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Indifferential topology, thejet bundle is a certain construction that makes a newsmoothfiber bundle out of a given smooth fiber bundle. It makes it possible to writedifferential equations onsections of a fiber bundle in an invariant form.Jets may also be seen as the coordinate free versions ofTaylor expansions.

Historically, jet bundles are attributed toCharles Ehresmann, and were an advance on the method (prolongation) ofÉlie Cartan, of dealinggeometrically withhigher derivatives, by imposingdifferential form conditions on newly introduced formal variables. Jet bundles are sometimes calledsprays, althoughsprays usually refer more specifically to the associatedvector field induced on the corresponding bundle (e.g., thegeodesic spray onFinsler manifolds.)

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 thecalculus of variations.^[1] Consequently, the jet bundle is now recognized as the correct domain for ageometrical covariant field theory and much work is done ingeneral relativistic formulations of fields using this approach.

SupposeM is anm-dimensionalmanifold and that (E, π,M) is afiber bundle. Forp ∈M, let Γ(p) denote the set of all local sections whose domain containsp. Let⁠${\displaystyle I=(I(1),I(2),...,I(m))}$⁠ be amulti-index (anm-tuple of non-negative integers, not necessarily in ascending order), then define:

Define the local sections σ, η ∈ Γ(p) to have the samer-jet atp if

${\displaystyle {\frac{{\mathrm{\partial }}^{|I|}{\sigma }^{\alpha }}{\mathrm{\partial }{x}^I}|}_p={\frac{{\mathrm{\partial }}^{|I|}{\eta }^{\alpha }}{\mathrm{\partial }{x}^I}|}_p,0\le |I|\le r.}$

The relation that two maps have the samer-jet is anequivalence relation. Anr-jet is anequivalence class under this relation, and ther-jet with representative σ is denoted${\displaystyle j_p^r\sigma }$. The integerr is also called theorder of the jet,p is itssource and σ(p) is itstarget.

Ther-th jet manifold of π is the set

${\displaystyle J^r(\pi )=\left\{j_p^r\sigma :p\in M,\sigma \in \mathrm{\Gamma }(p)\right\}.}$

We may define projectionsπ_r andπ_r,0 called thesource and target projections respectively, by

${\displaystyle \{\begin{array}{l}{\pi }_r:J^r(\pi )\to M\\ j_p^r\sigma \mapsto p\end{array},\{\begin{array}{l}{\pi }_{r,0}:J^r(\pi )\to E\\ j_p^r\sigma \mapsto \sigma (p)\end{array}}$

If 1 ≤k ≤r, then thek-jet projection is the functionπ_r,k defined by

$${\displaystyle \{\begin{array}{l}{\pi }_{r,k}:J^r(\pi )\to J^k(\pi )\\ j_p^r\sigma \mapsto j_p^k\sigma \end{array}}$$

From this definition, it is clear thatπ_r =π oπ_r,0 and that if 0 ≤m ≤k, thenπ_r,m =π_k,m oπ_r,k. It is conventional to regardπ_r,r as theidentity map onJ ^r(π) and to identifyJ ⁰(π) withE.

The functionsπ_r,k,π_r,0 andπ_r aresmoothsurjectivesubmersions.

Acoordinate system onE will generate a coordinate system onJ ^r(π). Let (U,u) be an adaptedcoordinate chart onE, whereu = (xⁱ,u^α). Theinduced coordinate chart (U^r,u^r) onJ ^r(π) is defined by

where

and the${\displaystyle n\left((\genfrac{}{}{0ex}{}{m+r}{r})-1\right)}$ functions known as thederivative coordinates:

$${\displaystyle \{\begin{array}{l}{u}_I^{\alpha }:U^k\to \mathbf{R}\\ u_I^{\alpha }\left(j_p^r\sigma \right)={\frac{{\mathrm{\partial }}^{|I|}{\sigma }^{\alpha }}{\mathrm{\partial }{x}^I}|}_p\end{array}}$$

Given an atlas of adapted charts (U,u) onE, the corresponding collection of charts (U ^r,u ^r) is afinite-dimensionalC^∞ atlas onJ ^r(π).

Since the atlas on each${\displaystyle J^r(\pi )}$ defines a manifold, the triples${\displaystyle (J^r(\pi ),{\pi }_{r,k},J^k(\pi ))}$,${\displaystyle (J^r(\pi ),{\pi }_{r,0},E)}$ and${\displaystyle (J^r(\pi ),{\pi }_r,M)}$ all define fibered manifolds. In particular, if${\displaystyle (E,\pi ,M)}$is a fiber bundle, the triple${\displaystyle (J^r(\pi ),{\pi }_r,M)}$ defines ther-th jet bundle of π.

IfW ⊂M is an open submanifold, then

${\displaystyle J^r\left(\pi {|}_{{\pi }^{-1}(W)}\right)\cong {\pi }_r^{-1}(W).}$

Ifp ∈M, then the fiber${\displaystyle {\pi }_r^{-1}(p)}$ is denoted${\displaystyle J_p^r(\pi )}$.

Let σ be a local section of π with domainW ⊂M. Ther-th jet prolongation of σ is the map${\displaystyle j^r\sigma :W\to J^r(\pi )}$ defined by

${\displaystyle (j^r\sigma )(p)=j_p^r\sigma .}$

Note that${\displaystyle {\pi }_r\circ j^r\sigma ={\mathbb{i}\mathbb{d}}_W}$, so${\displaystyle j^r\sigma }$ really is a section. In local coordinates,${\displaystyle j^r\sigma }$ is given by

${\displaystyle \left({\sigma }^{\alpha },\frac{{\mathrm{\partial }}^{|I|}{\sigma }^{\alpha }}{\mathrm{\partial }{x}^I}\right)1\le |I|\le r.}$

We identify${\displaystyle j^0\sigma }$ with${\displaystyle \sigma }$ .

An independently motivated construction of the sheaf of sections${\displaystyle \mathrm{\Gamma }{J}^k\left({\pi }_{TM}\right)}$ is given.

Consider a diagonal map${\mathrm{\Delta }}_n:M\to \prod _{i=1}^{n+1}M$, where the smooth manifold${\displaystyle M}$ is alocally ringed space by${\displaystyle C^k(U)}$ for each open${\displaystyle U}$. Let${\displaystyle \mathcal{I}}$ be theideal sheaf of${\displaystyle {\mathrm{\Delta }}_n(M)}$, equivalently let${\displaystyle \mathcal{I}}$ be thesheaf of smoothgerms which vanish on${\displaystyle {\mathrm{\Delta }}_n(M)}$ for all. Thepullback of thequotient sheaf${\displaystyle {{\mathrm{\Delta }}_n}^{\ast }\left(\mathcal{I}/{\mathcal{I}}^{n+1}\right)}$ from$\prod _{i=1}^{n+1}M$ to${\displaystyle M}$ by${\displaystyle {\mathrm{\Delta }}_n}$ is the sheaf of k-jets.^[2]

Thedirect limit of the sequence of injections given by the canonical inclusions${\displaystyle {\mathcal{I}}^{n+1}\hookrightarrow {\mathcal{I}}^n}$ of sheaves, gives rise to theinfinite jet sheaf${\displaystyle {\mathcal{J}}^{\mathrm{\infty }}(TM)}$. Observe that by the direct limit construction it is a filtered ring.

If π is thetrivial bundle (M ×R, pr₁,M), then there is a canonicaldiffeomorphism between the first jet bundle${\displaystyle J^1(\pi )}$ andT*M ×R. To construct this diffeomorphism, for each σ in${\displaystyle {\mathrm{\Gamma }}_M(\pi )}$ write${\displaystyle \overline{\sigma }=p{r}_2\circ \sigma \in C^{\mathrm{\infty }}(M)}$.

Then, wheneverp ∈M

${\displaystyle j_p^1\sigma =\left\{\psi :\psi \in {\mathrm{\Gamma }}_p(\pi );\overline{\psi }(p)=\overline{\sigma }(p);d{\overline{\psi }}_p=d{\overline{\sigma }}_p\right\}.}$

Consequently, the mapping

${\displaystyle \{\begin{array}{l}{J}^1(\pi )\to T^{\ast }M\times \mathbf{R}\\ j_p^1\sigma \mapsto \left(d{\overline{\sigma }}_p,\overline{\sigma }(p)\right)\end{array}}$

is well-defined and is clearlyinjective. Writing it out in coordinates shows that it is a diffeomorphism, because if(xⁱ, u) are coordinates onM ×R, whereu = id_R is the identity coordinate, then the derivative coordinatesu_i onJ¹(π) correspond to the coordinates ∂_i onT*M.

Likewise, if π is the trivial bundle (R ×M, pr₁,R), then there exists a canonical diffeomorphism between${\displaystyle J^1(\pi )}$andR ×TM.

The spaceJ^r(π) carries a naturaldistribution, that is, a sub-bundle of thetangent bundleTJ^r(π)), called theCartan distribution. The Cartan distribution is spanned by all tangent planes to graphs of holonomic sections; that is, sections of the formj^rφ forφ a section of π.

The annihilator of the Cartan distribution is a space ofdifferential one-forms calledcontact forms, onJ^r(π). The space of differential one-forms onJ^r(π) is denoted by${\displaystyle {\mathrm{\Lambda }}^1{J}^r(\pi )}$ and the space of contact forms is denoted by${\displaystyle {\mathrm{\Lambda }}_C^r\pi }$. A one form is a contact form provided itspullback along every prolongation is zero. In other words,${\displaystyle \theta \in {\mathrm{\Lambda }}^1{J}^r\pi }$ is a contact form if and only if

${\displaystyle {\left(j^{r+1}\sigma \right)}^{\ast }\theta =0}$

for all local sections σ of π overM.

The Cartan distribution is the main geometrical structure on jet spaces and plays an important role in the geometric theory ofpartial differential equations. The Cartan distributions are completely non-integrable. In particular, they are notinvolutive. The dimension of the Cartan distribution grows with the order of the jet space. However, on the space of infinite jetsJ^∞ the Cartan distribution becomes involutive and finite-dimensional: its dimension coincides with the dimension of the base manifoldM.

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

for allp ∈M and σ in Γ_p(π). A general 1-form onJ¹(π) takes the form

${\displaystyle \theta =a(x,u,u_1)dx+b(x,u,u_1)du+c(x,u,u_1)d{u}_1}$

A section σ in Γ_p(π) has first prolongation

${\displaystyle j^1\sigma =(u,u_1)=\left(\sigma (p),{\frac{\mathrm{\partial }\sigma }{\mathrm{\partial }x}|}_p\right).}$

Hence,(j¹σ)*θ can be calculated as

This will vanish for all sections σ if and only ifc = 0 anda = −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 spaceJ²(π) with additional coordinateu₂, such that

${\displaystyle u_2(j_p^2\sigma )={\frac{{\mathrm{\partial }}^2\sigma }{\mathrm{\partial }{x}^2}|}_p={\sigma }^{″}(x)}$

a general 1-form has the construction

${\displaystyle \theta =a(x,u,u_1,u_2)dx+b(x,u,u_1,u_2)du+c(x,u,u_1,u_2)d{u}_1+e(x,u,u_1,u_2)d{u}_2}$

This is a contact form if and only if

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

${\displaystyle \theta =b(x,\sigma (x),{\sigma }^{\prime }(x)){\theta }_0+c(x,\sigma (x),{\sigma }^{\prime }(x)){\theta }_1,}$

where θ₁ =du₁ −u₂dx is the next basic contact form (Note that here we are identifying the form θ₀ with its pull-back${\displaystyle {\left({\pi }_{2,1}\right)}^{\ast }{\theta }_0}$ toJ²(π)).

In general, providingx, u ∈R, a contact form onJ^r+1(π) can be written as alinear combination of the basic contact forms

${\displaystyle {\theta }_k=d{u}_k-u_{k+1}dxk=0,\dots ,r-1}$

where

${\displaystyle u_k\left(j^k\sigma \right)={\frac{{\mathrm{\partial }}^k\sigma }{\mathrm{\partial }{x}^k}|}_p.}$

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

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

${\displaystyle \theta =\sum _{|I|=0}^r{P}_{\alpha }^I{\theta }_I^{\alpha }}$

with smooth coefficients${\displaystyle P_i^{\alpha }(x^i,u^{\alpha },u_I^{\alpha })}$ of the basic contact forms

${\displaystyle {\theta }_I^{\alpha }=d{u}_I^{\alpha }-u_{I,i}^{\alpha }d{x}^i}$

|I| is known as theorder of the contact form${\displaystyle {\theta }_i^{\alpha }}$. Note that contact forms onJ^r+1(π) have orders at mostr. Contact forms provide a characterization of those local sections ofπ_r+1 which are prolongations of sections of π.

Let ψ ∈ Γ_W(π_r+1), thenψ =j^r+1σ where σ ∈ Γ_W(π) if and only if${\displaystyle {\psi }^{\ast }(\theta {|}_W)=0,\mathrm{\forall }\theta \in {\mathrm{\Lambda }}_C^1{\pi }_{r+1,r}.}$

A generalvector field on the total spaceE, coordinated by${\displaystyle (x,u)\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\left(x^i,u^{\alpha }\right)}$, is

${\displaystyle V\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}{\rho }^i(x,u)\frac{\mathrm{\partial }}{\mathrm{\partial }{x}^i}+{\varphi }^{\alpha }(x,u)\frac{\mathrm{\partial }}{\mathrm{\partial }{u}^{\alpha }}.}$

A vector field is calledhorizontal, meaning that all the vertical coefficients vanish, if${\displaystyle {\varphi }^{\alpha }}$ = 0.

A vector field is calledvertical, meaning that all the horizontal coefficients vanish, ifρⁱ = 0.

For fixed(x, u), we identify

${\displaystyle V_{(x,u)}\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}{\rho }^i(x,u)\frac{\mathrm{\partial }}{\mathrm{\partial }{x}^i}+{\varphi }^{\alpha }(x,u)\frac{\mathrm{\partial }}{\mathrm{\partial }{u}^{\alpha }}}$

having coordinates(x, u, ρⁱ, φ^α), with an element in the fiberT_xuE ofTE over(x, u) inE, calledatangent vector inTE. A section

${\displaystyle \{\begin{array}{l}\psi :E\to TE\\ (x,u)\mapsto \psi (x,u)=V\end{array}}$

is calleda vector field onE with

${\displaystyle V={\rho }^i(x,u)\frac{\mathrm{\partial }}{\mathrm{\partial }{x}^i}+{\varphi }^{\alpha }(x,u)\frac{\mathrm{\partial }}{\mathrm{\partial }{u}^{\alpha }}}$

and ψ inΓ(TE).

The jet bundleJ^r(π) is coordinated by${\displaystyle (x,u,w)\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\left(x^i,u^{\alpha },w_i^{\alpha }\right)}$. For fixed(x, u, w), identify

${\displaystyle V_{(x,u,w)}\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}{V}^i(x,u,w)\frac{\mathrm{\partial }}{\mathrm{\partial }{x}^i}+V^{\alpha }(x,u,w)\frac{\mathrm{\partial }}{\mathrm{\partial }{u}^{\alpha }}+V_i^{\alpha }(x,u,w)\frac{\mathrm{\partial }}{\mathrm{\partial }{w}_i^{\alpha }}+V_{i_1{i}_2}^{\alpha }(x,u,w)\frac{\mathrm{\partial }}{\mathrm{\partial }{w}_{i_1{i}_2}^{\alpha }}+\cdots +V_{i_1\cdots i_r}^{\alpha }(x,u,w)\frac{\mathrm{\partial }}{\mathrm{\partial }{w}_{i_1\cdots i_r}^{\alpha }}}$

having coordinates

${\displaystyle \left(x,u,w,v_i^{\alpha },v_{i_1{i}_2}^{\alpha },\cdots ,v_{i_1\cdots i_r}^{\alpha }\right),}$

with an element in the fiber${\displaystyle T_{xuw}(J^r\pi )}$ ofTJ^r(π) over(x, u, w) ∈J^r(π), calleda tangent vector inTJ^r(π). Here,

${\displaystyle v_i^{\alpha },v_{i_1{i}_2}^{\alpha },\dots ,v_{i_1\cdots i_r}^{\alpha }}$

are real-valued functions onJ^r(π). A section

${\displaystyle \{\begin{array}{l}\mathrm{\Psi }:J^r(\pi )\to T{J}^r(\pi )\\ (x,u,w)\mapsto \mathrm{\Psi }(u,w)=V\end{array}}$

isa vector field onJ^r(π), and we say${\displaystyle \mathrm{\Psi }\in \mathrm{\Gamma }(T\left(J^r\pi \right)).}$

Let(E, π, M) be a fiber bundle. Anr-th orderpartial differential equation on π is aclosedembedded submanifoldS of the jet manifoldJ^r(π). A solution is a local section σ ∈ Γ_W(π) satisfying${\displaystyle j_p^r\sigma \in S}$, for allp inM.

Consider an example of a first order partial differential equation.

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

${\displaystyle F=u_1^1{u}_2^1-2x^2{u}^1}$

gives rise to the differential equation

${\displaystyle S=\left\{j_p^1\sigma \in J^1\pi :\left(u_1^1{u}_2^1-2x^2{u}^1\right)\left(j_p^1\sigma \right)=0\right\}}$

which can be written

${\displaystyle \frac{\mathrm{\partial }\sigma }{\mathrm{\partial }{x}^1}\frac{\mathrm{\partial }\sigma }{\mathrm{\partial }{x}^2}-2x^2\sigma =0.}$

The particular

${\displaystyle \{\begin{array}{l}\sigma :{\mathbf{R}}^2\to {\mathbf{R}}^2\times \mathbf{R}\\ \sigma (p_1,p_2)=\left(p^1,p^2,p^1(p^2{)}^2\right)\end{array}}$

has first prolongation given by

${\displaystyle j^1\sigma \left(p_1,p_2\right)=\left(p^1,p^2,p^1{\left(p^2\right)}^2,{\left(p^2\right)}^2,2p^1{p}^2\right)}$

and is a solution of this differential equation, because

and so${\displaystyle j_p^1\sigma \in S}$ foreveryp ∈R².

A local diffeomorphismψ :J^r(π) →J^r(π) defines a contact transformation of orderr if it preserves the contact ideal, meaning that if θ is any contact form onJ^r(π), thenψ*θ is also a contact form.

The flow generated by a vector fieldV^r on the jet spaceJ^r(π) forms a one-parameter group of contact transformations if and only if theLie derivative${\displaystyle {\mathcal{L}}_{V^r}(\theta )}$ of any contact form θ preserves the contact ideal.

Let us begin with the first order case. Consider a general vector fieldV¹ onJ¹(π), given by

${\displaystyle V^1\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}{\rho }^i\left(x^i,u^{\alpha },u_I^{\alpha }\right)\frac{\mathrm{\partial }}{\mathrm{\partial }{x}^i}+{\varphi }^{\alpha }\left(x^i,u^{\alpha },u_I^{\alpha }\right)\frac{\mathrm{\partial }}{\mathrm{\partial }{u}^{\alpha }}+{\chi }_i^{\alpha }\left(x^i,u^{\alpha },u_I^{\alpha }\right)\frac{\mathrm{\partial }}{\mathrm{\partial }{u}_i^{\alpha }}.}$

We now apply${\displaystyle {\mathcal{L}}_{V^1}}$ to the basic contact forms${\displaystyle {\theta }_0^{\alpha }=d{u}^{\alpha }-u_i^{\alpha }d{x}^i,}$ and expand theexterior derivative of the functions in terms of their coordinates to obtain:

Therefore,V¹ determines a contact transformation if and only if the coefficients ofdxⁱ and${\displaystyle d{u}_i^k}$ in the formula vanish. The latter requirements imply thecontact conditions

${\displaystyle \frac{\mathrm{\partial }{\varphi }^{\alpha }}{\mathrm{\partial }{u}_i^k}-u_l^{\alpha }\frac{\mathrm{\partial }{\rho }^l}{\mathrm{\partial }{u}_i^k}=0}$

The former requirements provide explicit formulae for the coefficients of the first derivative terms inV¹:

${\displaystyle {\chi }_i^{\alpha }={\hat{D}}_i{\varphi }^{\alpha }-u_l^{\alpha }\left({\hat{D}}_i{\rho }^l\right)}$

where

${\displaystyle {\hat{D}}_i=\frac{\mathrm{\partial }}{\mathrm{\partial }{x}^i}+u_i^k\frac{\mathrm{\partial }}{\mathrm{\partial }{u}^k}}$

denotes the zeroth order truncation of the total derivativeD_i.

Thus, the contact conditions uniquely prescribe the prolongation of any point or contact vector field. That is, if${\displaystyle {\mathcal{L}}_{V^r}}$ satisfies these equations,V^r is called ther-th prolongation ofV to a vector field onJ^r(π).

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

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

for allp ∈M andσ in Γ_p(π). A contact form onJ¹(π) has the form

${\displaystyle \theta =du-u_{1}dx}$

Consider a vectorV onE, having the form

${\displaystyle V=x\frac{\mathrm{\partial }}{\mathrm{\partial }u}-u\frac{\mathrm{\partial }}{\mathrm{\partial }x}}$

Then, the first prolongation of this vector field toJ¹(π) is

If we now take the Lie derivative of the contact form with respect to this prolonged vector field,${\displaystyle {\mathcal{L}}_{V^1}(\theta ),}$ we obtain

Hence, for preservation of the contact ideal, we require

${\displaystyle 1+u_1{u}_1-\rho (x,u,u_1)=0\iff \rho (x,u,u_1)=1+u_1{u}_1.}$

And so the first prolongation ofV to a vector field onJ¹(π) is

${\displaystyle V^1=x\frac{\mathrm{\partial }}{\mathrm{\partial }u}-u\frac{\mathrm{\partial }}{\mathrm{\partial }x}+(1+u_1{u}_1)\frac{\mathrm{\partial }}{\mathrm{\partial }{u}_1}.}$

Let us also calculate the second prolongation ofV to a vector field onJ²(π). We have${\displaystyle \{x,u,u_1,u_2\}}$ as coordinates onJ²(π). Hence, the prolonged vector has the form

${\displaystyle V^2=x\frac{\mathrm{\partial }}{\mathrm{\partial }u}-u\frac{\mathrm{\partial }}{\mathrm{\partial }x}+\rho (x,u,u_1,u_2)\frac{\mathrm{\partial }}{\mathrm{\partial }{u}_1}+\varphi (x,u,u_1,u_2)\frac{\mathrm{\partial }}{\mathrm{\partial }{u}_2}.}$

The contact forms are

To preserve the contact ideal, we require

Now,θ has nou₂ dependency. Hence, from this equation we will pick up the formula forρ, which will necessarily be the same result as we found forV¹. Therefore, the problem is analogous to prolonging the vector fieldV¹ toJ²(π). That is to say, we may generate ther-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

${\displaystyle \rho (x,u,u_1)=1+u_1{u}_1}$

and so

Therefore, the Lie derivative of the second contact form with respect toV² is

Hence, for${\displaystyle {\mathcal{L}}_{V^2}({\theta }_1)}$ to preserve the contact ideal, we require

${\displaystyle 3u_1{u}_2-\varphi (x,u,u_1,u_2)=0\iff \varphi (x,u,u_1,u_2)=3u_1{u}_2.}$

And so the second prolongation ofV to a vector field onJ²(π) is

${\displaystyle V^2=x\frac{\mathrm{\partial }}{\mathrm{\partial }u}-u\frac{\mathrm{\partial }}{\mathrm{\partial }x}+(1+u_1{u}_1)\frac{\mathrm{\partial }}{\mathrm{\partial }{u}_1}+3u_1{u}_2\frac{\mathrm{\partial }}{\mathrm{\partial }{u}_2}.}$

Note that the first prolongation ofV can be recovered by omitting the second derivative terms inV², or by projecting back toJ¹(π).

Theinverse limit of the sequence of projections${\displaystyle {\pi }_{k+1,k}:J^{k+1}(\pi )\to J^k(\pi )}$ gives rise to theinfinite jet spaceJ^∞(π). A point${\displaystyle j_p^{\mathrm{\infty }}(\sigma )}$ is the equivalence class of sections of π that have the samek-jet inp as σ for all values ofk. The natural projection π_∞ maps${\displaystyle j_p^{\mathrm{\infty }}(\sigma )}$ intop.

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 onJ^∞(π), not relying on differentiable charts, is given by thedifferential calculus over commutative algebras. Dual to the sequence of projections${\displaystyle {\pi }_{k+1,k}:J^{k+1}(\pi )\to J^k(\pi )}$ of manifolds is the sequence of injections${\displaystyle {\pi }_{k+1,k}^{\ast }:C^{\mathrm{\infty }}(J^k(\pi ))\to C^{\mathrm{\infty }}\left(J^{k+1}(\pi )\right)}$ of commutative algebras. Let's denote${\displaystyle C^{\mathrm{\infty }}(J^k(\pi ))}$ simply by${\displaystyle {\mathcal{F}}_k(\pi )}$. Take now thedirect limit${\displaystyle \mathcal{F}(\pi )}$ of the${\displaystyle {\mathcal{F}}_k(\pi )}$'s. It will be a commutative algebra, which can be assumed to be the smooth functions algebra over the geometric objectJ^∞(π). Observe that${\displaystyle \mathcal{F}(\pi )}$, being born as a direct limit, carries an additional structure: it is a filtered commutative algebra.

Roughly speaking, a concrete element${\displaystyle \phi \in \mathcal{F}(\pi )}$ will always belong to some${\displaystyle {\mathcal{F}}_k(\pi )}$, so it is a smooth function on the finite-dimensional manifoldJ^k(π) in the usual sense.

Given ak-th order system of PDEsE ⊆J^k(π), the collectionI(E) of vanishing onE smooth functions onJ^∞(π) is anideal in the algebra${\displaystyle {\mathcal{F}}_k(\pi )}$, and hence in the direct limit${\displaystyle \mathcal{F}(\pi )}$ too.

EnhanceI(E) by adding all the possible compositions oftotal derivatives applied to all its elements. This way we get a new idealI of${\displaystyle \mathcal{F}(\pi )}$ which is now closed under the operation of taking total derivative. The submanifoldE_(∞) ofJ^∞(π) cut out byI is called theinfinite prolongation ofE.

Geometrically,E_(∞) is the manifold offormal solutions ofE. A point${\displaystyle j_p^{\mathrm{\infty }}(\sigma )}$ ofE_(∞) can be easily seen to be represented by a section σ whosek-jet's graph is tangent toE at the point${\displaystyle j_p^k(\sigma )}$ with arbitrarily high order of tangency.

Analytically, ifE is given by φ = 0, a formal solution can be understood as the set of Taylor coefficients of a section σ in a pointp that make vanish theTaylor series of${\displaystyle \phi \circ j^k(\sigma )}$ at the pointp.

Most importantly, the closure properties ofI imply thatE_(∞) is tangent to theinfinite-order contact structure${\displaystyle \mathcal{C}}$ onJ^∞(π), so that by restricting${\displaystyle \mathcal{C}}$ toE_(∞) one gets thediffiety${\displaystyle (E_{(\mathrm{\infty })},\mathcal{C}{|}_{E_{(\mathrm{\infty })}})}$, and can study the associatedVinogradov (C-spectral) sequence.