In differential geometry, the jet bundle is a certain construction that makes a new smooth fiber bundle out of a given smooth fiber bundle. It makes it possible to write differential equations on sections of a fiber bundle in an invariant form. Jets may also be seen as the coordinate free versions of Taylor expansions.
Historically, jet bundles are attributed to Ehresmann, and were an advance on the method (prolongation) of Élie Cartan, of dealing geometrically with higher derivatives, by imposing differential form conditions on newly introduced formal variables. Jet bundles are sometimes called sprays, although sprays usually refer more specifically to the associated vector field induced on the corresponding bundle (e.g., the geodesic spray on Finsler manifolds.)
More recently, jet bundles have appeared as a concise way to describe phenomena associated with the derivatives of maps, particularly those associated with the calculus of variations. Consequently, the jet bundle is now recognized as the correct domain for a geometrical covariant field theory and much work is done in general relativistic formulations of fields using this approach.
Contents

Jets 1

Jet manifolds 2

Jet bundles 3

Contact structure 4

Vector fields 5

Partial differential equations 6

Jet Prolongation 7

Infinite Jet Spaces 8

Infinitely prolonged PDEs 8.1

Remark 9

See also 10

References 11
Jets
Suppose M is an mdimensional manifold and that (E, π, M) is a fiber bundle. For p ∈ M, let Γ(π) denote the set of all local sections whose domain contains p. Let I = (I(1), I(2), ..., I(m)) be a multiindex (an ordered mtuple of integers), then

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)}.
Define the local sections σ, η ∈ Γ(π) to have the same rjet at p if

\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.
The relation that two maps have the same rjet is an equivalence relation. An rjet is an equivalence class under this relation, and the rjet with representative σ is denoted j^r_p\sigma . The integer r is also called the order of the jet, p is its source and σ(p) is its target.
Jet manifolds
The rth jet manifold of π is the set

\{j^{r}_{p}\sigma:p \in M, \sigma \in \Gamma(\pi)\}
and is denoted J^{r}(π). We may define projections π_{r} and π_{r,0} called the source and target projections respectively, by

\begin{cases} \pi_r: J^{r}(\pi) \to M \\ j^{r}_{p}\sigma \mapsto p \end{cases}

\begin{cases} \pi_{r, 0}: J^{r}(\pi) \to E \\ j^{r}_{p}\sigma \mapsto \sigma(p) \end{cases}
If 1 ≤ k ≤ r, then the kjet projection is the function π_{r,k} defined by

\begin{cases} \pi_{r, k}: J^{r}(\pi) \to J^{k}(\pi)\\ j^{r}_{p}\sigma \mapsto j^{k}_{p}\sigma \end{cases}
From this definition, it is clear that π_{r} = π o π_{r,0} and that if 0 ≤ m ≤ k, then π_{r,m} = π_{k,m} o π_{r,k}. It is conventional to regard π_{r,r} = id_{Jr(π)}, the identity map on J^{r}(π) and to identify J^{0}(π) with E.
The functions π_{r,k}, π_{r,0} and π_{r} are smooth surjective submersions.
A coordinate system on E will generate a coordinate system on J^{r}(π). Let (U, u) be an adapted coordinate chart on E, where u = (x^{i}, u^{α}). The induced coordinate chart (U^{r}, u^{r}) on J^{r}(π) is defined by

U^{r} = \{ j^{r}_{p}\sigma: \sigma(p) \in U \} \,

u^{r} = (x^{i}, u^{\alpha}, u^{\alpha}_{I})\,
where

x^{i}(j^{r}_{p}\sigma) = x^{i}(p)

u^{\alpha}(j^{r}_{p}\sigma) = u^{\alpha}(\sigma(p))
and the n \left( {}^{m+r}C_{r} 1\right)\, functions

u^{\alpha}_{I}:U^{k} \to \mathbf{R}\,
are specified by

u^{\alpha}_{I}(j^{r}_{p}\sigma) = \left.\frac{\partial^{I} \sigma^{\alpha}}{\partial x^{I}}\right_{p}
and are known as the derivative coordinates.
Given an atlas of adapted charts (U, u) on E, the corresponding collection of charts (U^{r}, u^{r}) is a finitedimensional C^{∞} atlas on J^{r}(π).
Jet bundles
Since the atlas on each J^{r}(π) defines a manifold, the triples (J^{r}(π), π_{r,k}, J^{k}(π)), (J^{r}(π), π_{r,0}, E) and (J^{r}(π), π_{r}, M) all define fibered manifolds. In particular, if (E, π, M) is a fiber bundle, the triple (J^{r}(π), π_{r}, M) defines the rth jet bundle of π.
If W ⊂ M is an open submanifold, then

J^{r}\left(\pi_{\pi^{1}(W)}\right) \cong \pi^{1}_{r}(W).\,
If p ∈ M, then the fiber \pi^{1}_{r}(p)\, is denoted J^{r}_{p}(\pi).
Let σ be a local section of π with domain W ⊂ M. The rth jet prolongation of σ is the map j^{r}σ: W → J^{r}(π) defined by

(j^{r}\sigma)(p) = j^{r}_{p}\sigma. \,
Note that π_{r} o j^{r}σ = id_{W}, so j^{r}σ really is a section. In local coordinates, j^{r}σ is given by

\left(\sigma^{\alpha}, \frac{\partial^{I} \sigma^{\alpha}}{\partial x^{I}}\right) \qquad 1 \leq I \leq r. \,
We identify j^{0}σ with σ.
Example
If π is the trivial bundle (M × R, pr_{1}, M), then there is a canonical diffeomorphism between the first jet bundle J^{1}(π) and T*M × R. To construct this diffeomorphism, for each σ in Γ_{M}(π) write \bar{\sigma} = pr_{2} \circ \sigma \in C^{\infty}(M)\,.
Then, whenever p ∈ M

j^{1}_{p}\sigma = \{ \psi : \psi \in \Gamma_{p}(\pi); \bar{\psi}(p) = \bar{\sigma}(p); d\bar{\psi}_{p} = d\bar{\sigma}_{p} \}. \,
Consequently, the mapping

\begin{cases} J^{1}(\pi) \to T^*M \times \mathbf{R} \\ j^{1}_{p}\sigma \mapsto (d\bar{\sigma}_{p},\bar{\sigma}(p)) \end{cases}
is welldefined and is clearly injective. Writing it out in coordinates shows that it is a diffeomorphism, because if (x^{i}, u) are coordinates on M × R, where u = id_{R} is the identity coordinate, then the derivative coordinates u_{i} on J^{1}(π) correspond to the coordinates ∂_{i} on T*M.
Likewise, if π is the trivial bundle (R × M, pr_{1}, R), then there exists a canonical diffeomorphism between J^{1}(π) and R × TM.
Contact structure
The space J^{r}(π) carries a natural distribution, that is, a subbundle of the tangent bundle TJ^{r}(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 j^{r}φ for φ a section of π.
The annihilator of the Cartan distribution is a space of differential oneforms called contact forms, on J^{r}(π). The space of differential oneforms on J^{r}(π) is denoted by \Lambda^1J^r(\pi) and the space of contact forms is denoted by \Lambda_C^r\pi. A one form is a contact form provided its pullback along every prolongation is zero. In other words, \theta\in\Lambda^1J^r\pi is a contact form if and only if

(j^{r+1}\sigma)^*\theta = 0
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 partial differential equations. The Cartan distributions are completely nonintegrable. In particular, they are not involutive. 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 finitedimensional: its dimension coincides with the dimension of the base manifold M.
Example
Let us consider the case (E, π, M), where E ≃ R^{2} and M ≃ R. Then, (J^{1}(π), π, M) defines the first jet bundle, and may be coordinated by (x, u, u_{1}), where

x(j^{1}_{p}\sigma)

= x(p) = x\,

u(j^{1}_{p}\sigma)

= u(\sigma(p)) = u(\sigma(x)) = \sigma(x) \,

u_{1}(j^{1}_{p}\sigma)

= \left.\frac{\partial \sigma}{\partial x}\right_{p} = \sigma'(x)

for all p ∈ M and σ in Γ_{p}(π). A general 1form on J^{1}(π) takes the form

\theta = a(x, u, u_{1})dx + b(x, u, u_{1})du + c(x, u,u_{1})du_{1}\,
A section σ in Γ_{p}(π) has first prolongation

j^{1}\sigma = (u,u_{1}) = \left(\sigma(p), \left. \frac{\partial \sigma}{\partial x} \right_{p} \right).
Hence, (j^{1}σ)*θ can be calculated as

(j^{1}_{p}\sigma)^{*} \theta \,

= \theta \circ j^{1}_{p}\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 \,

This will vanish for all sections σ if and only if c = 0 and a = −bσ′(x). Hence, θ = b(x, u, u_{1})θ_{0} must necessarily be a multiple of the basic contact form θ_{0} = du − u_{1}dx. Proceeding to the second jet space J^{2}(π) with additional coordinate u_{2}, such that

u_{2}(j^{2}_{p}\sigma)=\left.\frac{\partial^{2} \sigma}{\partial x^{2}}\right_{p} = \sigma''(x)\,
a general 1form has the construction

\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}\,
This is a contact form if and only if

(j^{2}_{p}\sigma)^{*} \theta \,

= \theta \circ j^{2}_{p}\sigma \,


= a(x, \sigma(x), \sigma'(x),\sigma''(x))dx + b(x, \sigma(x),\sigma'(x),\sigma''(x))d(\sigma(x))+ \,


+ 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\,

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

\theta = b(x, \sigma(x), \sigma'(x))\theta_{0} + c(x, \sigma(x), \sigma'(x))\theta_{1}\,
where θ_{1} = du_{1} − u_{2}dx is the next basic contact form (Note that here we are identifying the form θ_{0} with its pullback (\pi_{2,1})^{*}\theta_{0}\, to J^{2}(π)).
In general, providing x, u ∈ R, a contact form on J^{r+1}(π) can be written as a linear combination of the basic contact forms

\theta_{k} = du_{k}  u_{k+1}dx \qquad k=0, \ldots, r1\,
where u_{k}(j^{k}\sigma)= \left.\frac{\partial^{k} \sigma}{\partial x^{k}}\right_{p}\,.
Similar arguments lead to a complete characterization of all contact forms.
In local coordinates, every contact oneform on J^{r+1}(π) can be written as a linear combination

\theta = \sum_{I=0}^{r} P_{\alpha}^{I}\theta_{I}^{\alpha}\,
with smooth coefficients P^{\alpha}_{I}(x^{i},u^{\alpha})\, of the basic contact forms

\theta_{I}^{\alpha} = du^{\alpha}_{I}  u^{\alpha}_{I,i}dx^{i}\,
I is known as the order of the contact form \theta_{I}^{\alpha}. Note that contact forms on J^{r+1}(π) have orders at most r. Contact forms provide a characterization of those local sections of π_{r+1} which are prolongations of sections of π.
Let ψ ∈ Γ_{W}(π_{r+1}), then ψ = j^{r+1}σ where σ ∈ Γ_{W}(π) if and only if \psi^{*}(\theta_{W})=0, \forall \theta \in \Lambda_{C}^{1}\pi_{r+1,r}.\,
Vector fields
A general vector field on the total space E, coordinated by (x,u) \ \stackrel{\mathrm{def}}{=}\ (x^{i},u^{\alpha})\,, is

V \ \stackrel{\mathrm{def}}{=}\ \rho^{i}(x,u)\frac{\partial}{\partial x^{i}} + \phi^{\alpha}(x,u)\frac{\partial}{\partial u^{\alpha}}.\,
A vector field is called horizontal, meaning that all the vertical coefficients vanish, i.e. φ^{α} = 0.
A vector field is called vertical, meaning that all the horizontal coefficients vanish, i.e. ρ^{i} = 0.
For fixed (x, u), we identify

V_{(xu)} \ \stackrel{\mathrm{def}}{=}\ \rho^{i}(x,u) \frac{\partial}{\partial x^{i}} + \phi^{\alpha}(x,u) \frac{\partial}{\partial u^{\alpha}}\,
having coordinates (x, u, ρ^{i}, φ^{α}), with an element in the fiber T_{xu}E of TE over (x,u) in E, called a tangent vector in TE. A section

\begin{cases} \psi : E \to TE \\ (x,u) \mapsto \psi(x,u) = V \end{cases}
is called a vector field on E' with

V = \rho^{i}(x,u) \frac{\partial}{\partial x^{i}} + \phi^{\alpha}(x,u) \frac{\partial}{\partial u^{\alpha}}\,
and ψ in Γ(TE).
The jet bundle J^{r}(π) is coordinated by (x,u,w) \ \stackrel{\mathrm{def}}{=}\ (x^{i},u^{\alpha},w_{i}^{\alpha})\,. For fixed (x,u,w), identify

V_{(xuw)} \ \stackrel{\mathrm{def}}{=}\ \,

V^{i}(x,u,w) \frac{\partial}{\partial x^{i}} + V^{\alpha}(x,u,w) \frac{\partial}{\partial u^{\alpha}} \ + \ V^{\alpha}_{i}(x,u,w) \frac{\partial}{\partial w^{\alpha}_{i}} +\,


\qquad + \ V^{\alpha}_{i_{1}i_{2}}(x,u,w) \frac{\partial}{\partial w^{\alpha}_{i_{1}i_{2}}} + \cdots \ + \ \cdots + V^{\alpha}_{i_{1}i_{2} \cdots i_{r}}(x,u,w) \frac{\partial}{\partial w^{\alpha}_{i_{1}i_{2} \cdots i_{r}}}\,

having coordinates (x,u,w,v^{\alpha}_{i}, v^{\alpha}_{i_{1} i_{2}},\ldots,v^{ \alpha}_{i_{1}i_{2} \cdots i_{r}})\,, with an element in the fiber T_{xuw}(J^{r}\pi)\, of TJ^{r}(π) over (x, u, w) ∈ J^{r}(π), called a tangent vector in TJ^{r}(π). Here,

v^{\alpha}_{i}, v^{\alpha}_{i_{1}i_{2}},\ldots,v^{\alpha}_{i_{1}i_{2} \cdots i_{r}}\,
are realvalued functions on J^{r}(π). A section

\begin{cases} \Psi : J^{r}(\pi) \to TJ^{r}(\pi) \\ (x,u,w) \mapsto \Psi(u,w) = V \end{cases}
is a vector field on J^{r}(π), and we say \Psi \in \Gamma(T(J^{r}\pi))\,.
Partial differential equations
Let (E, π, M) be a fiber bundle. An rth order partial differential equation on π is a closed embedded submanifold S of the jet manifold J^{r}(π). A solution is a local section σ ∈ Γ_{W}(π) satisfying j^{r}_{p}\sigma \in S, for all p in M.
Let us consider an example of a first order partial differential equation.
Example
Let π be the trivial bundle (R^{2} × R, pr_{1}, R^{2}) with global coordinates (x^{1}, x^{2}, u^{1}). Then the map F : J^{1}(π) → R defined by

F = u^{1}_{1}u^{1}_{2}  2x^{2}u^{1}\,
gives rise to the differential equation

S = \{ j^{1}_{p}\sigma \in J^{1}\pi : (u^{1}_{1}u^{1}_{2}  2x^{2}u^{1})(j^{1}_{p}\sigma)=0 \} \,
which can be written

\frac{\partial \sigma}{\partial x^{1}}\frac{\partial \sigma}{\partial x^{2}}  2x^{2}\sigma = 0. \,
The particular section σ: R^{2} → R^{2} × R defined by

\sigma(p_{1},p_{2}) = (p^{1},p^{2},p^{1}(p^{2})^{2}) \,
has first prolongation given by

j^{1}\sigma(p_{1},p_{2}) = (p^{1},p^{2},p^{1}(p^{2})^{2},(p^{2})^{2},2p^{1}p^{2}) \,
and is a solution of this differential equation, because

(u^{1}_{1}u^{1}_{2}  2x^{2}u^{1})(j^{1}_{p}\sigma) \,

= u^{1}_{1}(j^{1}_{p}\sigma)u^{1}_{2}(j^{1}_{p}\sigma)  2x^{2}(j^{1}_{p}\sigma)u^{1}(j^{1}_{p}\sigma) \,


= (p^{2})^{2} \cdot 2p^{1}p^{2}  2 \cdot p^{2} \cdot p^{1}(p^{2})^{2} \,


= 2p^{1}(p^{2})^3  2p^{1}(p^{2})^3 \,


= 0 \,

and so j^{1}_{p}\sigma \in S for every p ∈ R^{2}.
Jet Prolongation
A local diffeomorphism ψ: J^{r}(π) → J^{r}(π) defines a contact transformation of order r if it preserves the contact ideal, meaning that if θ is any contact form on J^{r}(π), then ψ*θ is also a contact form.
The flow generated by a vector field V^{r} on the jet space J^{r}(π) forms a oneparameter group of contact transformations if and only if the Lie derivative \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 field V^{1} on J^{1}(π), given by

V^1 \ \stackrel{\mathrm{def}}{=}\ \rho^{i}(u^{1})\frac{\partial}{\partial x^{i}} + \phi^{\alpha}(u^{1})\frac{\partial}{\partial u^{\alpha}} + \chi^{\alpha}_{i}(u^{1})\frac{\partial}{\partial u^{\alpha}_{i}}.
We now apply \mathcal{L}_{V^{1}} to the basic contact forms \theta^{\alpha} = du^{\alpha}  u_{i}^{\alpha}dx^{i}\,, and obtain

\mathcal{L}_{V^{1}}(\theta^{\alpha})

= \mathcal{L}_{V^{1}}(du^{\alpha}  u_{i}^{\alpha}dx^{i})


= \mathcal{L}_{V^{1}}du^{\alpha}  (\mathcal{L}_{V^{1}}u_{i}^{\alpha})dx^{i}  u_{i}^{\alpha}(\mathcal{L}_{V^{1}}dx^{i}) \,


= d(V^{1}u^{\alpha})  V^{1}u_{i}^{\alpha}dx^{i}  u_{i}^{\alpha}d(V^{1}x^{i}) \,


= d\phi^{\alpha}  \chi^{\alpha}_{i}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^{k}_{i}}\, du^{k}_{i}  \chi^{\alpha}_{i}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^{k}_{m}}\, du^{k}_{m} \right ] \,

where we have expanded the exterior derivative of the functions in terms of their coordinates. Next, we note that

\theta^{k} = du^{k}  u_{i}^{k}dx^{i} \quad \Longrightarrow \quad du^{k} = \theta^{k} + u_{i}^{k}dx^{i} \,
and so we may write

\mathcal{L}_{V^{1}}(\theta^{\alpha}) \,

= \frac{\partial \phi^{\alpha}}{\partial x^{i}}\, dx^{i} + \frac{\partial \phi^{\alpha}}{\partial u^{k}}\, (\theta^{k} + u_{i}^{k}dx^{i}) + \frac{\partial \phi^{\alpha}}{\partial u^{k}_{i}}\, du^{k}_{i}  \chi^{\alpha}_{i}dx^{i}  \,


u_{l}^{\alpha} \left[ \frac{\partial \rho^{l}}{\partial x^{i}}\, dx^{i} + \frac{\partial \rho^{l}}{\partial u^{k}}\, (\theta^{k} + u_{i}^{k}dx^{i}) + \frac{\partial \rho^{l}}{\partial u^{k}_{i}}\, du^{k}_{i} \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^{\alpha}_{i}\right]\, dx^{i} + \left[ \frac{\partial \phi^{\alpha}}{\partial u^{k}_{i}}  u_{l}^{\alpha}\frac{\partial \rho^{l}}{\partial u^{k}_{i}}\right]\, du^{k}_{i} + \,


+ \left( \frac{\partial \phi^{\alpha}}{\partial u^{k}}  u_{l}^{\alpha}\frac{\partial \rho^{l}}{\partial u^{k}} \right)\theta^{k}.\,

Therefore, V^{1} determines a contact transformation if and only if the coefficients of dx^{i} and du^{k}_{i}\, in the formula vanish. The latter requirements imply the contact conditions

\frac{\partial \phi^{\alpha}}{\partial u^{k}_{i}}  u^{\alpha}_{l} \frac{\partial \rho^{l}}{\partial u^{k}_{i}} = 0\,
The former requirements provide explicit formulae for the coefficients of the first derivative terms in V^{1}:

\chi^{\alpha}_{i} = \widehat{D}_{i} \phi^{\alpha}  u^{\alpha}_{l}(\widehat{D}_{i}\rho^{l})
where

\widehat{D}_{i} = \frac{\partial}{\partial x^{i}} + u^{k}_{i}\frac{\partial}{\partial u^{k}}
denotes the zeroth order truncation of the total derivative D_{i}.
Thus, the contact conditions uniquely prescribe the prolongation of any point or contact vector field. That is, if \mathcal{L}_{V^{r}}\, satisfies these equations, V^{r} is called the rth prolongation of V to a vector field on J^{r}(π).
These results are best understood when applied to a particular example. Hence, let us examine the following.
Example
Let us consider the case (E, π, M), where E ≅ R^{2} and M ≃ R. Then, (J^{1}(π), π, E) defines the first jet bundle, and may be coordinated by (x, u, u_{1}), where

x(j^{1}_{p}\sigma) \,

= x(p) = x \,

u(j^{1}_{p}\sigma) \,

= u(\sigma(p)) = u(\sigma(x)) = \sigma(x) \,

u_{1}(j^{1}_{p}\sigma) \,

= \left.\frac{\partial \sigma}{\partial x}\right_{p} = \dot{\sigma}(x) \,

for all p ∈ M and σ in Γ_{p}(π). A contact form on J^{1}(π) has the form

\theta = du  u_{1}dx \,
Let us consider a vector V on E, having the form

V = x \frac{\partial}{\partial u}  u \frac{\partial}{\partial x} \,
Then, the first prolongation of this vector field to J^{1}(π) is

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}} \,

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

\mathcal{L}_{V^{1}}(\theta) \,

= \mathcal{L}_{V^{1}}(du  u_{1}dx) \,


= \mathcal{L}_{V^{1}}du  (\mathcal{L}_{V^{1}}u_{1})dx  u_{1}(\mathcal{L}_{V^{1}}dx) \,


= d(V^{1}u)  V^{1}u_{1}dx  u_{1}d(V^{1}x) \,


= dx  \rho(x,u,u_{1})dx + u_{1}du \,


= (1  \rho(x,u,u_{1}) )dx + u_{1}du \,

But, we may identify du = θ + u_{1}dx. Thus, we get

\mathcal{L}_{V^{1}}(\theta) \,

= [\,1  \rho(x,u,u_{1})\,]dx + u_{1}(\theta + u_{1}dx) \,


= [\,1 + u_{1}u_{1}  \rho(x,u,u_{1})\,]dx + u_{1}\theta \,

Hence, for \mathcal{L}_{V^{1}}(\theta)\, to preserve the contact ideal, we require


1 + u_{1}u_{1}  \rho(x,u,u_{1}) = 0 \,

\Longrightarrow \quad \,

\rho(x,u,u_{1}) = 1 + u_{1}u_{1}\,

And so the first prolongation of V to a vector field on J^{1}(π) is

V^{1} = x \frac{\partial}{\partial u}  u \frac{\partial}{\partial x} + (1 + u_{1}u_{1})\frac{\partial}{\partial u_{1}} \,
Let us also calculate the second prolongation of V to a vector field on J^{2}(π). We have \{x,u,u_{1}, y_{2}\}\, as coordinates on J^{2}(π). Hence, the prolonged vector has the form

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}} \,
The contacts forms are

\theta \,

= du  u_{1}dx \,

\theta_{1} \,

= du_{1}  u_{2}dx \,

To preserve the contact ideal, we require

\mathcal{L}_{V^{2}}(\theta) \,

= 0\,

\mathcal{L}_{V^{2}}(\theta_{1}) \,

= 0 \,

Now, θ has no u_{2} dependency. Hence, from this equation we will pick up the formula for ρ, which will necessarily be the same result as we found for V^{1}. Therefore, the problem is analogous to prolonging the vector field V^{1} to J^{2}(π). That is to say, we may generate the rth 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

\rho(x,u,u_{1}) = 1 + u_{1}u_{1} \,
and so

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}} \,

Therefore, the Lie derivative of the second contact form with respect to V^{2} is

\mathcal{L}_{V^{2}}(\theta_{1}) \,

= \mathcal{L}_{V^{2}}(du_{1}  u_{2}dx) \,


= \mathcal{L}_{V^{2}}du_{1}  (\mathcal{L}_{V^{2}}u_{2})dx  u_{2}(\mathcal{L}_{V^{2}}dx) \,


= d(V^{2}u_{1})  V^{2}u_{2}dx  u_{2}d(V^{2}x) \,


= d(1u_{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 \,

Again, let us identify du = θ + u_{1}dx and du_{1} = θ_{1} + u_{2}dx. Then we have

\mathcal{L}_{V^{2}}(\theta_{1}) \,

= 2u_{1}(\theta_{1} + u_{2}dx)  \phi(x,u,u_{1},u_{2})dx + u_{2}(\theta + u_{1}dx) \,


= [\, 3u_{1}u_{2}  \phi(x,u,u_{1},u_{2})\,]dx + u_{2}\theta + 2u_{1}\theta_{1} \,

Hence, for \mathcal{L}_{V^{2}}(\theta_{1})\, to preserve the contact ideal, we require


3u_{1}u_{2}  \phi(x,u,u_{1},u_{2}) = 0 \,

\Longrightarrow \quad \,

\phi(x,u,u_{1},u_{2}) = 3u_{1}u_{2} \,

And so the second prolongation of V to a vector field on J^{2}(π) is

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}} \,
Note that the first prolongation of V can be recovered by omitting the second derivative terms in V^{2}, or by projecting back to J^{1}(π).
Infinite Jet Spaces
The inverse limit of the sequence of projections \pi_{k+1,k}:J^{k+1}(\pi)\to J^k(\pi) gives rise to the infinite jet space J^{∞}(π). A point j_p^\infty(\sigma) is the equivalence class of sections of π that have the same kjet in p as σ for all values of k. The natural projection π_{∞} maps j_p^\infty(\sigma) into p.
Just by thinking in terms of coordinates, J^{∞}(π) appears to be an infinitedimensional geometric object. In fact, the simplest way of introducing a differentiable structure on J^{∞}(π), not relying on differentiable charts, is given by the differential calculus over commutative algebras. Dual to the sequence of projections \pi_{k+1,k}:J^{k+1}(\pi)\to J^k(\pi) of manifolds is the sequence of injections \pi_{k+1,k}^*: C^\infty(J^{k}(\pi))\to C^\infty(J^{k+1}(\pi)) of commutative algebras. Let's denote C^\infty(J^{k}(\pi)) simply by \mathcal{F}_k(\pi). Take now the direct limit \mathcal{F}(\pi) of the \mathcal{F}_k(\pi)'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 \mathcal{F}(\pi), being born as a direct limit, carries an additional structure: it is a filtered commutative algebra.
Roughly speaking, a concrete element \varphi\in\mathcal{F}(\pi) will always belong to some \mathcal{F}_k(\pi), so it is a smooth function on the finitedimensional manifold J^{k}(π) in the usual sense.
Infinitely prolonged PDEs
Given a kth order system of PDEs E ⊆ J^{k}(π), the collection I(E) of vanishing on E smooth functions on J^{∞}(π) is an ideal in the algebra \mathcal{F}_k(\pi), and hence in the direct limit \mathcal{F}(\pi) too.
Enhance I(E) by adding all the possible compositions of total derivatives applied to all its elements. This way we get a new ideal I of \mathcal{F}(\pi) 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 j_p^\infty(\sigma) of E_{(∞)} can be easily seen to be represented by a section σ whose kjet's graph is tangent to E at the point j_p^k(\sigma) 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 Taylor series of \varphi\circ j^k(\sigma) at the point p.
Most importantly, the closure properties of I imply that E_{(∞)} is tangent to the infiniteorder contact structure \mathcal{C} on J^{∞}(π), so that by restricting \mathcal{C} to E_{(∞)} one gets the diffiety (E_{(\infty)}, \mathcal{C}_{E_{(\infty)}}), and can study the associated Cspectral sequence.
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

gr_{f}: M → M × N

gr_{f}(p) = (p, f(p))
(gr_{f} is known as the graph of the function f) of the trivial bundle (M × N, π_{1}, M). However, this restriction does not simplify the theory, as the global triviality of π does not imply the global triviality of π_{1}.
See also
References

Ehresmann, C., "Introduction à la théorie des structures infinitésimales et des pseudogroupes de Lie." Geometrie Differentielle, Colloq. Inter. du Centre Nat. de la Recherche Scientifique, Strasbourg, 1953, 97127.

Kolář, I., Michor, P., Slovák, J., Natural operations in differential geometry. SpringerVerlag: Berlin Heidelberg, 1993. ISBN 3540562354, ISBN 0387562354.

Saunders, D. J., "The Geometry of Jet Bundles", Cambridge University Press, 1989, ISBN 0521369487

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, ISBN 082180958X.

Olver, P. J., "Equivalence, Invariants and Symmetry", Cambridge University Press, 1995, ISBN 0521478111

Giachetta, G., Mangiarotti, L., Sardanashvily, G., "Advanced Classical Field Theory", World Scientific, 2009, ISBN 9789812838957

Sardanashvily, G., Advanced Differential Geometry for Theoreticians. Fiber bundles, jet manifolds and Lagrangian theory", Lambert Academic Publishing, 2013, ISBN 9783659378157; arXiv: 0908.1886
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