The D_Dx command computes total derivatives, as diff does, but for D_Dx the input and/or the output can be in jet notation. Also, D_Dx can compute partial derivatives with respect to dependent variables of the jet space (functions in DepVars) or their derivatives. D_Dx also works with anticommutative variables set using the Physics package.

The first argument, expression, is the object being differentiated, followed by the differentiation variables, and then the last argument, $\mathrm{DepVars}$, is always an indication of the dependent variables of the problem, either as a function (could be that way when there is only one) or as a list of them.

The differentiation variables can include any name but also functions of the jet space, that is, functions in DepVars or their partial derivatives. When the differentiation variable is a name the total derivative is computed, so for instance $x$, $f\left(x\right)$ and $\mathrm{f\text{'}}\left(x\right)$ are all considered functions of $x$. When the differentiation variable is a function, however, a partial derivative is computed in that, for instance, $f\left(x\right)$ and $\mathrm{f\text{'}}\left(x\right)$ are considered independent objects. This permits computing in a natural way objects like

where the right-hand-side is entered as D_Dx(L(x(t), diff(x(t), t)), diff(x(t), t), t, [x(t)]) and means to differentiate $L\left(x\left(t\right),\stackrel{\mathbf{.}}{x}\left(t\right)\right)$ with respect to $\stackrel{\mathbf{.}}{x}\left(t\right)$ followed by taking the total derivative with respect to $t$, taking $\left[x\left(t\right)\right]$ as the dependent variable defining the jet space.

The output is by default expressed in the jetnotation found in the input expression, whenever expression is written using jetnotation, or otherwise it is expressed in jetvariables notation when expression is passed in standard function notation. This default can be changed by passing the optional argument jetnotation = ..., where the right hand side can be false (to return in function notation), jetnumbers, jetvariables, jetvariableswithbrackets or jetODE; see the jet notations available.

$\mathrm{with}\left(\mathrm{PDEtools}\,\mathrm{D\_Dx}\,\mathrm{ToJet}\,\mathrm{FromJet}\right)$

$\left[\mathrm{D\_Dx}\,\mathrm{ToJet}\,\mathrm{FromJet}\right]$

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

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

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

$\mathrm{PDE}≔\left(\frac{\partial }{\partial x}u\left(x\,t\right)\right)v\left(t\right)w\left(y\right)+\left(\frac{\partial }{\partial t}u\left(x\,t\right)\right)v\left(t\right)\left(\frac{\ⅆ}{\ⅆy}w\left(y\right)\right)+u\left(x\,t\right)\left(\frac{\ⅆ}{\ⅆt}v\left(t\right)\right)\left(\frac{\ⅆ}{\ⅆy}w\left(y\right)\right)$

$\mathrm{D\_Dx}\left(\mathrm{PDE}\,w\left(y\right)\,\mathrm{DepVars}\right)$

$v{u}_x$

$\mathrm{FromJet}\left(\,\mathrm{DepVars}\right)$

$v\left(t\right)\left(\frac{\partial }{\partial x}u\left(x\,t\right)\right)$

$\mathrm{D\_Dx}\left(\mathrm{PDE}\,w\left(y\right)\,\mathrm{DepVars}\,\mathrm{jetnotation}=\mathrm{false}\right)$

$v\left(t\right)\left(\frac{\partial }{\partial x}u\left(x\,t\right)\right)$

$\mathrm{D\_Dx}\left(\mathrm{PDE}\,u\left[x\right]\,t\,\mathrm{DepVars}\right)$

$v_{t}w$

$\mathrm{JPDE}≔\mathrm{ToJet}\left(\mathrm{PDE}\,\mathrm{DepVars}\right)$

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

$\mathrm{D\_Dx}\left(\mathrm{PDE}\,t\,\mathrm{DepVars}\right)$

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

$\mathrm{D\_Dx}\left(\mathrm{JPDE}\,t\,\mathrm{DepVars}\right)$

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

$-$

$0$