In differential topology, 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.

- Jets
- Jet manifolds
- Jet bundles
- Algebraic-geometric perspective
- Example
- Contact structure
- Example 2
- Vector fields
- Partial differential equations
- Example 3
- Jet prolongation
- Example 4
- Infinite jet spaces
- Infinitely prolonged PDEs
- Remark
- See also
- References
- Further reading

Historically, jet bundles are attributed to Charles 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.)

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 calculus of variations.^{ [1] } 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.

Suppose *M* is an *m*-dimensional manifold and that (*E*, π, *M*) is a fiber bundle. For *p* ∈ *M*, let Γ(p) denote the set of all local sections whose domain contains *p*. Let be a multi-index (an *m*-tuple of integers, not necessarily in ascending order), then define:

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

The relation that two maps have the same *r*-jet is an equivalence relation. An *r*-jet is an equivalence class under this relation, and the *r*-jet with representative σ is denoted . The integer *r* is also called the **order** of the jet, *p* is its **source** and σ(*p*) is its **target**.

The ** r-th jet manifold of π** is the set

We may define projections *π _{r}* and

If 1 ≤ *k* ≤ *r*, then the ** k-jet projection** is the function

From this definition, it is clear that *π _{r}* =

The functions *π _{r,k}*,

A coordinate system on *E* will generate a coordinate system on *J ^{r}*(

where

and the functions known as the **derivative coordinates**:

Given an atlas of adapted charts (*U*, *u*) on *E*, the corresponding collection of charts (*U ^{r}*,

Since the atlas on each defines a manifold, the triples *, ** and ** all define fibered manifolds. In particular, if **is a fiber bundle, the triple ** defines the *** r-th jet bundle of π**.

If *W* ⊂ *M* is an open submanifold, then

If *p* ∈ *M*, then the fiber is denoted .

Let σ be a local section of π with domain *W* ⊂ *M*. The ** r-th jet prolongation of σ** is the map defined by

Note that , so really is a section. In local coordinates, is given by

We identify * with .*

An independently motivated construction of the sheaf of sections * is given*.

*Consider a diagonal map , where the smooth manifold is a locally ringed space by for each open . Let be the ideal sheaf of , equivalently let be the sheaf of smooth germs which vanish on for all . The pullback of the quotient sheaf from to by is the sheaf of k-jets. ^{ [2] }*

*The direct limit of the sequence of injections given by the canonical inclusions of sheaves, gives rise to the infinite jet sheaf. Observe that by the direct limit construction it is a filtered ring.*

*If π is the trivial bundle ( M × R, pr_{1}, M), then there is a canonical diffeomorphism between the first jet bundle and T*M × R. To construct this diffeomorphism, for each σ in write .*

*Then, whenever p ∈ M*

*Consequently, the mapping*

*is well-defined 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 and R × TM.*

*The space J^{r}(π) carries a natural distribution, that is, a sub-bundle 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 one-forms called contact forms, on J^{r}(π). The space of differential one-forms on J^{r}(π) is denoted by and the space of contact forms is denoted by . A one form is a contact form provided its pullback along every prolongation is zero. In other words, is a contact form if and only if*

*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 non-integrable. 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 finite-dimensional: its dimension coincides with the dimension of the base manifold M.*

*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*

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

*A section σ in Γ _{p}(π) has first prolongation*

*Hence, (j^{1}σ)*θ can be calculated as*

*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*

*a general 1-form has the construction*

*This is a contact form if and only if*

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

*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 pull-back 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*

*where*

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

*In local coordinates, every contact one-form on J^{r+1}(π) can be written as a linear combination*

*with smooth coefficients of the basic contact forms*

*|I|* is known as the **order** of the contact form . Note that contact forms on *J ^{r+1}(π)* have orders at most

*Let ψ ∈ Γ _{W}(π_{r+1}), then ψ = j^{r+1}σ where σ ∈ Γ_{W}(π) if and only if *

*A general vector field on the total space E, coordinated by , is*

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

*A vector field is called vertical, meaning that all the horizontal coefficients vanish, if ρ^{i} = 0.*

*For fixed (x, u), we identify*

*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*

*is called a vector field on E with*

*and ψ in Γ(TE).*

*The jet bundle J^{r}(π) is coordinated by . For fixed (x, u, w), identify*

*having coordinates*

*with an element in the fiber of TJ^{r}(π) over (x, u, w) ∈ J^{r}(π), called a tangent vector in TJ^{r}(π). Here,*

*are real-valued functions on J^{r}(π). A section*

*is a vector field on J^{r}(π), and we say *

*Let (E, π, M) be a fiber bundle. An r-th 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 , for all p in M.*

*Consider an example of a first order partial differential equation.*

*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*

*gives rise to the differential equation*

*which can be written*

*The particular*

*has first prolongation given by*

*and is a solution of this differential equation, because*

*and so for everyp ∈ R^{2}.*

*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 one-parameter group of contact transformations if and only if the Lie derivative 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*

*We now apply to the basic contact forms and expand the exterior derivative of the functions in terms of their coordinates to obtain:*

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

*The former requirements provide explicit formulae for the coefficients of the first derivative terms in V^{1}:*

*where*

*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 satisfies these equations, V^{r} is called the r-th 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.*

*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*

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

*Consider a vector V on E, having the form*

*Then, the first prolongation of this vector field to J^{1}(π) is*

*If we now take the Lie derivative of the contact form with respect to this prolonged vector field, we obtain*

*Hence, for preservation of the contact ideal, we require*

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

*Let us also calculate the second prolongation of V to a vector field on J^{2}(π). We have as coordinates on J^{2}(π). Hence, the prolonged vector has the form*

*The contact forms are*

*To preserve the contact ideal, we require*

*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 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*

*and so*

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

*Hence, for to preserve the contact ideal, we require*

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

*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).*

*The inverse limit of the sequence of projections gives rise to the infinite jet spaceJ^{∞}(π). A point 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 into p.*

*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 differential calculus over commutative algebras. Dual to the sequence of projections of manifolds is the sequence of injections of commutative algebras. Let's denote simply by . Take now the direct limit of the '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 , being born as a direct limit, carries an additional structure: it is a filtered commutative algebra.*

*Roughly speaking, a concrete element will always belong to some , so it is a smooth function on the finite-dimensional manifold J^{k}(π) in the usual sense.*

*Given a k-th 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 , and hence in the direct limit 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 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 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 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 at the point p.*

*Most importantly, the closure properties of I imply that E_{(∞)} is tangent to the infinite-order contact structure on J^{∞}(π), so that by restricting to E_{(∞)} one gets the diffiety , and can study the associated Vinogradov (C-spectral) 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}.*

*In physics, the cross section is a measure of the probability that a specific process will take place when some kind of radiant excitation intersects a localized phenomenon. For example, the Rutherford cross-section is a measure of probability that an alpha-particle will be deflected by a given angle during a collision with an atomic nucleus. Cross section is typically denoted σ (sigma) and is expressed in units of area, more specific in barns. In a way, it can be thought of as the size of the object that the excitation must hit in order for the process to occur, but more exactly, it is a parameter of a stochastic process.*

*In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function of the form*

**Linear elasticity** is a mathematical model of how solid objects deform and become internally stressed due to prescribed loading conditions. It is a simplification of the more general nonlinear theory of elasticity and a branch of continuum mechanics.

*In mathematics, the Mellin transform is an integral transform that may be regarded as the multiplicative version of the two-sided Laplace transform. This integral transform is closely connected to the theory of Dirichlet series, and is often used in number theory, mathematical statistics, and the theory of asymptotic expansions; it is closely related to the Laplace transform and the Fourier transform, and the theory of the gamma function and allied special functions.*

*In information geometry, the Fisher information metric is a particular Riemannian metric which can be defined on a smooth statistical manifold, i.e., a smooth manifold whose points are probability measures defined on a common probability space. It can be used to calculate the informational difference between measurements.*

In mathematics, the **Radon transform** is the integral transform which takes a function *f* defined on the plane to a function *Rf* defined on the (two-dimensional) space of lines in the plane, whose value at a particular line is equal to the line integral of the function over that line. The transform was introduced in 1917 by Johann Radon, who also provided a formula for the inverse transform. Radon further included formulas for the transform in three dimensions, in which the integral is taken over planes. It was later generalized to higher-dimensional Euclidean spaces, and more broadly in the context of integral geometry. The complex analogue of the Radon transform is known as the Penrose transform. The Radon transform is widely applicable to tomography, the creation of an image from the projection data associated with cross-sectional scans of an object.

The **Gaussian integral**, also known as the **Euler–Poisson integral**, is the integral of the Gaussian function over the entire real line. Named after the German mathematician Carl Friedrich Gauss, the integral is

**Etendue** or **étendue** is a property of light in an optical system, which characterizes how "spread out" the light is in area and angle. It corresponds to the beam parameter product (BPP) in Gaussian beam optics.

*In Bayesian probability, the Jeffreys prior, named after Sir Harold Jeffreys, is a non-informative (objective) prior distribution for a parameter space; its density function is proportional to the square root of the determinant of the Fisher information matrix:*

*In general relativity, the Gibbons–Hawking–York boundary term is a term that needs to be added to the Einstein–Hilbert action when the underlying spacetime manifold has a boundary.*

*In calculus, the Leibniz integral rule for differentiation under the integral sign, named after Gottfried Leibniz, states that for an integral of the form*

*A ratio distribution is a probability distribution constructed as the distribution of the ratio of random variables having two other known distributions. Given two random variables X and Y, the distribution of the random variable Z that is formed as the ratio Z = X/Y is a ratio distribution.*

**Bending of plates**, or **plate bending**, refers to the deflection of a plate perpendicular to the plane of the plate under the action of external forces and moments. The amount of deflection can be determined by solving the differential equations of an appropriate plate theory. The stresses in the plate can be calculated from these deflections. Once the stresses are known, failure theories can be used to determine whether a plate will fail under a given load.

*A product distribution is a probability distribution constructed as the distribution of the product of random variables having two other known distributions. Given two statistically independent random variables X and Y, the distribution of the random variable Z that is formed as the product*

The **Uflyand-Mindlin theory** of vibrating plates is an extension of Kirchhoff–Love plate theory that takes into account shear deformations through-the-thickness of a plate. The theory was proposed in 1948 by Yakov Solomonovich Uflyand (1916-1991) and in 1951 by Raymond Mindlin with Mindlin making reference to Uflyand's work. Hence, this theory has to be referred to us Uflyand-Mindlin plate theory, as is done in the handbook by Elishakoff, and in papers by Andronov, Elishakoff, Hache and Challamel, Loktev, Rossikhin and Shitikova and Wojnar. In 1994, Elishakoff suggested to neglect the fourth-order time derivative in Uflyand-Mindlin equations. A similar, but not identical, theory in static setting, had been proposed earlier by Eric Reissner in 1945. Both theories are intended for thick plates in which the normal to the mid-surface remains straight but not necessarily perpendicular to the mid-surface. The Uflyand-Mindlin theory is used to calculate the deformations and stresses in a plate whose thickness is of the order of one tenth the planar dimensions while the Kirchhoff–Love theory is applicable to thinner plates.

In continuum mechanics, Whitham's **averaged Lagrangian** method – or in short **Whitham's method** – is used to study the Lagrangian dynamics of slowly-varying wave trains in an inhomogeneous (moving) medium. The method is applicable to both linear and non-linear systems. As a direct consequence of the averaging used in the method, wave action is a conserved property of the wave motion. In contrast, the wave energy is not necessarily conserved, due to the exchange of energy with the mean motion. However the total energy, the sum of the energies in the wave motion and the mean motion, will be conserved for a time-invariant Lagrangian. Further, the averaged Lagrangian has a strong relation to the dispersion relation of the system.

*In fluid dynamics, Taylor scraping flow is a type of two-dimensional corner flow occurring when one of the wall is sliding over the other with constant velocity, named after G. I. Taylor.*

*In representation theory of mathematics, the Waldspurger formula relates the special values of two L-functions of two related admissible irreducible representations. Let k be the base field, f be an automorphic form over k, π be the representation associated via the Jacquet–Langlands correspondence with f. Goro Shimura (1976) proved this formula, when and f is a cusp form; Günter Harder made the same discovery at the same time in an unpublished paper. Marie-France Vignéras (1980) proved this formula, when { and f is a newform. Jean-Loup Waldspurger, for whom the formula is named, reproved and generalized the result of Vignéras in 1985 via a totally different method which was widely used thereafter by mathematicians to prove similar formulas.*

**Magnetic topological insulators** are three dimensional magnetic materials with a non-trivial topological index protected by a symmetry other than time-reversal. In contrast with a non-magnetic topological insulator, a magnetic topological insulator can have naturally gapped surface states as long as the quantizing symmetry is broken at the surface. These gapped surfaces exhibit a topologically protected half-quantized surface anomalous Hall conductivity perpendicular to the surface. The sign of the half-quantized surface anomalous Hall conductivity depends on the specific surface termination.

*↑ Krupka, Demeter (2015).**Introduction to Global Variational Geometry*. Atlantis Press. ISBN 978-94-6239-073-7.*↑ Vakil, Ravi (August 25, 1998). "A beginner's guide to jet bundles from the point of view of algebraic geometry" (PDF). Retrieved June 25, 2017.*

*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.*Kolář, I., Michor, P., Slovák, J.,**Natural operations in differential geometry.*Springer-Verlag: Berlin Heidelberg, 1993. ISBN 3-540-56235-4, ISBN 0-387-56235-4.*Saunders, D. J., "The Geometry of Jet Bundles", Cambridge University Press, 1989, ISBN 0-521-36948-7**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 0-8218-0958-X.**Olver, P. J., "Equivalence, Invariants and Symmetry", Cambridge University Press, 1995, ISBN 0-521-47811-1**Giachetta, G., Mangiarotti, L., Sardanashvily, G., "Advanced Classical Field Theory", World Scientific, 2009, ISBN 978-981-283-895-7**Sardanashvily, G., Advanced Differential Geometry for Theoreticians. Fiber bundles, jet manifolds and Lagrangian theory", Lambert Academic Publishing, 2013, ISBN 978-3-659-37815-7; arXiv : 0908.1886*

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