Jet bundle and total derivative

Intuition Beginner

Suppose you want to track not just where a function sends a point, but also how fast it is changing there, and how fast that rate is changing, and so on. A jet is the bookkeeping device that records a function together with a finite stack of its derivatives at a single point. Two functions have the same jet of order two at a point when they agree in value, in first derivative, and in second derivative there. The jet forgets everything beyond that stack.

Why bother? Differential equations are conditions on a function and its derivatives. If you package the function-plus-derivatives into one geometric object, then a differential equation becomes an ordinary equation, a condition carving out a surface inside the space of all jets. The messy business of "this involves a second derivative here" turns into the clean business of "this point lies on that surface."

The total derivative is the tool that differentiates while staying inside this bookkeeping. When you move along the base and ask how a jet-quantity changes, the total derivative supplies the answer by promoting each recorded derivative to the next one up the stack. It is differentiation that knows the stack is there.

Visual Beginner

Picture a curve drawn over a horizontal axis. At one marked point, draw the tangent line: that is the first-order data. Then draw the small parabola that hugs the curve best at that point: that is the second-order data. The jet of order two is exactly this packet — the point, the tangent line, and the best-fitting parabola — with the rest of the curve erased.

The picture records the central idea: many different curves share the same tangent line and the same best-fitting parabola at the marked point, and the jet is precisely what they have in common. Sliding the marked point along the axis sweeps out a whole family of these packets, one over each base point, and that family is the jet bundle.

Worked example Beginner

Take the function that sends a number to its cube, and look at the point where the input equals two. The value there is eight, because two cubed is eight. The first derivative of the cubing function is three times the input squared, which at two equals three times four, namely twelve. The second derivative is six times the input, which at two equals twelve.

So the second-order jet of the cubing function at input two is the packet of three numbers: value eight, first derivative twelve, second derivative twelve. Any other function that also passes through eight at input two with the same first and second derivatives shares this jet.

Now test the total-derivative idea by hand. The first derivative, as a function of the input, is three times the input squared. Differentiate it once more and you get six times the input — which is the second derivative. The total derivative is the rule that says: to advance the stack, take the recorded first-derivative function and produce the recorded second-derivative function from it.

What this tells us: a jet is a short list of numbers read off a function at one point, and the total derivative is the rule that walks up that list by ordinary differentiation, never leaving the bookkeeping.

Check your understanding Beginner

Formal definition Intermediate+

Let $\pi :E\to X$ be a smooth fibred manifold with $\dim X=p$ and fibre dimension $q$, so that local coordinates split as $(x^i,u^{\alpha })$ with $1\le i\le p$ and $1\le \alpha \le q$. In the model case $E=X\times U$ a section is a smooth map $f:X\to U$, written $u=f(x)$. The definition follows Olver ^{[Olver Ch. 2]} and Saunders ^{[Saunders Ch. 4]}.

Two local sections $f,g$ defined near $x_0\in X$ are $k$-equivalent at $x_0$ when $f(x_0)=g(x_0)$ and all partial derivatives of $f$ and $g$ up to order $k$ agree at $x_0$. This is an equivalence relation; its class is the $k$-jet $j_{x_0}^{k}f$. The set of all $k$-jets of sections of $\pi$, as the base point and the section vary, is the $k$-jet bundle $J^k(E)=J^k(X,U)$.

A multi-index $J=(j_1,\dots ,j_p)$ with $|J|=j_1+\cdots +j_p$ records a partial-derivative order, and ${\partial }_J{f}^{\alpha }:={\partial }^{|J|}{f}^{\alpha }/\partial x_1^{j_1}\cdots \partial x_p^{j_p}$. The induced jet coordinates on $J^k(E)$ are

$$(x^i,u^{\alpha },u_J^{\alpha }),1\le |J|\le k,$$

where the coordinate $u_J^{\alpha }$ evaluated on $j_x^{k}f$ returns ${\partial }_J{f}^{\alpha }(x)$. The projection ${\pi }^k:J^k(E)\to X$ remembers only the base point; the projections ${\pi }_l^k:J^k(E)\to J^l(E)$ for $l<k$ truncate the derivative stack at order $l$.

Definition (prolongation of a section). The $k$-th prolongation of a section $f$ is the section $$ \mathrm{pr}^{(k)} f : X \to J^k(E), \qquad x \mapsto j^k_x f, $$ whose coordinate components are $u_J^{\alpha }\circ {\mathrm{pr}}^{(k)}f={\partial }_J{f}^{\alpha }$. A section of ${\pi }^k:J^k(E)\to X$ is holonomic when it equals ${\mathrm{pr}}^{(k)}f$ for some section $f$ of $\pi$; equivalently, its component functions are the iterated derivatives of its order-zero part.

Definition (contact forms). On $J^k(E)$ the contact (Cartan) forms are $$ \theta^\alpha_J := du^\alpha_J - u^\alpha_{J,i}, dx^i, \qquad |J| \le k - 1, $$ where $J,i$ denotes the multi-index $J$ with its $i$-th entry raised by one, and summation over the repeated index $i$ is implied. They generate the contact ideal ${\mathcal{C}}^k\subset {\Omega }^{\bullet }(J^k(E))$.

Definition (total derivative). The total derivative in the $x^i$ direction is the vector field on the infinite jet (truncated to whatever order is needed) $$ D_i := \frac{\partial}{\partial x^i} + \sum_{\alpha, J} u^\alpha_{J,i}, \frac{\partial}{\partial u^\alpha_J}, $$ the sum running over fibre indices $\alpha$ and multi-indices $J$. For a multi-index $J=(j_1,\dots ,j_p)$ write $D_J:=D_1^{j_1}\cdots D_p^{j_p}$; the operators $D_i$ commute, so $D_J$ is well-defined.

A non-example worth recording: the operator $\partial /\partial x^i$ alone is not the geometrically correct derivative along a prolonged section, because it ignores how the recorded derivatives $u_J^{\alpha }$ themselves change. The total derivative repairs this by adding the terms that advance each $u_J^{\alpha }$ to $u_{J,i}^{\alpha }$.

Counterexamples to common slips

• A section of $J^k(E)\to X$ need not be holonomic: assigning $u_x$ a value unrelated to ${\partial }_{x}u$ gives a perfectly smooth section of the jet bundle that is not the prolongation of anything. Holonomy is the contact condition ${\theta }_J^{\alpha }=0$, not an automatic feature.
• The total derivative $D_i$ is not a vector field on a finite jet bundle $J^k(E)$: applied to a coordinate $u_J^{\alpha }$ with $|J|=k$ it produces $u_{J,i}^{\alpha }$ of order $k+1$, which lives one level up. It is honestly a vector field only on $J^{k+1}(E)$ acting on functions pulled back from $J^k(E)$, or on the infinite jet $J^{\infty }(E)$.
• "Order $k$" counts derivatives including none: the value $u^{\alpha }$ is the $|J|=0$ datum. Off-by-one errors here corrupt the dimension count $\dim J^k(E)=p+q\left(\genfrac{}{}{0ex}{}{p+k}{k}\right)$.

Key theorem with proof Intermediate+

Theorem (total derivative computes prolonged derivatives). Let $f$ be a smooth section of $\pi :E\to X$ and let $P(x,u^{(k)})$ be a smooth function on $J^k(E)$, where $u^{(k)}$ denotes all jet coordinates up to order $k$. Then for each $i$, $$ \frac{\partial}{\partial x^i}\Bigl[ P\bigl(x, \mathrm{pr}^{(k)} f(x)\bigr) \Bigr] = \bigl(D_i P\bigr)\bigl(x, \mathrm{pr}^{(k+1)} f(x)\bigr). $$ In particular, taking $P=u_J^{\alpha }$ gives ${\partial }_i({\partial }_J{f}^{\alpha })=u_{J,i}^{\alpha }\circ {\mathrm{pr}}^{(k+1)}f$, the identity $D_i(\mathrm{pr}f)=\mathrm{pr}({\partial }_{i}f)$ in coordinate form.

Proof. Evaluate the composite $x\mapsto P(x,{\mathrm{pr}}^{(k)}f(x))$. By construction the jet coordinate $u_J^{\alpha }$ pulls back along ${\mathrm{pr}}^{(k)}f$ to the function ${\partial }_J{f}^{\alpha }$ on $X$. So the composite is the function $$ x \longmapsto P\bigl(x,; (\partial_J f^\alpha(x)){|J| \le k}\bigr). $$ Differentiate with respect to $x^i$ by the chain rule. The explicit $x^i$ dependence of $P$ contributes $\partial P/\partial x^i$. Each jet-coordinate slot $u_J^{\alpha }$ contributes $(\partial P/\partial u_J^{\alpha })$ times the $x^i$-derivative of the function occupying that slot, namely $\partial_i \partial_J f^\alpha = \partial{J,i} f^\alpha$.Thatlastquantityisexactlythevalueofthecoordinate$u^\alpha_{J,i}$on$\mathrm{pr}^{(k+1)} f$. Collecting, $$ \frac{\partial}{\partial x^i} P(x, \mathrm{pr}^{(k)} f) = \frac{\partial P}{\partial x^i} + \sum_{\alpha, J} \frac{\partial P}{\partial u^\alpha_J}, \bigl(u^\alpha_{J,i} \circ \mathrm{pr}^{(k+1)} f\bigr). $$ The right-hand side is the definition of $D_{i}P$ evaluated on ${\mathrm{pr}}^{(k+1)}f$, because $D_i={\partial }_{x^i}+\sum u_{J,i}^{\alpha }{\partial }_{u_J^{\alpha }}$. Since $P$ depends on jet coordinates only up to order $k$, the sum is finite and $D_{i}P$ is a well-defined function on $J^{k+1}(E)$. This proves the displayed identity; the special case $P=u_J^{\alpha }$ reads ${\partial }_i{\partial }_J{f}^{\alpha }=u_{J,i}^{\alpha }\circ {\mathrm{pr}}^{(k+1)}f$. $\square$

Bridge. This identity is the engine of jet geometry. It builds toward the prolongation formula for vector fields in the Advanced results, where the coefficient of ${\partial }_{u_J^{\alpha }}$ in a prolonged symmetry is computed as $D_J$ applied to a characteristic. It appears again in 05.00.04 (Noether's theorem), where the conserved current is extracted from a total-derivative divergence and the on-shell vanishing is read through the contact ideal. The same machinery underlies the cotangent-bundle action functional of 05.02.05, whose Euler-Lagrange derivation is a total-derivative integration by parts. The unifying content is that the total derivative is precisely the operator making the diagram "differentiate the base section, then prolong" agree with "prolong, then move by the total derivative" commute — and once that square commutes, every differential-equation condition becomes a contact-compatible condition on $J^k(E)$ that the prolongation formula can transport.

Exercises Intermediate+

Lean formalization Intermediate+

Mathlib has no jet-bundle object; the statement below is a skeleton fixing the intended signatures, with the gap detailed in Mathlib gap analysis.

import Mathlib.Geometry.Manifold.ContMDiff.Basic
import Mathlib.Analysis.Calculus.IteratedDeriv.Defs

variable {p q : ℕ}

/-- A multi-index of length `p`, recording a partial-derivative order. -/
abbrev MultiIndex (p : ℕ) := Fin p → ℕ

/-- Jet coordinates up to order `k`: base `x`, fibre `u`, derivatives `u_J`. -/
structure JetCoords (p q k : ℕ) where
  x  : Fin p → ℝ
  u  : Fin q → ℝ
  uJ : ∀ (J : MultiIndex p), (J.1.sum ≤ k) → (Fin q → ℝ)

/-- The total derivative `D_i` advances each coordinate `u_J` to `u_{J,i}`. -/
noncomputable def totalDeriv
    (i : Fin p) (P : JetCoords p q k → ℝ) : JetCoords p q (k+1) → ℝ :=
  fun _ => sorry  -- ∂P/∂xⁱ + Σ_{α,J} u^α_{J,i} · ∂P/∂u^α_J

/-- Key identity: D_i computes the xⁱ-derivative of P along a prolonged section. -/
theorem totalDeriv_prolong
    (f : (Fin p → ℝ) → (Fin q → ℝ)) (P : JetCoords p q k → ℝ) (i : Fin p) :
    True := by
  trivial  -- placeholder: ∂ᵢ(P ∘ pr^k f) = (D_i P) ∘ pr^{k+1} f

The sorry and placeholder stand on three Mathlib gaps: the jet manifold $J^k(E)$ as a bundle with adapted coordinates, the contact codistribution, and the total-derivative derivation together with the prolongation map ${\mathrm{pr}}^{(k)}$. All three sit directly on top of the existing ContMDiff and iteratedFDeriv libraries and are Mathlib-contribution-sized.

Advanced results Master

The total derivative organises the calculus of prolongations. The central operational result is the formula for prolonging a vector field, which converts an infinitesimal symmetry of $E$ into one of $J^k(E)$.

Prolongation of a vector field. Let $v={\xi }^i(x,u){\partial }_{x^i}+{\varphi }^{\alpha }(x,u){\partial }_{u^{\alpha }}$ be a vector field on $E$. Its $k$-th prolongation is the unique vector field $$ \mathrm{pr}^{(k)} v = \xi^i,\partial_{x^i} + \sum_{|J| \le k} \phi^\alpha_J,\partial_{u^\alpha_J} $$ on $J^k(E)$ whose flow is the prolonged flow of $v$, with coefficients given by Olver's formula ^{[Olver Ch. 2]} $$ \phi^\alpha_J = D_J!\left(\phi^\alpha - \xi^i u^\alpha_i\right) + \xi^i u^\alpha_{J,i}, \qquad Q^\alpha := \phi^\alpha - \xi^i u^\alpha_i, $$ where $Q^{\alpha }$ is the characteristic of $v$ and $D_J=D_1^{j_1}\cdots D_p^{j_p}$. The formula encodes that prolongation commutes with the total derivative: ${\varphi }_{J,i}^{\alpha }=D_i{\varphi }_J^{\alpha }-(D_i{\xi }^j)u_{J,j}^{\alpha }$, the recursion that one usually applies in practice.

Differential equations on the jet bundle. A system of $k$-th order PDEs is a closed embedded submanifold $\mathcal{S}\subset J^k(E)$ cut out by ${\Delta }_{\nu }(x,u^{(k)})=0$. A section $f$ solves the system when ${\mathrm{pr}}^{(k)}f(X)\subset \mathcal{S}$. The $l$-th prolongation ${\mathcal{S}}^{(l)}\subset J^{k+l}(E)$ is cut out by the original equations together with all their total derivatives $D_J{\Delta }_{\nu }=0$ for $|J|\le l$. A vector field $v$ is an infinitesimal symmetry of $\mathcal{S}$ when ${\mathrm{pr}}^{(k)}v$ is tangent to $\mathcal{S}$, i.e. ${\mathrm{pr}}^{(k)}v({\Delta }_{\nu })=0$ on $\mathcal{S}$. This is Lie's infinitesimal criterion, and it is a linear, algorithmic condition once the prolongation formula above is in hand.

The variational bicomplex. On the infinite jet $J^{\infty }(E)$ the de Rham complex splits into horizontal forms (built from $d{x}^i$) and vertical contact forms (built from ${\theta }_J^{\alpha }$), giving a bicomplex with horizontal differential $d_H=d{x}^i\wedge D_i$ and vertical differential $d_V$. The Euler operator (variational derivative) $$ \mathsf{E}\alpha(L) = \sum{J} (-D)_J,\frac{\partial L}{\partial u^\alpha_J}, \qquad (-D)J := (-1)^{|J|} D_J, $$ sends a Lagrangian density $L$ to its Euler-Lagrange expression, and the fundamental theorem of the variational bicomplex states that $\mathsf{E}\alpha(L) = 0$identicallyifandonlyif$L$isatotaldivergence$L = D_i R^i$[ref:TOD{O}_{R}EFAnderson].Thetotalderivativeisthedifferentialofthehorizontaledgecomplex;theEuleroperatoristheobstructiontohorizontalexactness.Vinogrado{v}^{\prime }s$\mathcal{C}$-spectral sequence ^{[Vinogradov 1984]} resolves this bicomplex systematically, producing conservation laws as cohomology classes.

Synthesis. The jet bundle converts analysis into geometry along three coupled axes, and the total derivative is the connective tissue of all three. First, it turns differentiation of base sections into an algebraic operation on jet coordinates, so that the identity $D_i(\mathrm{pr}f)=\mathrm{pr}({\partial }_{i}f)$ makes "the derivative of a solution" an intrinsic object. Second, it turns a differential equation into a subvariety $\mathcal{S}\subset J^k(E)$ whose prolongations ${\mathcal{S}}^{(l)}$ are generated by total derivatives, so that integrability and symmetry become tangency conditions computed by $D_J$. Third, it turns the calculus of variations into the horizontal edge of a bicomplex, with the Euler operator as the cohomological obstruction and conservation laws as horizontal-cohomology classes. Bringing these together, the prolongation formula ${\varphi }_J^{\alpha }=D_J{Q}^{\alpha }+{\xi }^i{u}_{J,i}^{\alpha }$ is one operator — the total derivative — applied to the characteristic, and it is the same operator that defines the contact ideal, prolongs the equation, and differentiates the Lagrangian; the unity of Lie symmetry analysis, formal integrability theory, and the variational bicomplex is the unity of $D_i$ acting in three guises.

Full proof set Master

Proposition (contact ideal characterises holonomic sections). A section $\sigma :X\to J^k(E)$ of ${\pi }^k$ is holonomic — equal to ${\mathrm{pr}}^{(k)}f$ for some section $f$ of $\pi$ — if and only if $\sigma^\theta^\alpha_J = 0$foreverycontactform$\theta^\alpha_J$with$|J| \le k-1$.*

Proof. Write $\sigma$ in coordinates as $x\mapsto (x,{\sigma }_J^{\alpha }(x))$ where ${\sigma }_J^{\alpha }$ is the function filling the $u_J^{\alpha }$ slot, with ${\sigma }^{\alpha }:={\sigma }_{\varnothing }^{\alpha }$ the order-zero part.

($\Rightarrow$) If $\sigma ={\mathrm{pr}}^{(k)}f$ then ${\sigma }_J^{\alpha }={\partial }_J{f}^{\alpha }$, so ${\sigma }^{\ast }(d{u}_J^{\alpha })=d({\partial }_J{f}^{\alpha })={\partial }_{J,i}{f}^{\alpha }d{x}^i$ and ${\sigma }^{\ast }(u_{J,i}^{\alpha }d{x}^i)={\partial }_{J,i}{f}^{\alpha }d{x}^i$. Hence ${\sigma }^{\ast }{\theta }_J^{\alpha }={\partial }_{J,i}{f}^{\alpha }d{x}^i-{\partial }_{J,i}{f}^{\alpha }d{x}^i=0$ for all $|J|\le k-1$.

($\Leftarrow$) Suppose ${\sigma }^{\ast }{\theta }_J^{\alpha }=0$ for all $|J|\le k-1$. Computing the pullback, $$ \sigma^*\theta^\alpha_J = d\sigma^\alpha_J - \sigma^\alpha_{J,i}, dx^i = \left(\frac{\partial \sigma^\alpha_J}{\partial x^i} - \sigma^\alpha_{J,i}\right) dx^i, $$ so vanishing forces $\partial {\sigma }_J^{\alpha }/\partial x^i={\sigma }_{J,i}^{\alpha }$ for every $i$ and every $|J|\le k-1$. Apply this with $|J|=0$: $\partial {\sigma }^{\alpha }/\partial x^i={\sigma }_i^{\alpha }$, so the order-one slots are the first derivatives of ${\sigma }^{\alpha }$. Set $f^{\alpha }:={\sigma }^{\alpha }$. Inductively, if ${\sigma }_J^{\alpha }={\partial }_J{f}^{\alpha }$ for all $|J|=m<k$, then for $|J|=m$ the relation gives ${\sigma }_{J,i}^{\alpha }={\partial }_i{\sigma }_J^{\alpha }={\partial }_i{\partial }_J{f}^{\alpha }={\partial }_{J,i}{f}^{\alpha }$, establishing the claim for all multi-indices of order $m+1\le k$. Hence ${\sigma }_J^{\alpha }={\partial }_J{f}^{\alpha }$ for all $|J|\le k$, which says $\sigma ={\mathrm{pr}}^{(k)}f$. $\square$

Proposition (prolongation respects total derivatives). The prolongation coefficients satisfy the recursion ${\varphi }_{J,i}^{\alpha }=D_i{\varphi }_J^{\alpha }-(D_i{\xi }^j)u_{J,j}^{\alpha }$, with base case ${\varphi }^{\alpha }={\varphi }^{\alpha }$ for $|J|=0$.

Proof. From Olver's formula ${\varphi }_J^{\alpha }=D_J{Q}^{\alpha }+{\xi }^j{u}_{J,j}^{\alpha }$ with $Q^{\alpha }={\varphi }^{\alpha }-{\xi }^j{u}_j^{\alpha }$. Apply $D_i$: $$ D_i \phi^\alpha_J = D_i D_J Q^\alpha + (D_i \xi^j) u^\alpha_{J,j} + \xi^j D_i u^\alpha_{J,j}. $$ The total derivatives commute, so $D_i{D}_J{Q}^{\alpha }=D_{J,i}{Q}^{\alpha }$, and $D_i{u}_{J,j}^{\alpha }=u_{J,j,i}^{\alpha }=u_{J,i,j}^{\alpha }$ by symmetry of the multi-index. Therefore $$ D_i \phi^\alpha_J = D_{J,i} Q^\alpha + \xi^j u^\alpha_{J,i,j} + (D_i \xi^j) u^\alpha_{J,j} = \phi^\alpha_{J,i} + (D_i \xi^j) u^\alpha_{J,j}, $$ using Olver's formula again at order $|J|+1$ to recognise ${\varphi }_{J,i}^{\alpha }=D_{J,i}{Q}^{\alpha }+{\xi }^j{u}_{J,i,j}^{\alpha }$. Rearranging gives the stated recursion. $\square$

Proposition (prolonged scaling field on $u_{xx}$). For $v=x{\partial }_x+u{\partial }_u$ on $J^0(\mathbb{R},\mathbb{R})$, the second prolongation is ${\mathrm{pr}}^{(2)}v=x{\partial }_x+u{\partial }_u-u_x{\partial }_{u_{xx}}$; in particular the $u_x$-coefficient vanishes and the $u_{xx}$-coefficient is $-u_{xx}$.

Proof. Here $\xi =x$, $\varphi =u$, characteristic $Q=u-x{u}_x$. The first-order coefficient is ${\varphi }^x=D_{x}Q+\xi u_{xx}$. Compute $D_{x}Q=D_x(u-x{u}_x)=u_x-(u_x+x{u}_{xx})=-x{u}_{xx}$, so ${\varphi }^x=-x{u}_{xx}+x{u}_{xx}=0$. For the second order, the recursion gives ${\varphi }^{xx}=D_x{\varphi }^x-(D_x\xi )u_{xx}=D_x(0)-(1)u_{xx}=-u_{xx}$. Hence ${\mathrm{pr}}^{(2)}v=x{\partial }_x+u{\partial }_u+0\cdot {\partial }_{u_x}-u_{xx}{\partial }_{u_{xx}}$. As a check on the equation $u_{xx}=u^3$: the symmetry condition ${\mathrm{pr}}^{(2)}v(u_{xx}-u^3)=-u_{xx}-3u^2\cdot u=-u^3-3u^3$, which is $-4u^3$; this is not a multiple of $u_{xx}-u^3$ vanishing on the equation, so plain scaling is not a symmetry of $u_{xx}=u^3$ — correctly diagnosing that $u_{xx}=u^3$ scales anisotropically and needs the weighted field $x{\partial }_x-u{\partial }_u$ instead. $\square$

Connections Master

The total derivative supplies the divergence operator in the geometric form of 05.00.04 (Noether's theorem): a variational symmetry with characteristic $Q^{\alpha }$ yields a conserved current whose vanishing divergence is a total-derivative identity on the jet bundle, and the Euler operator built from $D_J$ is what converts the symmetry condition into the conservation law.

The jet-bundle realisation of a differential equation as a submanifold $\mathcal{S}\subset J^k(E)$ specialises, in the symplectic setting, to the Hamilton-Jacobi equation of 05.05.04: the Hamilton-Jacobi PDE is a first-order equation on the jet bundle $J^1(X,\mathbb{R})$ whose contact structure recovers the canonical one-form, tying the contact ideal here to the Liouville form on the cotangent bundle.

The cotangent bundle of 05.02.05 is the prototype of a jet target: $J^1(X,\mathbb{R})$ projects to $T^{\ast }X\times \mathbb{R}$ with the contact form $du-p_{i}d{x}^i$, so the first-jet contact structure is the contactification of the canonical symplectic structure, and the total derivative restricts to the Reeb-transverse differentiation used in generating-function calculations.

Historical & philosophical context Master

Charles Ehresmann introduced jets in 1951 in a short note to the Comptes Rendus de l'Académie des Sciences (Les prolongements d'une variété différentiable, vol. 233) ^{[Ehresmann 1951]}, abstracting the local Taylor-coefficient data of a map into a coordinate-free geometric object and defining the jet bundles $J^k$ as the natural carriers of higher-order contact between submanifolds. The construction grew out of his programme on differentiable fibre bundles and connections; the contact (Cartan) forms trace back further to Élie Cartan's exterior-differential-systems treatment of partial differential equations from the 1900s-1920s.

Sophus Lie's nineteenth-century theory of continuous symmetry groups of differential equations was the motivating application, but Lie worked with prolongation formulae written out by hand in coordinates. The jet-bundle reformulation made Lie's infinitesimal criterion into a tangency condition on a manifold, and Peter Olver's 1986 monograph Applications of Lie Groups to Differential Equations (second edition 1993) ^{[Olver Ch. 2]} gave the modern algorithmic presentation of prolongation, the total derivative, and the characteristic, with the recursion that mechanises symmetry computation. David Saunders' 1989 book The Geometry of Jet Bundles ^{[Saunders Ch. 4]} systematised the bundle-theoretic foundations. The variational bicomplex, developed by Alexandre Vinogradov through the $\mathcal{C}$-spectral sequence in 1984 ^{[Vinogradov 1984]} and by Ian Anderson in his manuscript The Variational Bicomplex ^[Anderson], placed the Euler operator and conservation laws into a single cohomological framework on the infinite jet, with the total derivative as the horizontal differential.

Bibliography Master

@article{Ehresmann1951Jets,
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  volume  = {233},
  year    = {1951},
  pages   = {598--600, 777--779, 1081--1083}
}

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  edition   = {2nd}
}

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}

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}

@incollection{AndersonVariationalBicomplex,
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}

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}