1. Prolongation of Jet Spaces. Suppose that the command DGsetup has been used to initialize a jet space $J^{\mathrm{\ℓ}}\left(E\right)$. This means that the standard jet space coordinates $\(x^i\, u^{\mathrm{\α}}\, u_{i_{}}^{\mathrm{\α}}\, u_{i_{}j}^{\mathrm{\α}}$, ..., $u_{\mathrm{ij} \cdot \cdot \cdot \mathrm{\ℓ}}^{\mathrm{\α}}\)$are protected. The coordinate vector fields, coordinate 1-forms, and contact forms to order $\mathrm{\ℓ}\mathit{ }$are initialized and protected. The command Prolong(k), where $k\ge l\,$with extend these protections and definitions to order $k$. The result is same as making a call to DGsetup to initialize the jet space $J^k\left(E\right)\,$but is slightly faster since Prolong command only needs to define and protect the coordinates,vectors and 1 -forms from order $\mathrm{\ℓ} \+1$to $k\.$

 2. Prolongation of Vector Fields. Let $Z$ be a vector field on $J^k\left(E\right)\.$We say that $Z$ preserves the contact ideal on $J^k\left(E\right)$ if for any contact form $\mathrm{\Θ}\,$the Lie derivative ${\mathrm{\ℒ}}_Z\mathrm{\Θ}$is also a contact form. Let $X$be a projectable, point, contact, evolutionary, total,or generalized vector field with values in the tangent space E. (See AssignVectorType for the definitions of these types of vector fields.) Then, for each $k$, there is a unique vector field $Z$on $J^k\left(E\right)$ which preserves the contact ideal on $J^k\left(E\right)$and which projects pointwise to $X\.$ This vector field Z is called the prolongation of $X$ to order $k$. and is denoted by ${\mathrm{pr}}^k\left(X\right)$. The explicit formula for vector field prolongation is given below. The second calling sequence Prolong(X, k) computes the prolongation of the vector field $X$to order $k\.$

 3. Prolongation of Transformations. Let $E\to M$and $F\to N$be two fiber bundles. We say that a transformation $\mathrm{\ψ} \: J^{\mathrm{\ℓ}}\left(E\right) \to J^n\left(F\right)$is a generalized contact transformation if for every contact form $\mathrm{\Θ}$ on $J^n\left(F\right)$, the pullback ${\mathrm{\ψ}}^{\*}\left(\mathrm{\Θ}\right)$is a contact form on $J^{\ell }\left(E\right)$. Let $\mathrm{\φ}$ be a projectable transformation, a point transformation, a contact transformation, a differential substitution or a generalized differential substitution. These maps are defined as mappings from $J^p\left(E\right)$to$J^q\left(F\right)$for the appropriate values of $p\,$$q\.$ (See AssignTransformationType for the definitions of these different types of transformations.) Then, for each $k$, there is a unique generalized contact transformation $\mathrm{\psi } \: J^{p\+k}\left(E\right) \to J^{q\+k}\left(F\right)$which covers $\mathrm{\φ}$. This transformation $\mathrm{\Ψ}$is called the prolongation of $\mathrm{\φ}$to order$k$ and it denoted by ${\mathrm{pr}}^k\left(\mathrm{\φ}\right)$.$$The third calling sequence Prolong($\mathbf{\φ}\mathbf{ }\,$k) computes the prolongation of $\mathrm{\φ}$to order $k\.$

 4. Prolongation of Differential Equations. A system of $\mathrm{\ℓ}$-th order differential equations can defined as the zero set of a collection $\mathrm{\Δ}$ of functions $F_a\:J^{\mathrm{\ℓ}}\left(E\right) \to \mathrm{\ℝ}$. The $k-$th order prolongation of $\mathrm{\Δ}\,$denote by ${\mathrm{pr}}^k\left(\mathrm{\Δ}\right)$is the system of ($\mathrm{\ℓ}$ +$k$)-th order differential equations defined as the zero set of the functions$F_a$and all their total derivatives ${\mathrm{D}}_{i_1}{\mathrm{D}}_{i_2}\cdot \cdot \cdot {\mathrm{D}}_{i_t}{F}_a$ to order $t\le k$. The fourth calling sequence Prolong(Delta, k) computes the prolongation of a system of differential equations $\mathrm{\Δ}$to order $k$. Use the command DifferentialEquationData to convert a list of functions$F_a\:J^{\mathrm{\ℓ}}\left(E\right) \to \mathrm{\ℝ}$into a differential equation data structure that can be passed to the Prolong command. The result is a new differential equation data structure representing the prolongation of the differential equations.

If $X \= A^j \frac{\partial }{\partial x^j} \+B^{\mathrm{\α} }\frac{\partial }{\partial u^{\mathrm{\α}}}$is a generalized vector field on $E$, then the $k$-th prolongation of X is the vector field

${\mathrm{pr}}^{k}X \= A^j \frac{\partial }{\partial x^i} \+B^{\mathrm{\alpha } }\frac{\partial }{\partial u^{\mathrm{\alpha }}}\+{\mathrm{\ζ}}_i^{\mathrm{\α}}\frac{\partial }{\partial u_i^{\mathrm{\α}}} \+{\mathrm{\zeta }}_{\mathrm{ij}}^{\mathrm{\alpha }}\frac{\partial }{\partial u_{\mathrm{ij}}^{\mathrm{\α} }}\+\cdot \cdot \cdot \+{\mathrm{\zeta }}_{i_1{i}_2\cdot \cdot \cdot i_k}^{\mathrm{\alpha }}\frac{\partial }{\partial u_{i_1{i}_2\cdot \cdot \cdot i_k}^{\mathrm{\α}}}$where  ${\mathrm{\zeta }}_{i_1{i}_2\cdot \cdot \cdot i_{\mathrm{\ℓ} }}^{\mathrm{\alpha }}\={\mathrm{D}}_{i_1}{\mathrm{D}}_{i_2}\cdot \cdot \cdot {\mathrm{D}}_{i_{\mathrm{\ℓ}}} \left(B^{\mathrm{\alpha } }-A^i{u}_i^{\mathrm{\alpha }}\right) - u_{i_1{i}_2\cdot \cdot \cdot i_{\mathrm{\ℓ}}j}^{\mathrm{\α}}{A}^j$.

For further details see the either of the two books by P. J. Olver.

Example 1. Prolongation of Jet Spaces

Define the jet space $J^1\left(E\right)\,$where $E\={\mathrm{\ℝ}}^2\times \mathrm{\ℝ}\mathit{ }$with coordinates $\left(x\,y\,u\right) \to \left(x\,y\right)\.$

Display the jet coordinates, the coordinate vector fields, the 1-forms, and the contact 1-forms.

Prolong the jet space $J^1\left(E\right)$to $J^3\left(E\right)$.

Again display the jet coordinates, the coordinate vector fields, the 1-forms, and the contact 1-forms.

Example 2. Prolongation of Vector Fields

Define the jet space $J^2\left(E_2\right)\,$where $E_2 \= \mathrm{\ℝ}\mathit{ }\times \mathrm{\ℝ}\mathit{ }\mathit{\to }\mathit{ }\mathrm{\ℝ}$ with coordinates $\left(x\, u\right) \to \left(x\right)$.

Define an arbitrary point vector field $X_1$ on $E_2$.

Prolong $X_1$ to order 1--this agrees with the standard prolongation formula found in all texts.

Define the infinitesimal generator $X_2$for a rotation in the $x$-$u$ plane.

Prolong $X_2$to order 1.

Prolong $X_2$to order to 2--we can achieve the same result by prolonging $\mathrm{pr1X2}$.

Define the jet space $J^1\left(E_3\right)$, where $E_3 \={\mathrm{\ℝ}}^{2 }\times {\mathrm{\ℝ}}^2 \to {\mathrm{\ℝ}}^{2 }$with coordinates $\left(x\,y\,u\,v\right)\to \left(x\, y\right)$.

Define a vector field $X_{3 }$whose flow simultaneously scales the coordinates $x\, y\, u\,v$.

Example 3. Prolongation of Transformations

Define the jet space $J^1\left(F\right)$where $F \=\mathrm{\ℝ} \times \mathrm{\ℝ}\mathit{ }\mathit{\to }\mathrm{\ℝ}$ with coordinates $\left(y\, v\right) \to \left(v\right)$.

Define a projectable transformation from $E_2$ to $F$.

Prolong phi1 to order 2.

Define a differential substitution from J${}^2\left(E_2\right)$ to $F\.$

Prolong phi2 to order 2.

Example 4. Prolongation of Differential Equations

Define a second order ode on $E_2$(coordinates $x\, u\)$

Calculate the second order prolongation of DE1. Note that the list of jet variables to be solved for is also prolonged.

Define a system of over-determined partial differential equations in 2 independent variables $x\,y$ and 1 dependent variable $u$.

The second prolongation of DE2 is an overdetermined system of Frobenius type--all the 4-th order derivatives of which can be solved for.