The ToJet and FromJet commands rewrite back and forth mathematical expressions using jet and function notation respectively. ToJet is also used to rewrite an expression already in jet notation using a different jet notation.

Given a set of $m$ smooth functions $Y_m$ depending on $n$ independent variables $X_n$, the associated "jet space" prolonged to order k is the Euclidean space you construct taking each of these $n$ independent and $m$ dependent variables, as well as the partial derivatives of the $Y_m$ with respect to the $X_n$ up to order $k$, all as independent variables in equal footing. Recalling, a function of $n$ variables has $\left(\genfrac{}{}{0ex}{}{n+k}{k}\right)$ partial derivatives of order k, including order 0. Thus, the dimension of the $k$th prolongation of the jet space is $n+m\left(\genfrac{}{}{0ex}{}{n+k}{k}\right)$.

The four alternative jet notations handled by ToJet and FromJet and understood by all the symmetry commands of PDEtools  are: jetvariables (also used in DifferentialAlgebra), jetvariableswithbrackets (also used in DifferentialAlgebra and the deprecated diffalg), jetnumbers (also used in the Maple D differentiation operator) and jetODE (also used in DEtools[Lie]). You transform from one notation to the other one using ToJet with the option jetnotation = ... or the synonym notation = ..., as shown in the Examples section.

The default jet notation used by ToJet is jetvariables, which is frequently used in textbooks and easier to read than the other notations. The table relating function <-> jetvariables notation can be summarized around an example - say the function $u\left(x\&comma;t\right)$, as follows:

Both the DifferentialAlgebra and its preceding deprecated diffalg packages also use a similar notation, jetvariableswithbrackets, with only one difference: $u\left(x\&comma;t\right)=u\left[\right]$, so with brackets [] after the name of the dependent variable only when representing the derivative of order zero. To have these brackets in the output of ToJet use the option jetnotation = jetvariableswithbrackets.

An alternative jet notation, frequently used for programming purposes, is the jetnumbers notation, that follows the conventions used by the Maple D differentiation operator. The equivalence between function and jetnumbers notation is thus as follows:

When the dependent variables involved have different dependencies, the enumeration used to represent the independent variables is displayed on the screen (see first example in the Examples section). To avoid the display of this information use the quiet option.

Finally, the jetODE notation can be used when there is only one dependent variable of only one independent variable, as for instance $u\left(x\right)$. The table relating function <-> jetODE notation is

Note the use of $\mathrm{\_y}$ to prefix the representation of derivatives regardless of whether the unknown (in this example, $u\left(x\right)$) has the name $y$.

By default FromJet returns the derivatives found in expr expressed using diff; to make it return using the D notation specify the optional argument differentiationnotation = D.

To avoid having to remember the optional keywords, if you type the keyword misspelled, or just a portion of it, a matching against the correct keywords is performed, and when there is only one match, the input is automatically corrected.

$\mathrm{with}\left(\mathrm{PDEtools}\&comma;\mathrm{D\_Dx}\&comma;\mathrm{declare}\&comma;\mathrm{ToJet}\&comma;\mathrm{FromJet}\right)$

$\left[\mathrm{D\_Dx}\&comma;\mathrm{declare}\&comma;\mathrm{ToJet}\&comma;\mathrm{FromJet}\right]$

$\mathrm{DepVars}≔\left[u\left(x\&comma;t\right)\&comma;v\left(t\right)\&comma;w\left(y\right)\right]$

$\mathrm{DepVars}≔\left[u\left(x\&comma;t\right)\&comma;v\left(t\right)\&comma;w\left(y\right)\right]$

$\mathrm{PDE}≔\mathrm{diff}\left(u\left(x\&comma;t\right)v\left(t\right)w\left(y\right)\&comma;x\right)+\mathrm{diff}\left(u\left(x\&comma;t\right)v\left(t\right)w\left(y\right)\&comma;t\&comma;y\right)$

$\mathrm{PDE}≔\left(\frac{\partial }{\partial x}u\left(x\&comma;t\right)\right)v\left(t\right)w\left(y\right)+\left(\frac{\partial }{\partial t}u\left(x\&comma;t\right)\right)v\left(t\right)\left(\frac{\&DifferentialD;}{\&DifferentialD;y}w\left(y\right)\right)+u\left(x\&comma;t\right)\left(\frac{\&DifferentialD;}{\&DifferentialD;t}v\left(t\right)\right)\left(\frac{\&DifferentialD;}{\&DifferentialD;y}w\left(y\right)\right)$

$\mathrm{jetPDE}≔\mathrm{ToJet}\left(\mathrm{PDE}\&comma;\mathrm{DepVars}\right)$

$\mathrm{jetPDE}≔u{v}_t{w}_y+vw{u}_x+v{u}_t{w}_y$

$\mathrm{jetnumbersPDE}≔\mathrm{ToJet}\left(\mathrm{PDE}\&comma;\mathrm{DepVars}\&comma;\mathrm{notation}=\mathrm{jetnumbers}\right)$

$\mathrm{jet\; notation:\; [1\; =\; x,\; 2\; =\; t,\; 3\; =\; y]}$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\mathrm{jetnumbersPDE}≔u\left[\right]v_2{w}_3+u_{1}v\left[\right]w\left[\right]+u_{2}v\left[\right]w_3$

$\mathrm{FromJet}\left(\mathrm{jetnumbersPDE}\&comma;\mathrm{DepVars}\right)$

$\left(\frac{\partial }{\partial x}u\left(x\&comma;t\right)\right)v\left(t\right)w\left(y\right)+\left(\frac{\partial }{\partial t}u\left(x\&comma;t\right)\right)v\left(t\right)\left(\frac{\&DifferentialD;}{\&DifferentialD;y}w\left(y\right)\right)+u\left(x\&comma;t\right)\left(\frac{\&DifferentialD;}{\&DifferentialD;t}v\left(t\right)\right)\left(\frac{\&DifferentialD;}{\&DifferentialD;y}w\left(y\right)\right)$

$\mathrm{FromJet}\left(\mathrm{jetPDE}\&comma;\mathrm{DepVars}\&comma;\mathrm{differentiationnotation}=\mathrm{D}\right)$

${\mathrm{D}}_1\left(u\right)\left(x\&comma;t\right)v\left(t\right)w\left(y\right)+{\mathrm{D}}_2\left(u\right)\left(x\&comma;t\right)v\left(t\right)\mathrm{D}\left(w\right)\left(y\right)+u\left(x\&comma;t\right)\mathrm{D}\left(v\right)\left(t\right)\mathrm{D}\left(w\right)\left(y\right)$

$\mathrm{D\_Dx}\left(\mathrm{jetnumbersPDE}\&comma;t\&comma;\mathrm{DepVars}\right)$

$u\left[\right]v_{2,2}{w}_3+u_1{v}_{2}w\left[\right]+2u_2{v}_2{w}_3+u_{1,2}v\left[\right]w\left[\right]+u_{2,2}v\left[\right]w_3$

$\mathrm{FromJet}\left(\&comma;\mathrm{DepVars}\right)$

$\left(\frac{{\partial }^2}{\partial t\partial x}u\left(x\&comma;t\right)\right)v\left(t\right)w\left(y\right)+\left(\frac{\partial }{\partial x}u\left(x\&comma;t\right)\right)\left(\frac{\&DifferentialD;}{\&DifferentialD;t}v\left(t\right)\right)w\left(y\right)+\left(\frac{{\partial }^2}{\partial t^2}u\left(x\&comma;t\right)\right)v\left(t\right)\left(\frac{\&DifferentialD;}{\&DifferentialD;y}w\left(y\right)\right)+2\left(\frac{\partial }{\partial t}u\left(x\&comma;t\right)\right)\left(\frac{\&DifferentialD;}{\&DifferentialD;t}v\left(t\right)\right)\left(\frac{\&DifferentialD;}{\&DifferentialD;y}w\left(y\right)\right)+u\left(x\&comma;t\right)\left(\frac{{\&DifferentialD;}^2}{\&DifferentialD;t^2}v\left(t\right)\right)\left(\frac{\&DifferentialD;}{\&DifferentialD;y}w\left(y\right)\right)$

$\mathrm{diff}\left(\mathrm{PDE}\&comma;t\right)-$

$0$

$\mathrm{D\_Dx}\left(\mathrm{jetPDE}\&comma;t\&comma;\mathrm{DepVars}\right)$

$u{v}_{t,t}{w}_y+vw{u}_{x,t}+v{u}_{t,t}{w}_y+w{u}_x{v}_t+2u_t{v}_t{w}_y$

$\mathrm{JetODE}≔\mathrm{\_y2}=f\left(x\&comma;y\&comma;\mathrm{\_y1}\right)$

$\mathrm{JetODE}≔\mathrm{\_y2}=f\left(x\&comma;y\&comma;\mathrm{\_y1}\right)$

$\mathrm{D\_Dx}\left(\mathrm{JetODE}\&comma;x\&comma;y\left(x\right)\right)$

$\mathrm{\_y3}=\frac{\partial }{\partial x}f\left(x\&comma;y\&comma;\mathrm{\_y1}\right)+\left(\frac{\partial }{\partial y}f\left(x\&comma;y\&comma;\mathrm{\_y1}\right)\right)\mathrm{\_y1}+\left(\frac{\partial }{\partial \mathrm{\_y1}}f\left(x\&comma;y\&comma;\mathrm{\_y1}\right)\right)\mathrm{\_y2}$

$\mathrm{FromJet}\left(\mathrm{JetODE}\&comma;y\left(x\right)\right)$

$\frac{{\&DifferentialD;}^2}{\&DifferentialD;x^2}y\left(x\right)=f\left(x\&comma;y\left(x\right)\&comma;\frac{\&DifferentialD;}{\&DifferentialD;x}y\left(x\right)\right)$

$\mathrm{ToJet}\left(\&comma;y\left(x\right)\right)$

$y_{x,x}=f\left(x\&comma;y\&comma;y_x\right)$

$\mathrm{ToJet}\left(\&comma;y\left(x\right)\&comma;\mathrm{notation}=\mathrm{jetODE}\right)$

$\mathrm{\_y2}=f\left(x\&comma;y\&comma;\mathrm{\_y1}\right)$

$\mathrm{ToJet}\left(\&comma;y\left(x\right)\&comma;\mathrm{notation}=\mathrm{jetvariableswithbrackets}\right)$

$y_{x,x}=f\left(x\&comma;y\left[\right]\&comma;y_x\right)$

$\mathrm{ToJet}\left(\mathrm{JetODE}\&comma;y\left(x\right)\right)$

$y_{x,x}=f\left(x\&comma;y\&comma;y_x\right)$

$\mathrm{ee}≔\mathrm{\_y2}\mathrm{\_y1}+uy+y\left[x\right]u\left(x\&comma;t\right)+u\left[1\right]y\left(x\right)+u\left[x\right]u\left[t\right]$

$\mathrm{ee}≔\mathrm{\_y2}\mathrm{\_y1}+uy+y_{x}u\left(x\&comma;t\right)+u_{1}y\left(x\right)+u_x{u}_t$

$\mathrm{DepVars}≔\left[y\left(x\right)\&comma;u\left(x\&comma;t\right)\right]$

$\mathrm{DepVars}≔\left[y\left(x\right)\&comma;u\left(x\&comma;t\right)\right]$

$\mathrm{ToJet}\left(\mathrm{ee}\&comma;\mathrm{DepVars}\right)$

$uy+u{y}_x+y{u}_x+u_x{u}_t+y_{x,x}{y}_x$

$\mathrm{FromJet}\left(\mathrm{ee}\&comma;\mathrm{DepVars}\right)$

$\left(\frac{{\&DifferentialD;}^2}{\&DifferentialD;x^2}y\left(x\right)\right)\left(\frac{\&DifferentialD;}{\&DifferentialD;x}y\left(x\right)\right)+u\left(x\&comma;t\right)y\left(x\right)+\left(\frac{\&DifferentialD;}{\&DifferentialD;x}y\left(x\right)\right)u\left(x\&comma;t\right)+\left(\frac{\partial }{\partial x}u\left(x\&comma;t\right)\right)y\left(x\right)+\left(\frac{\partial }{\partial x}u\left(x\&comma;t\right)\right)\left(\frac{\partial }{\partial t}u\left(x\&comma;t\right)\right)$