2></div>

A general <a href="Vector_field" title="Vector field">vector field</a> on the total space E, coordinated by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,u)\mathrel {\stackrel {\mathrm {def} }{=}} \left(x^{i},u^{\alpha }\right)\,}">

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<annotation encoding="application/x-tex">{\displaystyle (x,u)\mathrel {\stackrel {\mathrm {def} }{=}} \left(x^{i},u^{\alpha }\right)\,}</annotation>

</semantics>

</math><img src="./4ee1a666bfdca48e00cdd716daa639d95725fef3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.343ex; height:4.009ex;" alt="{\displaystyle (x,u)\mathrel {\stackrel {\mathrm {def} }{=}} \left(x^{i},u^{\alpha }\right)\,}" loading="lazy">, is

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<annotation encoding="application/x-tex">{\displaystyle V\mathrel {\stackrel {\mathrm {def} }{=}} \rho ^{i}(x,u){\frac {\partial }{\partial x^{i}}}+\phi ^{\alpha }(x,u){\frac {\partial }{\partial u^{\alpha }}}.\,}</annotation>

</semantics>

</math><img src="./5a73aaad771c8319d08e7efeb33b67c63f2fc468.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:33.936ex; height:5.676ex;" alt="{\displaystyle V\mathrel {\stackrel {\mathrm {def} }{=}} \rho ^{i}(x,u){\frac {\partial }{\partial x^{i}}}+\phi ^{\alpha }(x,u){\frac {\partial }{\partial u^{\alpha }}}.\,}" loading="lazy"></dd></dl>

A vector field is called horizontal, meaning that all the vertical coefficients vanish, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi ^{\alpha }}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \phi ^{\alpha }}</annotation>

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</math><img src="./2856edf6ea22b822d25e5f639bc7cddfcab7975d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.67ex; height:2.676ex;" alt="{\displaystyle \phi ^{\alpha }}" loading="lazy"> = 0.

A vector field is called vertical, meaning that all the horizontal coefficients vanish, if ρi = 0.

For fixed (x, u), we identify

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<annotation encoding="application/x-tex">{\displaystyle V_{(x,u)}\mathrel {\stackrel {\mathrm {def} }{=}} \rho ^{i}(x,u){\frac {\partial }{\partial x^{i}}}+\phi ^{\alpha }(x,u){\frac {\partial }{\partial u^{\alpha }}}\,}</annotation>

</semantics>

</math><img src="./d80387b3c6dc6a833ea3180c0ad408bf77b452ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:36.706ex; height:5.676ex;" alt="{\displaystyle V_{(x,u)}\mathrel {\stackrel {\mathrm {def} }{=}} \rho ^{i}(x,u){\frac {\partial }{\partial x^{i}}}+\phi ^{\alpha }(x,u){\frac {\partial }{\partial u^{\alpha }}}\,}" loading="lazy"></dd></dl>

having coordinates (x, u, ρi, φα), with an element in the fiber TxuE of TE over (x, u) in E, called a <a href="Tangent_vector" title="Tangent vector">tangent vector</a> in TE. A section

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<annotation encoding="application/x-tex">{\displaystyle {\begin{cases}\psi :E\to TE\\(x,u)\mapsto \psi (x,u)=V\end{cases}}}</annotation>

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</math><img src="./b75f306d161062fd0a72e356973e83e7bd389cf1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:23.513ex; height:6.176ex;" alt="{\displaystyle {\begin{cases}\psi :E\to TE\\(x,u)\mapsto \psi (x,u)=V\end{cases}}}" loading="lazy"></dd></dl>

is called a vector field on E with

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<annotation encoding="application/x-tex">{\displaystyle V=\rho ^{i}(x,u){\frac {\partial }{\partial x^{i}}}+\phi ^{\alpha }(x,u){\frac {\partial }{\partial u^{\alpha }}}}</annotation>

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</math><img src="./dd484b5d55f851f0993c2863016562003fa69135.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:32.454ex; height:5.676ex;" alt="{\displaystyle V=\rho ^{i}(x,u){\frac {\partial }{\partial x^{i}}}+\phi ^{\alpha }(x,u){\frac {\partial }{\partial u^{\alpha }}}}" loading="lazy"></dd></dl>

and ψ in Γ(TE).

The jet bundle Jr(π) is coordinated by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,u,w)\mathrel {\stackrel {\mathrm {def} }{=}} \left(x^{i},u^{\alpha },w_{i}^{\alpha }\right)\,}">

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<annotation encoding="application/x-tex">{\displaystyle (x,u,w)\mathrel {\stackrel {\mathrm {def} }{=}} \left(x^{i},u^{\alpha },w_{i}^{\alpha }\right)\,}</annotation>

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</math><img src="./33336a711412f9c0583b6277a38e692dc09ea8ca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.023ex; height:4.009ex;" alt="{\displaystyle (x,u,w)\mathrel {\stackrel {\mathrm {def} }{=}} \left(x^{i},u^{\alpha },w_{i}^{\alpha }\right)\,}" loading="lazy">. For fixed (x, u, w), identify

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<annotation encoding="application/x-tex">{\displaystyle V_{(x,u,w)}\mathrel {\stackrel {\mathrm {def} }{=}} V^{i}(x,u,w){\frac {\partial }{\partial x^{i}}}+V^{\alpha }(x,u,w){\frac {\partial }{\partial u^{\alpha }}}+V_{i}^{\alpha }(x,u,w){\frac {\partial }{\partial w_{i}^{\alpha }}}+V_{i_{1}i_{2}}^{\alpha }(x,u,w){\frac {\partial }{\partial w_{i_{1}i_{2}}^{\alpha }}}+\cdots +V_{i_{1}\cdots i_{r}}^{\alpha }(x,u,w){\frac {\partial }{\partial w_{i_{1}\cdots i_{r}}^{\alpha }}}}</annotation>

</semantics>

</math><img src="./0b17af58f76716b0d26090ea4a57af3af3b9010e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:117.775ex; height:6.509ex;" alt="{\displaystyle V_{(x,u,w)}\mathrel {\stackrel {\mathrm {def} }{=}} V^{i}(x,u,w){\frac {\partial }{\partial x^{i}}}+V^{\alpha }(x,u,w){\frac {\partial }{\partial u^{\alpha }}}+V_{i}^{\alpha }(x,u,w){\frac {\partial }{\partial w_{i}^{\alpha }}}+V_{i_{1}i_{2}}^{\alpha }(x,u,w){\frac {\partial }{\partial w_{i_{1}i_{2}}^{\alpha }}}+\cdots +V_{i_{1}\cdots i_{r}}^{\alpha }(x,u,w){\frac {\partial }{\partial w_{i_{1}\cdots i_{r}}^{\alpha }}}}" loading="lazy"></dd></dl>

having coordinates

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<annotation encoding="application/x-tex">{\displaystyle \left(x,u,w,v_{i}^{\alpha },v_{i_{1}i_{2}}^{\alpha },\cdots ,v_{i_{1}\cdots i_{r}}^{\alpha }\right),}</annotation>

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</math><img src="./66ce232921bc624c621561f78dc5c54219711d4f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:30.035ex; height:4.843ex;" alt="{\displaystyle \left(x,u,w,v_{i}^{\alpha },v_{i_{1}i_{2}}^{\alpha },\cdots ,v_{i_{1}\cdots i_{r}}^{\alpha }\right),}" loading="lazy"></dd></dl>

with an element in the fiber <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{xuw}(J^{r}\pi )}">

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<annotation encoding="application/x-tex">{\displaystyle T_{xuw}(J^{r}\pi )}</annotation>

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</math><img src="./0a48a8b44869e9286f98fe57146ee0dd0edf128a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.288ex; height:2.843ex;" alt="{\displaystyle T_{xuw}(J^{r}\pi )}" loading="lazy"> of TJr(π) over (x, u, w) ∈ Jr(π), called a tangent vector in TJr(π). Here,

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<annotation encoding="application/x-tex">{\displaystyle v_{i}^{\alpha },v_{i_{1}i_{2}}^{\alpha },\ldots ,v_{i_{1}\cdots i_{r}}^{\alpha }}</annotation>

</semantics>

</math><img src="./1fcad68d44201f6945c84f8307fa1fef9f6d8ed3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:18.8ex; height:3.176ex;" alt="{\displaystyle v_{i}^{\alpha },v_{i_{1}i_{2}}^{\alpha },\ldots ,v_{i_{1}\cdots i_{r}}^{\alpha }}" loading="lazy"></dd></dl>

are real-valued functions on Jr(π). A section

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<annotation encoding="application/x-tex">{\displaystyle {\begin{cases}\Psi :J^{r}(\pi )\to TJ^{r}(\pi )\\(x,u,w)\mapsto \Psi (u,w)=V\end{cases}}}</annotation>

</semantics>

</math><img src="./365d238d36b29300cfd08f1bf5c59388c922d2c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.84ex; height:6.176ex;" alt="{\displaystyle {\begin{cases}\Psi :J^{r}(\pi )\to TJ^{r}(\pi )\\(x,u,w)\mapsto \Psi (u,w)=V\end{cases}}}" loading="lazy"></dd></dl>

is a vector field on Jr(π), and we say <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi \in \Gamma (T\left(J^{r}\pi \right)).}">

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<annotation encoding="application/x-tex">{\displaystyle \Psi \in \Gamma (T\left(J^{r}\pi \right)).}</annotation>

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</math><img src="./20c7b63f0bdfb87e8f083a5683c7f4391410eb44.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.222ex; height:2.843ex;" alt="{\displaystyle \Psi \in \Gamma (T\left(J^{r}\pi \right)).}" loading="lazy">

<div class="mw-heading mw-heading2"><h2 id="Partial_differential_equations">Partial differential equations</h2></div>

Let (E, π, M) be a fiber bundle. An r-th order <a href="Partial_differential_equation" title="Partial differential equation">partial differential equation</a> on π is a <a href="Closed_manifold" title="Closed manifold">closed</a> <a href="Embedding" title="Embedding">embedded</a> submanifold S of the jet manifold Jr(π). A solution is a local section σ ∈ ΓW(π) satisfying <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j_{p}^{r}\sigma \in S}">

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<annotation encoding="application/x-tex">{\displaystyle j_{p}^{r}\sigma \in S}</annotation>

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</math><img src="./56e780d58e2f845e2d3db41159e1624a1f65e7ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.027ex; width:7.714ex; height:2.843ex;" alt="{\displaystyle j_{p}^{r}\sigma \in S}" loading="lazy">, for all p in M.

Consider an example of a first order partial differential equation.

<div class="mw-heading mw-heading3"><h3 id="Example_3">Example</h3></div>

Let π be the trivial bundle (R2 × R, pr1, R2) with global coordinates (x1, x2, u1). Then the map F : J1(π) → R defined by

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<annotation encoding="application/x-tex">{\displaystyle F=u_{1}^{1}u_{2}^{1}-2x^{2}u^{1}}</annotation>

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</math><img src="./cc33047b46d54b0d94cf0a0894f812aa9b91b995.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.378ex; height:3.343ex;" alt="{\displaystyle F=u_{1}^{1}u_{2}^{1}-2x^{2}u^{1}}" loading="lazy"></dd></dl>

gives rise to the differential equation

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<annotation encoding="application/x-tex">{\displaystyle S=\left\{j_{p}^{1}\sigma \in J^{1}\pi \ :\ \left(u_{1}^{1}u_{2}^{1}-2x^{2}u^{1}\right)\left(j_{p}^{1}\sigma \right)=0\right\}}</annotation>

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</math><img src="./b4c34c8158758f3c92c4172579191cb284c519da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:46.686ex; height:3.343ex;" alt="{\displaystyle S=\left\{j_{p}^{1}\sigma \in J^{1}\pi \ :\ \left(u_{1}^{1}u_{2}^{1}-2x^{2}u^{1}\right)\left(j_{p}^{1}\sigma \right)=0\right\}}" loading="lazy"></dd></dl>

which can be written

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \sigma }{\partial x^{1}}}{\frac {\partial \sigma }{\partial x^{2}}}-2x^{2}\sigma =0.}</annotation>

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</math><img src="./2c0f465f26ade7b890d58ce281a5cc5f228c2c42.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:21.7ex; height:5.676ex;" alt="{\displaystyle {\frac {\partial \sigma }{\partial x^{1}}}{\frac {\partial \sigma }{\partial x^{2}}}-2x^{2}\sigma =0.}" loading="lazy"></dd></dl>

The particular

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<annotation encoding="application/x-tex">{\displaystyle {\begin{cases}\sigma :\mathbf {R} ^{2}\to \mathbf {R} ^{2}\times \mathbf {R} \\\sigma (p_{1},p_{2})=\left(p^{1},p^{2},p^{1}(p^{2})^{2}\right)\end{cases}}}</annotation>

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</math><img src="./170021c4379b635226ae1d552741a699a8f7c223.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.169ex; height:6.176ex;" alt="{\displaystyle {\begin{cases}\sigma :\mathbf {R} ^{2}\to \mathbf {R} ^{2}\times \mathbf {R} \\\sigma (p_{1},p_{2})=\left(p^{1},p^{2},p^{1}(p^{2})^{2}\right)\end{cases}}}" loading="lazy"></dd></dl>

has first prolongation given by

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<annotation encoding="application/x-tex">{\displaystyle j^{1}\sigma \left(p_{1},p_{2}\right)=\left(p^{1},p^{2},p^{1}\left(p^{2}\right)^{2},\left(p^{2}\right)^{2},2p^{1}p^{2}\right)}</annotation>

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</math><img src="./d59d02a2fd320dc0f08f8a7e64331bf9e76bdbca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; margin-left: -0.027ex; width:44.152ex; height:4.843ex;" alt="{\displaystyle j^{1}\sigma \left(p_{1},p_{2}\right)=\left(p^{1},p^{2},p^{1}\left(p^{2}\right)^{2},\left(p^{2}\right)^{2},2p^{1}p^{2}\right)}" loading="lazy"></dd></dl>

and is a solution of this differential equation, because

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\left(u_{1}^{1}u_{2}^{1}-2x^{2}u^{1}\right)\left(j_{p}^{1}\sigma \right)&=u_{1}^{1}\left(j_{p}^{1}\sigma \right)u_{2}^{1}\left(j_{p}^{1}\sigma \right)-2x^{2}\left(j_{p}^{1}\sigma \right)u^{1}\left(j_{p}^{1}\sigma \right)\\&=\left(p^{2}\right)^{2}\cdot 2p^{1}p^{2}-2\cdot p^{2}\cdot p^{1}\left(p^{2}\right)^{2}\\&=2p^{1}\left(p^{2}\right)^{3}-2p^{1}\left(p^{2}\right)^{3}\\&=0\end{aligned}}}</annotation>

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</math><img src="./9a8abb33c28c47df71b734e175f724b8e9b918ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.671ex; width:63.15ex; height:14.509ex;" alt="{\displaystyle {\begin{aligned}\left(u_{1}^{1}u_{2}^{1}-2x^{2}u^{1}\right)\left(j_{p}^{1}\sigma \right)&=u_{1}^{1}\left(j_{p}^{1}\sigma \right)u_{2}^{1}\left(j_{p}^{1}\sigma \right)-2x^{2}\left(j_{p}^{1}\sigma \right)u^{1}\left(j_{p}^{1}\sigma \right)\\&=\left(p^{2}\right)^{2}\cdot 2p^{1}p^{2}-2\cdot p^{2}\cdot p^{1}\left(p^{2}\right)^{2}\\&=2p^{1}\left(p^{2}\right)^{3}-2p^{1}\left(p^{2}\right)^{3}\\&=0\end{aligned}}}" loading="lazy"></dd></dl>

and so <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j_{p}^{1}\sigma \in S}">

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<annotation encoding="application/x-tex">{\displaystyle j_{p}^{1}\sigma \in S}</annotation>

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</math><img src="./0d2d0390b368873022e383cf3d76945c08fb3824.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.027ex; width:7.714ex; height:3.176ex;" alt="{\displaystyle j_{p}^{1}\sigma \in S}" loading="lazy"> for every p ∈ R2.

<div class="mw-heading mw-heading2"><h2 id="Jet_prolongation">Jet prolongation</h2></div>

A local diffeomorphism ψ : Jr(π) → Jr(π) defines a contact transformation of order r if it preserves the contact ideal, meaning that if θ is any contact form on Jr(π), then ψ*θ is also a contact form.

The flow generated by a vector field Vr on the jet space Jr(π) forms a one-parameter group of contact transformations if and only if the <a href="Lie_derivative" title="Lie derivative">Lie derivative</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}_{V^{r}}(\theta )}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}_{V^{r}}(\theta )}</annotation>

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</math><img src="./da41c585af74ebe065a19aae0a40390fcc2503d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.858ex; height:2.843ex;" alt="{\displaystyle {\mathcal {L}}_{V^{r}}(\theta )}" loading="lazy"> of any contact form θ preserves the contact ideal.

Let us begin with the first order case. Consider a general vector field V1 on J1(π), given by

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<annotation encoding="application/x-tex">{\displaystyle V^{1}\ {\stackrel {\mathrm {def} }{=}}\ \rho ^{i}\left(x^{i},u^{\alpha },u_{I}^{\alpha }\right){\frac {\partial }{\partial x^{i}}}+\phi ^{\alpha }\left(x^{i},u^{\alpha },u_{I}^{\alpha }\right){\frac {\partial }{\partial u^{\alpha }}}+\chi _{i}^{\alpha }\left(x^{i},u^{\alpha },u_{I}^{\alpha }\right){\frac {\partial }{\partial u_{i}^{\alpha }}}.}</annotation>

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</math><img src="./f2d90c08392bfa2c820e09d7e47f9c5c8b85dd5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:70.933ex; height:6.176ex;" alt="{\displaystyle V^{1}\ {\stackrel {\mathrm {def} }{=}}\ \rho ^{i}\left(x^{i},u^{\alpha },u_{I}^{\alpha }\right){\frac {\partial }{\partial x^{i}}}+\phi ^{\alpha }\left(x^{i},u^{\alpha },u_{I}^{\alpha }\right){\frac {\partial }{\partial u^{\alpha }}}+\chi _{i}^{\alpha }\left(x^{i},u^{\alpha },u_{I}^{\alpha }\right){\frac {\partial }{\partial u_{i}^{\alpha }}}.}" loading="lazy"></dd></dl>

We now apply <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}_{V^{1}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}_{V^{1}}}</annotation>

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</math><img src="./13db42475e3424b8c95897371a06540c2025a187.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.023ex; height:2.843ex;" alt="{\displaystyle {\mathcal {L}}_{V^{1}}}" loading="lazy"> to the basic contact forms <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{0}^{\alpha }=du^{\alpha }-u_{i}^{\alpha }dx^{i},}">

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<annotation encoding="application/x-tex">{\displaystyle \theta _{0}^{\alpha }=du^{\alpha }-u_{i}^{\alpha }dx^{i},}</annotation>

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</math><img src="./a950abdb54352bc12bc66e4856dce41f1461cc46.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.749ex; height:3.343ex;" alt="{\displaystyle \theta _{0}^{\alpha }=du^{\alpha }-u_{i}^{\alpha }dx^{i},}" loading="lazy"> and expand the <a href="Exterior_derivative" title="Exterior derivative">exterior derivative</a> of the functions in terms of their coordinates to obtain:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\mathcal {L}}_{V^{1}}\left(\theta _{0}^{\alpha }\right)&={\mathcal {L}}_{V^{1}}\left(du^{\alpha }-u_{i}^{\alpha }dx^{i}\right)\\&={\mathcal {L}}_{V^{1}}du^{\alpha }-\left({\mathcal {L}}_{V^{1}}u_{i}^{\alpha }\right)dx^{i}-u_{i}^{\alpha }\left({\mathcal {L}}_{V^{1}}dx^{i}\right)\\&=d\left(V^{1}u^{\alpha }\right)-V^{1}u_{i}^{\alpha }dx^{i}-u_{i}^{\alpha }d\left(V^{1}x^{i}\right)\\&=d\phi ^{\alpha }-\chi _{i}^{\alpha }dx^{i}-u_{i}^{\alpha }d\rho ^{i}\\&={\frac {\partial \phi ^{\alpha }}{\partial x^{i}}}dx^{i}+{\frac {\partial \phi ^{\alpha }}{\partial u^{k}}}du^{k}+{\frac {\partial \phi ^{\alpha }}{\partial u_{i}^{k}}}du_{i}^{k}-\chi _{i}^{\alpha }dx^{i}-u_{i}^{\alpha }\left[{\frac {\partial \rho ^{i}}{\partial x^{m}}}dx^{m}+{\frac {\partial \rho ^{i}}{\partial u^{k}}}du^{k}+{\frac {\partial \rho ^{i}}{\partial u_{m}^{k}}}du_{m}^{k}\right]\\&={\frac {\partial \phi ^{\alpha }}{\partial x^{i}}}dx^{i}+{\frac {\partial \phi ^{\alpha }}{\partial u^{k}}}\left(\theta ^{k}+u_{i}^{k}dx^{i}\right)+{\frac {\partial \phi ^{\alpha }}{\partial u_{i}^{k}}}du_{i}^{k}-\chi _{i}^{\alpha }dx^{i}-u_{l}^{\alpha }\left[{\frac {\partial \rho ^{l}}{\partial x^{i}}}dx^{i}+{\frac {\partial \rho ^{l}}{\partial u^{k}}}\left(\theta ^{k}+u_{i}^{k}dx^{i}\right)+{\frac {\partial \rho ^{l}}{\partial u_{i}^{k}}}du_{i}^{k}\right]\\&=\left[{\frac {\partial \phi ^{\alpha }}{\partial x^{i}}}+{\frac {\partial \phi ^{\alpha }}{\partial u^{k}}}u_{i}^{k}-u_{l}^{\alpha }\left({\frac {\partial \rho ^{l}}{\partial x^{i}}}+{\frac {\partial \rho ^{l}}{\partial u^{k}}}u_{i}^{k}\right)-\chi _{i}^{\alpha }\right]dx^{i}+\left[{\frac {\partial \phi ^{\alpha }}{\partial u_{i}^{k}}}-u_{l}^{\alpha }{\frac {\partial \rho ^{l}}{\partial u_{i}^{k}}}\right]du_{i}^{k}+\left({\frac {\partial \phi ^{\alpha }}{\partial u^{k}}}-u_{l}^{\alpha }{\frac {\partial \rho ^{l}}{\partial u^{k}}}\right)\theta ^{k}\end{aligned}}}</annotation>

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</math><img src="./0151abaeb6a895d4256232079c3335749fb77ea7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -17.671ex; width:111.693ex; height:36.509ex;" alt="{\displaystyle {\begin{aligned}{\mathcal {L}}_{V^{1}}\left(\theta _{0}^{\alpha }\right)&={\mathcal {L}}_{V^{1}}\left(du^{\alpha }-u_{i}^{\alpha }dx^{i}\right)\\&={\mathcal {L}}_{V^{1}}du^{\alpha }-\left({\mathcal {L}}_{V^{1}}u_{i}^{\alpha }\right)dx^{i}-u_{i}^{\alpha }\left({\mathcal {L}}_{V^{1}}dx^{i}\right)\\&=d\left(V^{1}u^{\alpha }\right)-V^{1}u_{i}^{\alpha }dx^{i}-u_{i}^{\alpha }d\left(V^{1}x^{i}\right)\\&=d\phi ^{\alpha }-\chi _{i}^{\alpha }dx^{i}-u_{i}^{\alpha }d\rho ^{i}\\&={\frac {\partial \phi ^{\alpha }}{\partial x^{i}}}dx^{i}+{\frac {\partial \phi ^{\alpha }}{\partial u^{k}}}du^{k}+{\frac {\partial \phi ^{\alpha }}{\partial u_{i}^{k}}}du_{i}^{k}-\chi _{i}^{\alpha }dx^{i}-u_{i}^{\alpha }\left[{\frac {\partial \rho ^{i}}{\partial x^{m}}}dx^{m}+{\frac {\partial \rho ^{i}}{\partial u^{k}}}du^{k}+{\frac {\partial \rho ^{i}}{\partial u_{m}^{k}}}du_{m}^{k}\right]\\&={\frac {\partial \phi ^{\alpha }}{\partial x^{i}}}dx^{i}+{\frac {\partial \phi ^{\alpha }}{\partial u^{k}}}\left(\theta ^{k}+u_{i}^{k}dx^{i}\right)+{\frac {\partial \phi ^{\alpha }}{\partial u_{i}^{k}}}du_{i}^{k}-\chi _{i}^{\alpha }dx^{i}-u_{l}^{\alpha }\left[{\frac {\partial \rho ^{l}}{\partial x^{i}}}dx^{i}+{\frac {\partial \rho ^{l}}{\partial u^{k}}}\left(\theta ^{k}+u_{i}^{k}dx^{i}\right)+{\frac {\partial \rho ^{l}}{\partial u_{i}^{k}}}du_{i}^{k}\right]\\&=\left[{\frac {\partial \phi ^{\alpha }}{\partial x^{i}}}+{\frac {\partial \phi ^{\alpha }}{\partial u^{k}}}u_{i}^{k}-u_{l}^{\alpha }\left({\frac {\partial \rho ^{l}}{\partial x^{i}}}+{\frac {\partial \rho ^{l}}{\partial u^{k}}}u_{i}^{k}\right)-\chi _{i}^{\alpha }\right]dx^{i}+\left[{\frac {\partial \phi ^{\alpha }}{\partial u_{i}^{k}}}-u_{l}^{\alpha }{\frac {\partial \rho ^{l}}{\partial u_{i}^{k}}}\right]du_{i}^{k}+\left({\frac {\partial \phi ^{\alpha }}{\partial u^{k}}}-u_{l}^{\alpha }{\frac {\partial \rho ^{l}}{\partial u^{k}}}\right)\theta ^{k}\end{aligned}}}" loading="lazy"></dd></dl>

Therefore, V1 determines a contact transformation if and only if the coefficients of dxi and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle du_{i}^{k}}">

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \phi ^{\alpha }}{\partial u_{i}^{k}}}-u_{l}^{\alpha }{\frac {\partial \rho ^{l}}{\partial u_{i}^{k}}}=0}</annotation>

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</math><img src="./78520e3adeafba37d51abee3abbf5344f2953d0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:19.112ex; height:7.009ex;" alt="{\displaystyle {\frac {\partial \phi ^{\alpha }}{\partial u_{i}^{k}}}-u_{l}^{\alpha }{\frac {\partial \rho ^{l}}{\partial u_{i}^{k}}}=0}" loading="lazy"></dd></dl>

The former requirements provide explicit formulae for the coefficients of the first derivative terms in V1:

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<annotation encoding="application/x-tex">{\displaystyle \chi _{i}^{\alpha }={\widehat {D}}_{i}\phi ^{\alpha }-u_{l}^{\alpha }\left({\widehat {D}}_{i}\rho ^{l}\right)}</annotation>

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</math><img src="./0602083a0406a3b8341eb8fd56c39a6bd0f0f47d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:25.554ex; height:4.843ex;" alt="{\displaystyle \chi _{i}^{\alpha }={\widehat {D}}_{i}\phi ^{\alpha }-u_{l}^{\alpha }\left({\widehat {D}}_{i}\rho ^{l}\right)}" loading="lazy"></dd></dl>

where

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<annotation encoding="application/x-tex">{\displaystyle {\widehat {D}}_{i}={\frac {\partial }{\partial x^{i}}}+u_{i}^{k}{\frac {\partial }{\partial u^{k}}}}</annotation>

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</math><img src="./f49c20d3006f48f83340dfc3e1de7d584ef5d0ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:20.466ex; height:5.676ex;" alt="{\displaystyle {\widehat {D}}_{i}={\frac {\partial }{\partial x^{i}}}+u_{i}^{k}{\frac {\partial }{\partial u^{k}}}}" loading="lazy"></dd></dl>

denotes the zeroth order truncation of the total derivative Di.

Thus, the contact conditions uniquely prescribe the prolongation of any point or contact vector field. That is, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}_{V^{r}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}_{V^{r}}}</annotation>

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</math><img src="./5b347dd1d6fd1f6887336d7f4bec26c2045a540e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.958ex; height:2.509ex;" alt="{\displaystyle {\mathcal {L}}_{V^{r}}}" loading="lazy"> satisfies these equations, Vr is called the r-th prolongation of V to a vector field on Jr(π).

These results are best understood when applied to a particular example. Hence, let us examine the following.

<div class="mw-heading mw-heading3"><h3 id="Example_4">Example</h3></div>

Consider the case (E, π, M), where E ≅ R2 and M ≃ R. Then, (J1(π), π, E) defines the first jet bundle, and may be coordinated by (x, u, u1), where

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x(j_{p}^{1}\sigma )&=x(p)=x\\u(j_{p}^{1}\sigma )&=u(\sigma (p))=u(\sigma (x))=\sigma (x)\\u_{1}(j_{p}^{1}\sigma )&=\left.{\frac {\partial \sigma }{\partial x}}\right|_{p}={\dot {\sigma }}(x)\end{aligned}}}</annotation>

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</math><img src="./18a9646596fd51f38acc6348c3d4bc1f71968ab8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:37.111ex; height:12.843ex;" alt="{\displaystyle {\begin{aligned}x(j_{p}^{1}\sigma )&=x(p)=x\\u(j_{p}^{1}\sigma )&=u(\sigma (p))=u(\sigma (x))=\sigma (x)\\u_{1}(j_{p}^{1}\sigma )&=\left.{\frac {\partial \sigma }{\partial x}}\right|_{p}={\dot {\sigma }}(x)\end{aligned}}}" loading="lazy"></dd></dl>

for all p ∈ M and σ in Γp(π). A contact form on J1(π) has the form

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<annotation encoding="application/x-tex">{\displaystyle \theta =du-u_{1}dx}</annotation>

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</math><img src="./f9e4469c52e1511cdd8cb6dc7f86a0af2ae6fb4d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.504ex; height:2.509ex;" alt="{\displaystyle \theta =du-u_{1}dx}" loading="lazy"></dd></dl>

Consider a vector V on E, having the form

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<annotation encoding="application/x-tex">{\displaystyle V=x{\frac {\partial }{\partial u}}-u{\frac {\partial }{\partial x}}}</annotation>

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</math><img src="./c4e868b22c2b89795a8f12ed86289d5f32d8bb8e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.353ex; height:5.509ex;" alt="{\displaystyle V=x{\frac {\partial }{\partial u}}-u{\frac {\partial }{\partial x}}}" loading="lazy"></dd></dl>

Then, the first prolongation of this vector field to J1(π) is

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}V^{1}&=V+Z\\&=x{\frac {\partial }{\partial u}}-u{\frac {\partial }{\partial x}}+Z\\&=x{\frac {\partial }{\partial u}}-u{\frac {\partial }{\partial x}}+\rho (x,u,u_{1}){\frac {\partial }{\partial u_{1}}}\end{aligned}}}</annotation>

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</math><img src="./a656ececaf1e564f6a015e4e407daa39f47ddd7d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.838ex; margin-top: -0.324ex; width:36.79ex; height:14.843ex;" alt="{\displaystyle {\begin{aligned}V^{1}&=V+Z\\&=x{\frac {\partial }{\partial u}}-u{\frac {\partial }{\partial x}}+Z\\&=x{\frac {\partial }{\partial u}}-u{\frac {\partial }{\partial x}}+\rho (x,u,u_{1}){\frac {\partial }{\partial u_{1}}}\end{aligned}}}" loading="lazy"></dd></dl>

If we now take the Lie derivative of the contact form with respect to this prolonged vector field, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}_{V^{1}}(\theta ),}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}_{V^{1}}(\theta ),}</annotation>

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</math><img src="./f148c0ff7d76d809b36fe5241e43e5d86785a513.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.57ex; height:3.009ex;" alt="{\displaystyle {\mathcal {L}}_{V^{1}}(\theta ),}" loading="lazy"> we obtain

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\mathcal {L}}_{V^{1}}(\theta )&={\mathcal {L}}_{V^{1}}(du-u_{1}dx)\\&={\mathcal {L}}_{V^{1}}du-\left({\mathcal {L}}_{V^{1}}u_{1}\right)dx-u_{1}\left({\mathcal {L}}_{V^{1}}dx\right)\\&=d\left(V^{1}u\right)-V^{1}u_{1}dx-u_{1}d\left(V^{1}x\right)\\&=dx-\rho (x,u,u_{1})dx+u_{1}du\\&=(1-\rho (x,u,u_{1}))dx+u_{1}du\\&=[1-\rho (x,u,u_{1})]dx+u_{1}(\theta +u_{1}dx)&&du=\theta +u_{1}dx\\&=[1+u_{1}u_{1}-\rho (x,u,u_{1})]dx+u_{1}\theta \end{aligned}}}</annotation>

</semantics>

</math><img src="./73ccafd43ff5860a459d84a5765ab7982ff42a0f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.838ex; width:65.111ex; height:22.843ex;" alt="{\displaystyle {\begin{aligned}{\mathcal {L}}_{V^{1}}(\theta )&={\mathcal {L}}_{V^{1}}(du-u_{1}dx)\\&={\mathcal {L}}_{V^{1}}du-\left({\mathcal {L}}_{V^{1}}u_{1}\right)dx-u_{1}\left({\mathcal {L}}_{V^{1}}dx\right)\\&=d\left(V^{1}u\right)-V^{1}u_{1}dx-u_{1}d\left(V^{1}x\right)\\&=dx-\rho (x,u,u_{1})dx+u_{1}du\\&=(1-\rho (x,u,u_{1}))dx+u_{1}du\\&=[1-\rho (x,u,u_{1})]dx+u_{1}(\theta +u_{1}dx)&&du=\theta +u_{1}dx\\&=[1+u_{1}u_{1}-\rho (x,u,u_{1})]dx+u_{1}\theta \end{aligned}}}" loading="lazy"></dd></dl>

Hence, for preservation of the contact ideal, we require

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<annotation encoding="application/x-tex">{\displaystyle 1+u_{1}u_{1}-\rho (x,u,u_{1})=0\quad \Leftrightarrow \quad \rho (x,u,u_{1})=1+u_{1}u_{1}.}</annotation>

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</math><img src="./d99d3431c449549722d6cfa26a8ee1794c3642f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:56.892ex; height:2.843ex;" alt="{\displaystyle 1+u_{1}u_{1}-\rho (x,u,u_{1})=0\quad \Leftrightarrow \quad \rho (x,u,u_{1})=1+u_{1}u_{1}.}" loading="lazy"></dd></dl>

And so the first prolongation of V to a vector field on J1(π) is

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<annotation encoding="application/x-tex">{\displaystyle V^{1}=x{\frac {\partial }{\partial u}}-u{\frac {\partial }{\partial x}}+(1+u_{1}u_{1}){\frac {\partial }{\partial u_{1}}}.}</annotation>

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</math><img src="./4477e139b382a565d17f2978ae67fb00213d65e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:37.143ex; height:5.843ex;" alt="{\displaystyle V^{1}=x{\frac {\partial }{\partial u}}-u{\frac {\partial }{\partial x}}+(1+u_{1}u_{1}){\frac {\partial }{\partial u_{1}}}.}" loading="lazy"></dd></dl>

Let us also calculate the second prolongation of V to a vector field on J2(π). We have <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{x,u,u_{1},u_{2}\}}">

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<annotation encoding="application/x-tex">{\displaystyle \{x,u,u_{1},u_{2}\}}</annotation>

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</math><img src="./59d525e60aa01b73c79223dec9f73e12060509a3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.854ex; height:2.843ex;" alt="{\displaystyle \{x,u,u_{1},u_{2}\}}" loading="lazy"> as coordinates on J2(π). Hence, the prolonged vector has the form

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<annotation encoding="application/x-tex">{\displaystyle V^{2}=x{\frac {\partial }{\partial u}}-u{\frac {\partial }{\partial x}}+\rho (x,u,u_{1},u_{2}){\frac {\partial }{\partial u_{1}}}+\phi (x,u,u_{1},u_{2}){\frac {\partial }{\partial u_{2}}}.}</annotation>

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</math><img src="./6bd56c22c00b9e70077ad51c14de87a54aaa57d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:61.205ex; height:5.843ex;" alt="{\displaystyle V^{2}=x{\frac {\partial }{\partial u}}-u{\frac {\partial }{\partial x}}+\rho (x,u,u_{1},u_{2}){\frac {\partial }{\partial u_{1}}}+\phi (x,u,u_{1},u_{2}){\frac {\partial }{\partial u_{2}}}.}" loading="lazy"></dd></dl>

The contact forms are

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\theta &=du-u_{1}dx\\\theta _{1}&=du_{1}-u_{2}dx\end{aligned}}}</annotation>

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</math><img src="./f126609b42c558c8deb7ad53d00ffd26f52a9aa1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:17.364ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}\theta &=du-u_{1}dx\\\theta _{1}&=du_{1}-u_{2}dx\end{aligned}}}" loading="lazy"></dd></dl>

To preserve the contact ideal, we require

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\mathcal {L}}_{V^{2}}(\theta )&=0\\{\mathcal {L}}_{V^{2}}(\theta _{1})&=0\end{aligned}}}</annotation>

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</math><img src="./039edb408e71bb2fd2b762d0df85fdf8f54421eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:12.989ex; height:6.509ex;" alt="{\displaystyle {\begin{aligned}{\mathcal {L}}_{V^{2}}(\theta )&=0\\{\mathcal {L}}_{V^{2}}(\theta _{1})&=0\end{aligned}}}" loading="lazy"></dd></dl>

Now, θ has no u2 dependency. Hence, from this equation we will pick up the formula for ρ, which will necessarily be the same result as we found for V1. Therefore, the problem is analogous to prolonging the vector field V1 to J2(π). That is to say, we may generate the r-th prolongation of a vector field by recursively applying the Lie derivative of the contact forms with respect to the prolonged vector fields, r times. So, we have

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<annotation encoding="application/x-tex">{\displaystyle \rho (x,u,u_{1})=1+u_{1}u_{1}}</annotation>

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</math><img src="./efafed278196964ed09a8cf5ce424e7ed4415582.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.992ex; height:2.843ex;" alt="{\displaystyle \rho (x,u,u_{1})=1+u_{1}u_{1}}" loading="lazy"></dd></dl>

and so

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}V^{2}&=V^{1}+\phi (x,u,u_{1},u_{2}){\frac {\partial }{\partial u_{2}}}\\&=x{\frac {\partial }{\partial u}}-u{\frac {\partial }{\partial x}}+(1+u_{1}u_{1}){\frac {\partial }{\partial u_{1}}}+\phi (x,u,u_{1},u_{2}){\frac {\partial }{\partial u_{2}}}\end{aligned}}}</annotation>

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</math><img src="./08cc7398d0d6dee00b4e44e2784b10b645ea9ffd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.338ex; width:58.349ex; height:11.843ex;" alt="{\displaystyle {\begin{aligned}V^{2}&=V^{1}+\phi (x,u,u_{1},u_{2}){\frac {\partial }{\partial u_{2}}}\\&=x{\frac {\partial }{\partial u}}-u{\frac {\partial }{\partial x}}+(1+u_{1}u_{1}){\frac {\partial }{\partial u_{1}}}+\phi (x,u,u_{1},u_{2}){\frac {\partial }{\partial u_{2}}}\end{aligned}}}" loading="lazy"></dd></dl>

Therefore, the Lie derivative of the second contact form with respect to V2 is

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\mathcal {L}}_{V^{2}}(\theta _{1})&={\mathcal {L}}_{V^{2}}(du_{1}-u_{2}dx)\\&={\mathcal {L}}_{V^{2}}du_{1}-\left({\mathcal {L}}_{V^{2}}u_{2}\right)dx-u_{2}\left({\mathcal {L}}_{V^{2}}dx\right)\\&=d(V^{2}u_{1})-V^{2}u_{2}dx-u_{2}d(V^{2}x)\\&=d(1+u_{1}u_{1})-\phi (x,u,u_{1},u_{2})dx+u_{2}du\\&=2u_{1}du_{1}-\phi (x,u,u_{1},u_{2})dx+u_{2}du\\&=2u_{1}du_{1}-\phi (x,u,u_{1},u_{2})dx+u_{2}(\theta +u_{1}dx)&du&=\theta +u_{1}dx\\&=2u_{1}(\theta _{1}+u_{2}dx)-\phi (x,u,u_{1},u_{2})dx+u_{2}(\theta +u_{1}dx)&du_{1}&=\theta _{1}+u_{2}dx\\&=[3u_{1}u_{2}-\phi (x,u,u_{1},u_{2})]dx+u_{2}\theta +2u_{1}\theta _{1}\end{aligned}}}</annotation>

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</math><img src="./78252b9150284f554d25bad58168a88c161a3b05.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -12.171ex; width:83.359ex; height:25.509ex;" alt="{\displaystyle {\begin{aligned}{\mathcal {L}}_{V^{2}}(\theta _{1})&={\mathcal {L}}_{V^{2}}(du_{1}-u_{2}dx)\\&={\mathcal {L}}_{V^{2}}du_{1}-\left({\mathcal {L}}_{V^{2}}u_{2}\right)dx-u_{2}\left({\mathcal {L}}_{V^{2}}dx\right)\\&=d(V^{2}u_{1})-V^{2}u_{2}dx-u_{2}d(V^{2}x)\\&=d(1+u_{1}u_{1})-\phi (x,u,u_{1},u_{2})dx+u_{2}du\\&=2u_{1}du_{1}-\phi (x,u,u_{1},u_{2})dx+u_{2}du\\&=2u_{1}du_{1}-\phi (x,u,u_{1},u_{2})dx+u_{2}(\theta +u_{1}dx)&du&=\theta +u_{1}dx\\&=2u_{1}(\theta _{1}+u_{2}dx)-\phi (x,u,u_{1},u_{2})dx+u_{2}(\theta +u_{1}dx)&du_{1}&=\theta _{1}+u_{2}dx\\&=[3u_{1}u_{2}-\phi (x,u,u_{1},u_{2})]dx+u_{2}\theta +2u_{1}\theta _{1}\end{aligned}}}" loading="lazy"></dd></dl>

Hence, for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}_{V^{2}}(\theta _{1})}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}_{V^{2}}(\theta _{1})}</annotation>

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</math><img src="./b3de9b2d4228e3d891d5dc9a878f473aaa37bae4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.977ex; height:3.009ex;" alt="{\displaystyle {\mathcal {L}}_{V^{2}}(\theta _{1})}" loading="lazy"> to preserve the contact ideal, we require

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<annotation encoding="application/x-tex">{\displaystyle 3u_{1}u_{2}-\phi (x,u,u_{1},u_{2})=0\quad \Leftrightarrow \quad \phi (x,u,u_{1},u_{2})=3u_{1}u_{2}.}</annotation>

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</math><img src="./829e3e18c424071a4cc55f31a9b516c3175c18a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:58.414ex; height:2.843ex;" alt="{\displaystyle 3u_{1}u_{2}-\phi (x,u,u_{1},u_{2})=0\quad \Leftrightarrow \quad \phi (x,u,u_{1},u_{2})=3u_{1}u_{2}.}" loading="lazy"></dd></dl>

And so the second prolongation of V to a vector field on J2(π) is

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<annotation encoding="application/x-tex">{\displaystyle V^{2}=x{\frac {\partial }{\partial u}}-u{\frac {\partial }{\partial x}}+(1+u_{1}u_{1}){\frac {\partial }{\partial u_{1}}}+3u_{1}u_{2}{\frac {\partial }{\partial u_{2}}}.}</annotation>

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</math><img src="./c5575a7f3dfb5dbe8498c4ed1371d24405f44597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:50.451ex; height:5.843ex;" alt="{\displaystyle V^{2}=x{\frac {\partial }{\partial u}}-u{\frac {\partial }{\partial x}}+(1+u_{1}u_{1}){\frac {\partial }{\partial u_{1}}}+3u_{1}u_{2}{\frac {\partial }{\partial u_{2}}}.}" loading="lazy"></dd></dl>

Note that the first prolongation of V can be recovered by omitting the second derivative terms in V2, or by projecting back to J1(π).

<div class="mw-heading mw-heading2"><h2 id="Infinite_jet_spaces">Infinite jet spaces</h2></div>

The <a href="Inverse_limit" title="Inverse limit">inverse limit</a> of the sequence of projections <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{k+1,k}:J^{k+1}(\pi )\to J^{k}(\pi )}">

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<annotation encoding="application/x-tex">{\displaystyle \pi _{k+1,k}:J^{k+1}(\pi )\to J^{k}(\pi )}</annotation>

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</math><img src="./d0081fbf187f0a7525318ed15d50964a745fa9f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.992ex; height:3.343ex;" alt="{\displaystyle \pi _{k+1,k}:J^{k+1}(\pi )\to J^{k}(\pi )}" loading="lazy"> gives rise to the infinite jet space J∞(π). A point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j_{p}^{\infty }(\sigma )}">

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<annotation encoding="application/x-tex">{\displaystyle j_{p}^{\infty }(\sigma )}</annotation>

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</math><img src="./78726219f3979255a09edffcafa51d8191eac247.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.027ex; width:5.999ex; height:3.009ex;" alt="{\displaystyle j_{p}^{\infty }(\sigma )}" loading="lazy"> is the equivalence class of sections of π that have the same k-jet in p as σ for all values of k. The natural projection π∞ maps <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j_{p}^{\infty }(\sigma )}">

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<annotation encoding="application/x-tex">{\displaystyle j_{p}^{\infty }(\sigma )}</annotation>

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Just by thinking in terms of coordinates, J∞(π) appears to be an infinite-dimensional geometric object. In fact, the simplest way of introducing a differentiable structure on J∞(π), not relying on differentiable charts, is given by the <a href="Differential_calculus_over_commutative_algebras" title="Differential calculus over commutative algebras">differential calculus over commutative algebras</a>. Dual to the sequence of projections <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{k+1,k}:J^{k+1}(\pi )\to J^{k}(\pi )}">

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<annotation encoding="application/x-tex">{\displaystyle \pi _{k+1,k}:J^{k+1}(\pi )\to J^{k}(\pi )}</annotation>

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</math><img src="./d0081fbf187f0a7525318ed15d50964a745fa9f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.992ex; height:3.343ex;" alt="{\displaystyle \pi _{k+1,k}:J^{k+1}(\pi )\to J^{k}(\pi )}" loading="lazy"> of manifolds is the sequence of injections <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{k+1,k}^{*}:C^{\infty }(J^{k}(\pi ))\to C^{\infty }\left(J^{k+1}(\pi )\right)}">

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<annotation encoding="application/x-tex">{\displaystyle \pi _{k+1,k}^{*}:C^{\infty }(J^{k}(\pi ))\to C^{\infty }\left(J^{k+1}(\pi )\right)}</annotation>

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</math><img src="./291cebf1d9ddb3ceda005120933190f760eb48eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:36.664ex; height:3.676ex;" alt="{\displaystyle \pi _{k+1,k}^{*}:C^{\infty }(J^{k}(\pi ))\to C^{\infty }\left(J^{k+1}(\pi )\right)}" loading="lazy"> of commutative algebras. Let's denote <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{\infty }(J^{k}(\pi ))}">

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<annotation encoding="application/x-tex">{\displaystyle C^{\infty }(J^{k}(\pi ))}</annotation>

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</math><img src="./5ce780f91c152883b4fb4d075eb3563a2ae16ba2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.238ex; height:3.176ex;" alt="{\displaystyle C^{\infty }(J^{k}(\pi ))}" loading="lazy"> simply by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}_{k}(\pi )}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}_{k}(\pi )}</annotation>

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</math><img src="./9a90d0c6b3544c68bc6c90f4381011bde9bb11bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.901ex; height:2.843ex;" alt="{\displaystyle {\mathcal {F}}_{k}(\pi )}" loading="lazy">. Take now the <a href="Direct_limit" title="Direct limit">direct limit</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}(\pi )}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}(\pi )}</annotation>

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</math><img src="./de2066d556294c1840f11837e7c2eada0ea930ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.068ex; height:2.843ex;" alt="{\displaystyle {\mathcal {F}}(\pi )}" loading="lazy"> of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}_{k}(\pi )}">

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</math><img src="./9a90d0c6b3544c68bc6c90f4381011bde9bb11bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.901ex; height:2.843ex;" alt="{\displaystyle {\mathcal {F}}_{k}(\pi )}" loading="lazy">'s. It will be a commutative algebra, which can be assumed to be the smooth functions algebra over the geometric object J∞(π). Observe that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}(\pi )}">

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</math><img src="./de2066d556294c1840f11837e7c2eada0ea930ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.068ex; height:2.843ex;" alt="{\displaystyle {\mathcal {F}}(\pi )}" loading="lazy">, being born as a direct limit, carries an additional structure: it is a filtered commutative algebra.

Roughly speaking, a concrete element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi \in {\mathcal {F}}(\pi )}">

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<div class="mw-heading mw-heading3"><h3 id="Infinitely_prolonged_PDEs">Infinitely prolonged PDEs</h3></div>

Given a k-th order system of PDEs E ⊆ Jk(π), the collection I(E) of vanishing on E smooth functions on J∞(π) is an <a href="Ideal_(ring_theory)" title="Ideal (ring theory)">ideal</a> in the algebra <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}_{k}(\pi )}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}_{k}(\pi )}</annotation>

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</math><img src="./9a90d0c6b3544c68bc6c90f4381011bde9bb11bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.901ex; height:2.843ex;" alt="{\displaystyle {\mathcal {F}}_{k}(\pi )}" loading="lazy">, and hence in the direct limit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}(\pi )}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}(\pi )}</annotation>

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Enhance I(E) by adding all the possible compositions of <a href="Total_derivative" title="Total derivative">total derivatives</a> applied to all its elements. This way we get a new ideal I of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}(\pi )}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}(\pi )}</annotation>

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</math><img src="./de2066d556294c1840f11837e7c2eada0ea930ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.068ex; height:2.843ex;" alt="{\displaystyle {\mathcal {F}}(\pi )}" loading="lazy"> which is now closed under the operation of taking total derivative. The submanifold E(∞) of J∞(π) cut out by I is called the infinite prolongation of E.

Geometrically, E(∞) is the manifold of formal solutions of E. A point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j_{p}^{\infty }(\sigma )}">

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</math><img src="./78726219f3979255a09edffcafa51d8191eac247.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.027ex; width:5.999ex; height:3.009ex;" alt="{\displaystyle j_{p}^{\infty }(\sigma )}" loading="lazy"> of E(∞) can be easily seen to be represented by a section σ whose k-jet's graph is tangent to E at the point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j_{p}^{k}(\sigma )}">

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</math><img src="./1eac42fc1c71d3c16154e69867db0ac87e276268.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.027ex; width:5.212ex; height:3.176ex;" alt="{\displaystyle j_{p}^{k}(\sigma )}" loading="lazy"> with arbitrarily high order of tangency.

Analytically, if E is given by φ = 0, a formal solution can be understood as the set of Taylor coefficients of a section σ in a point p that make vanish the <a href="Taylor_series" title="Taylor series">Taylor series</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi \circ j^{k}(\sigma )}">

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</math><img src="./f68cb99c486291512062ef1a4b635bdbc0195641.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.901ex; height:3.176ex;" alt="{\displaystyle \varphi \circ j^{k}(\sigma )}" loading="lazy"> at the point p.

Most importantly, the closure properties of I imply that E(∞) is tangent to the infinite-order contact structure <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}}</annotation>

</semantics>

</math><img src="./e7b3edab7022ca9e2976651bc59c489513ee9019.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.239ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}}" loading="lazy"> on J∞(π), so that by restricting <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}}</annotation>

</semantics>

</math><img src="./e7b3edab7022ca9e2976651bc59c489513ee9019.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.239ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}}" loading="lazy"> to E(∞) one gets the <a href="Diffiety" title="Diffiety">diffiety</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (E_{(\infty )},{\mathcal {C}}|_{E_{(\infty )}})}">

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (E_{(\infty )},{\mathcal {C}}|_{E_{(\infty )}})}</annotation>

</semantics>

</math><img src="./ec46ad5e0ab94c8c7ea5a61048950cede609dbab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:13.581ex; height:3.676ex;" alt="{\displaystyle (E_{(\infty )},{\mathcal {C}}|_{E_{(\infty )}})}" loading="lazy">, and can study the associated <a href="Diffiety#Vinogradov_sequence" title="Diffiety">Vinogradov (C-spectral) sequence</a>.

<div class="mw-heading mw-heading2"><h2 id="Remark">Remark</h2></div>

This article has defined jets of local sections of a bundle, but it is possible to define jets of functions f: M → N, where M and N are manifolds; the jet of f then just corresponds to the jet of the section

(grf is known as the graph of the function f) of the trivial bundle (M × N, π1, M). However, this restriction does not simplify the theory, as the global triviality of π does not imply the global triviality of π1.

<ul><li><a href="Jet_group" title="Jet group">Jet group</a></li>

<li><a href="Jet_(mathematics)" title="Jet (mathematics)">Jet (mathematics)</a></li>

<li><a href="Lagrangian_system" title="Lagrangian system">Lagrangian system</a></li>

<li><a href="Variational_bicomplex" title="Variational bicomplex">Variational bicomplex</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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/* end https://en.wikipedia.org/ */

</style><cite id="CITEREFKrupka2015" class="citation book cs1">Krupka, Demeter (2015). <a rel="nofollow" class="external text" href="https://www.springer.com/it/book/9789462390720">Introduction to Global Variational Geometry</a>. Atlantis Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-94-6239-073-7</bdi>.</cite>

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite id="CITEREFVakil1998" class="citation web cs1">Vakil, Ravi (August 25, 1998). <a rel="nofollow" class="external text" href="http://math.stanford.edu/~vakil/files/jets.pdf">"A beginner's guide to jet bundles from the point of view of algebraic geometry"</a> (PDF). Retrieved June 25, 2017.</cite>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li>Ehresmann, C., "Introduction à la théorie des structures infinitésimales et des pseudo-groupes de Lie." Geometrie Differentielle, Colloq. Inter. du Centre Nat. de la Recherche Scientifique, Strasbourg, 1953, 97-127.</li>

<li>Kolář, I., Michor, P., Slovák, J., <a rel="nofollow" class="external text" href="http://www.emis.de/monographs/KSM/">Natural operations in differential geometry.</a> Springer-Verlag: Berlin Heidelberg, 1993. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>3-540-56235-4</bdi>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-387-56235-4</bdi>.</li>

<li>Saunders, D. J., "The Geometry of Jet Bundles", Cambridge University Press, 1989, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-521-36948-7</bdi></li>

<li>Krasil'shchik, I. S., Vinogradov, A. M., [et al.], "Symmetries and conservation laws for differential equations of mathematical physics", Amer. Math. Soc., Providence, RI, 1999, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-8218-0958-X</bdi>.</li>

<li><a href="Peter_J._Olver" title="Peter J. Olver">Olver, P. J.</a>, "Equivalence, Invariants and Symmetry", Cambridge University Press, 1995, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-521-47811-1</bdi></li></ul>

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</style><div id="Manifolds_(Glossary,_List,_Category)274" style="font-size:114%;margin:0 4em"><a href="Manifold" title="Manifold">Manifolds</a> (<a href="Glossary_of_differential_geometry_and_topology" title="Glossary of differential geometry and topology">Glossary</a>, <a href="List_of_manifolds" title="List of manifolds">List</a>, Category)</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Basic concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Topological_manifold" title="Topological manifold">Topological manifold</a>

<ul><li><a href="Atlas_(topology)" title="Atlas (topology)">Atlas</a></li></ul></li>

<li><a href="Differentiable_manifold" title="Differentiable manifold">Differentiable/Smooth manifold</a>

<ul><li><a href="Differential_structure" title="Differential structure">Differential structure</a></li>

<li><a href="Smooth_structure" title="Smooth structure">Smooth atlas</a></li></ul></li>

<li><a href="Submanifold" title="Submanifold">Submanifold</a></li>

<li><a href="Riemannian_manifold" title="Riemannian manifold">Riemannian manifold</a></li>

<li><a href="Smoothness" title="Smoothness">Smooth map</a></li>

<li><a href="Submersion_(mathematics)" title="Submersion (mathematics)">Submersion</a></li>

<li><a href="Pushforward_(differential)" title="Pushforward (differential)">Pushforward</a></li>

<li><a href="Tangent_space" title="Tangent space">Tangent space</a></li>

<li><a href="Differential_form" title="Differential form">Differential form</a></li>

<li><a href="Vector_field" title="Vector field">Vector field</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Main results (list)</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Atiyah%E2%80%93Singer_index_theorem" title="Atiyah–Singer index theorem">Atiyah–Singer index</a></li>

<li><a href="Darboux's_theorem" title="Darboux's theorem">Darboux's</a></li>

<li><a href="Frobenius_theorem_(differential_topology)" title="Frobenius theorem (differential topology)">Frobenius</a></li>

<li><a href="Generalized_Stokes_theorem" title="Generalized Stokes theorem">Generalized Stokes</a></li>

<li><a href="Hopf%E2%80%93Rinow_theorem" title="Hopf–Rinow theorem">Hopf–Rinow</a></li>

<li><a href="Noether's_theorem" title="Noether's theorem">Noether's</a></li>

<li><a href="Whitney_embedding_theorem" title="Whitney embedding theorem">Whitney embedding</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Smoothness" title="Smoothness">Maps</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Differentiable_curve" title="Differentiable curve">Curve</a></li>

<li><a href="Diffeomorphism" title="Diffeomorphism">Diffeomorphism</a>

<ul><li><a href="Local_diffeomorphism" title="Local diffeomorphism">Local</a></li></ul></li>

<li><a href="Geodesic" title="Geodesic">Geodesic</a></li>

<li><a href="Exponential_map_(Riemannian_geometry)" title="Exponential map (Riemannian geometry)">Exponential map</a>

<ul><li><a href="Exponential_map_(Lie_theory)" title="Exponential map (Lie theory)">in Lie theory</a></li></ul></li>

<li><a href="Foliation" title="Foliation">Foliation</a></li>

<li><a href="Immersion_(mathematics)" title="Immersion (mathematics)">Immersion</a></li>

<li><a href="Integral_curve" title="Integral curve">Integral curve</a></li>

<li><a href="Lie_derivative" title="Lie derivative">Lie derivative</a></li>

<li><a href="Section_(fiber_bundle)" title="Section (fiber bundle)">Section</a></li>

<li><a href="Submersion_(mathematics)" title="Submersion (mathematics)">Submersion</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of manifolds</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Closed_manifold" title="Closed manifold">Closed</a></li>

<li><a href="Collapsing_manifold" title="Collapsing manifold">Collapsing</a></li>

<li><a href="Complete_manifold" title="Complete manifold">Complete</a></li>

<li>(<a href="Almost_complex_manifold" title="Almost complex manifold">Almost</a>) <a href="Complex_manifold" title="Complex manifold">Complex</a></li>

<li>(<a href="Almost-contact_manifold" title="Almost-contact manifold">Almost</a>) <a href="Contact_manifold" class="mw-redirect" title="Contact manifold">Contact</a></li>

<li><a href="Fibered_manifold" title="Fibered manifold">Fibered</a></li>

<li><a href="Finsler_manifold" title="Finsler manifold">Finsler</a></li>

<li>(<a href="Almost_flat_manifold" title="Almost flat manifold">Almost</a>) <a href="Flat_manifold" title="Flat manifold">Flat</a></li>

<li><a href="G-structure_on_a_manifold" title="G-structure on a manifold">G-structure</a></li>

<li><a href="Hadamard_manifold" title="Hadamard manifold">Hadamard</a></li>

<li><a href="Hermitian_manifold" title="Hermitian manifold">Hermitian</a></li>

<li><a href="Hyperbolic_manifold" title="Hyperbolic manifold">Hyperbolic</a></li>

<li><a href="K%C3%A4hler_manifold" title="Kähler manifold">Kähler</a></li>

<li><a href="Kenmotsu_manifold" title="Kenmotsu manifold">Kenmotsu</a></li>

<li><a href="Lie_group" title="Lie group">Lie group</a>

<ul><li><a href="Lie_group%E2%80%93Lie_algebra_correspondence" title="Lie group–Lie algebra correspondence">Lie algebra</a></li></ul></li>

<li><a href="Manifold_with_boundary" class="mw-redirect" title="Manifold with boundary">Manifold with boundary</a></li>

<li><a href="Nilmanifold" title="Nilmanifold">Nilmanifold</a></li>

<li><a href="Orientability" title="Orientability">Oriented</a></li>

<li><a href="Parallelizable_manifold" title="Parallelizable manifold">Parallelizable</a></li>

<li><a href="Poisson_manifold" title="Poisson manifold">Poisson</a></li>

<li><a href="Prime_manifold" title="Prime manifold">Prime</a></li>

<li><a href="Quaternionic_manifold" title="Quaternionic manifold">Quaternionic</a></li>

<li><a href="Hypercomplex_manifold" title="Hypercomplex manifold">Hypercomplex</a></li>

<li>(<a href="Pseudo-Riemannian_manifold" title="Pseudo-Riemannian manifold">Pseudo-</a>, <a href="Sub-Riemannian_manifold" title="Sub-Riemannian manifold">Sub-</a>) <a href="Riemannian_manifold" title="Riemannian manifold">Riemannian</a></li>

<li><a href="Rizza_manifold" title="Rizza manifold">Rizza</a></li>

<li><a href="Stein_manifold" title="Stein manifold">Stein</a></li>

<li>(<a href="Almost_symplectic_manifold" title="Almost symplectic manifold">Almost</a>) <a href="Symplectic_manifold" title="Symplectic manifold">Symplectic</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Tensor" title="Tensor">Tensors</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Vectors</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Distribution_(differential_geometry)" title="Distribution (differential geometry)">Distribution</a></li>

<li><a href="Lie_bracket_of_vector_fields" title="Lie bracket of vector fields">Lie bracket</a></li>

<li><a href="Pushforward_(differential)" title="Pushforward (differential)">Pushforward</a></li>

<li><a href="Tangent_space" title="Tangent space">Tangent space</a>

<ul><li><a href="Tangent_bundle" title="Tangent bundle">bundle</a></li></ul></li>

<li><a href="Torsion_tensor" title="Torsion tensor">Torsion</a></li>

<li><a href="Vector_field" title="Vector field">Vector field</a></li>

<li><a href="Vector_flow" title="Vector flow">Vector flow</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Covectors</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Closed_and_exact_differential_forms" title="Closed and exact differential forms">Closed/Exact</a></li>

<li><a href="Covariant_derivative" title="Covariant derivative">Covariant derivative</a></li>

<li><a href="Cotangent_space" title="Cotangent space">Cotangent space</a>

<ul><li><a href="Cotangent_bundle" title="Cotangent bundle">bundle</a></li></ul></li>

<li><a href="De_Rham_cohomology" title="De Rham cohomology">De Rham cohomology</a></li>

<li><a href="Differential_form" title="Differential form">Differential form</a>

<ul><li><a href="Vector-valued_differential_form" title="Vector-valued differential form">Vector-valued</a></li></ul></li>

<li><a href="Exterior_derivative" title="Exterior derivative">Exterior derivative</a></li>

<li><a href="Interior_product" title="Interior product">Interior product</a></li>

<li><a href="Pullback_(differential_geometry)" title="Pullback (differential geometry)">Pullback</a></li>

<li><a href="Ricci_curvature" title="Ricci curvature">Ricci curvature</a>

<li><a href="Riemann_curvature_tensor" title="Riemann curvature tensor">Riemann curvature tensor</a></li>

<li><a href="Tensor_field" title="Tensor field">Tensor field</a>

<ul><li><a href="Tensor_density" title="Tensor density">density</a></li></ul></li>

<li><a href="Volume_form" title="Volume form">Volume form</a></li>

<li><a href="Wedge_product" class="mw-redirect" title="Wedge product">Wedge product</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Fiber_bundle" title="Fiber bundle">Bundles</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Adjoint_bundle" title="Adjoint bundle">Adjoint</a></li>

<li><a href="Affine_bundle" title="Affine bundle">Affine</a></li>

<li><a href="Associated_bundle" title="Associated bundle">Associated</a></li>

<li><a href="Cotangent_bundle" title="Cotangent bundle">Cotangent</a></li>

<li><a href="Fiber_bundle" title="Fiber bundle">Fiber</a></li>

<li>(<a href="Cofibration" title="Cofibration">Co-</a>) <a href="Fibration" title="Fibration">Fibration</a></li>

<li><a href="Lie_algebra_bundle" title="Lie algebra bundle">Lie algebra</a></li>

<li>(<a href="Stable_normal_bundle" title="Stable normal bundle">Stable</a>) <a href="Normal_bundle" title="Normal bundle">Normal</a></li>

<li><a href="Principal_bundle" title="Principal bundle">Principal</a></li>

<li><a href="Spinor_bundle" title="Spinor bundle">Spinor</a></li>

<li><a href="Subbundle" title="Subbundle">Subbundle</a></li>

<li><a href="Tangent_bundle" title="Tangent bundle">Tangent</a></li>

<li><a href="Tensor_bundle" title="Tensor bundle">Tensor</a></li>

<li><a href="Vector_bundle" title="Vector bundle">Vector</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Connection_(mathematics)" title="Connection (mathematics)">Connections</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Affine_connection" title="Affine connection">Affine</a></li>

<li><a href="Cartan_connection" title="Cartan connection">Cartan</a></li>

<li><a href="Ehresmann_connection" title="Ehresmann connection">Ehresmann</a></li>

<li><a href="Connection_(fibred_manifold)" title="Connection (fibred manifold)">Generalized</a></li>

<li><a href="Koszul_connection" class="mw-redirect" title="Koszul connection">Koszul</a></li>

<li><a href="Levi-Civita_connection" title="Levi-Civita connection">Levi-Civita</a></li>

<li><a href="Connection_(principal_bundle)" title="Connection (principal bundle)">Principal</a></li>

<li><a href="Connection_(vector_bundle)" title="Connection (vector bundle)">Vector</a></li>

<li><a href="Parallel_transport" title="Parallel transport">Parallel transport</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Classification_of_manifolds" title="Classification of manifolds">Classification of manifolds</a></li>

<li><a href="Gauge_theory_(mathematics)" title="Gauge theory (mathematics)">Gauge theory</a></li>

<li><a href="History_of_manifolds_and_varieties" title="History of manifolds and varieties">History</a></li>

<li><a href="Morse_theory" title="Morse theory">Morse theory</a></li>

<li><a href="Moving_frame" title="Moving frame">Moving frame</a></li>

<li><a href="Singularity_theory" title="Singularity theory">Singularity theory</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Generalizations</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Banach_manifold" title="Banach manifold">Banach manifold</a></li>

<li><a href="Diffeology" title="Diffeology">Diffeology</a></li>

<li><a href="Diffiety" title="Diffiety">Diffiety</a></li>

<li><a href="Fr%C3%A9chet_manifold" title="Fréchet manifold">Fréchet manifold</a></li>

<li><a href="K-theory" title="K-theory">K-theory</a></li>

<li><a href="Orbifold" title="Orbifold">Orbifold</a></li>

<li><a href="Secondary_calculus_and_cohomological_physics" title="Secondary calculus and cohomological physics">Secondary calculus</a>

<ul><li><a href="Differential_calculus_over_commutative_algebras" title="Differential calculus over commutative algebras">over commutative algebras</a></li></ul></li>

<li><a href="Sheaf_(mathematics)" title="Sheaf (mathematics)">Sheaf</a></li>

<li><a href="Stratifold" title="Stratifold">Stratifold</a></li>

<li><a href="Supermanifold" title="Supermanifold">Supermanifold</a></li>

<li><a href="Stratified_space" title="Stratified space">Stratified space</a></li></ul>

</div></td></tr></tbody></table></div>

<div class="navbox-styles"></div><div role="navigation" class="navbox authority-control" aria-labelledby="Authority_control_databases_frameless&#124;text-top&#124;10px&#124;alt=Edit_this_at_Wikidata&#124;link=https&#58;//www.wikidata.org/wiki/Q6188933#identifiers&#124;class=noprint&#124;Edit_this_at_Wikidata645" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Authority_control_databases_frameless&#124;text-top&#124;10px&#124;alt=Edit_this_at_Wikidata&#124;link=https&#58;//www.wikidata.org/wiki/Q6188933#identifiers&#124;class=noprint&#124;Edit_this_at_Wikidata645" style="font-size:114%;margin:0 4em">Authority control databases </div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">National</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><a rel="nofollow" class="external text" href="https://id.loc.gov/authorities/sh85070153">United States</a></li><li><a rel="nofollow" class="external text" href="https://www.nli.org.il/en/authorities/987007531580805171">Israel</a></li></ul></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><a rel="nofollow" class="external text" href="https://lux.collections.yale.edu/view/concept/dbf06631-695d-4bc6-9fa1-dcffd71d4e84">Yale LUX</a></li></ul></div></td></tr></tbody></table></div></div><div><div class="zim-footer">

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