take the total derivative of an expression, a differential form or a contact form - Maple Help

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JetCalculus[TotalDiff] - take the total derivative of an expression, a differential form or a contact form

Calling Sequences

TotalDiff(f, v)

Parameters

f         - a Maple expression, a differential form or a bi-form

v         - an independent variable, a positive integer or a list of positive integers

Description

 • The operation of total differentiation is a fundamental one in the study of jet spaces and their application to differential equations and the calculus of variations.  Informally, total differentiation of a function on a jet space with respect to an independent variable is the same as ordinary differentiation with respect to that variable if the jet coordinates are treated temporarily as functions of the independent variables.
 • Let be a fiber bundle with base dimension $n$ and fiber dimension $m$ and let be the $k$-th jet bundle. Introduce local coordinates , ..., where, as usual, if $s:M\to E$ is a section andis the $k$-jet of then

and dim$\left(M\right)$.

Then the total derivative of the jet coordinate  with respect to the independent variable  is . If  is a function on jet space, then by the chain rule

Similarly, the total derivatives of differential forms dand the contact form =- with respect to the independent variable are given by

${\mathrm{D}}_{k}$(dand  ${\mathrm{D}}_{k}$.

If and are 2 differential forms on jet space, then  . One can summarize all these formulas by saying that total differentiation with respect to the independent variable coincides with Lie differentiation with respect to the total vector field ${\mathrm{D}}_{k}$. Thus the total derivative with respect to ${x}^{k}$ commutes with the exterior derivative, the horizontal exterior derivative, and the vertical exterior derivative, that is, 

and .

 • If $f$ is a function or differential form on a jet space and $v$ an independent variable, then TotalDiff(f, v) calculates the total derivative of $f$ with respect to $v$. If $v$ is a list of $r$ positive integers, then the $r$-fold iterated total derivative is calculated.
 • The command TotalDiff is part of the DifferentialGeometry:-JetCalculus package.  It can be used in the form TotalDiff(...) only after executing the commands with(DifferentialGeometry) and with(JetCalculus), but can always be used by executing DifferentialGeometry:-JetCalculus:-TotalDiff(...).

Examples

 > $\mathrm{with}\left(\mathrm{DifferentialGeometry}\right):$$\mathrm{with}\left(\mathrm{JetCalculus}\right):$

Example 1.

First initialize the jet space for two independent variables and two dependent variables and prolong it to order 3.

 > $\mathrm{DGsetup}\left(\left[x,y\right],\left[u,v\right],E,3\right):$

Recall that represents the mixed 3rd derivative of once with respect to and twice with respect to $y.$The total derivative of  with respect to $x$ is which represents the 4th derivative of $u$, twice with respect to and twice with respect to $y$

The total derivative of $u\left[1,2,2\right]$ with respect to $y$ is $u\left[1,2,2,2\right]$ which represents the 4th derivative of $u$, once with respect to $x$ and 3 times with respect to $y$.

 E > $\mathrm{TotalDiff}\left({u}_{1,2,2},x\right)$
 ${{u}}_{{1}{,}{1}{,}{2}{,}{2}}$ (2.1)
 E > $\mathrm{TotalDiff}\left({u}_{1,2,2},y\right)$
 ${{u}}_{{1}{,}{2}{,}{2}{,}{2}}$ (2.2)

In place of the independent variables $x$ or the integer 1 or 2 can be used.

 E > $\mathrm{TotalDiff}\left({u}_{1,2,2},1\right)$
 ${{u}}_{{1}{,}{1}{,}{2}{,}{2}}$ (2.3)
 E > $\mathrm{TotalDiff}\left({u}_{1,2,2},2\right)$
 ${{u}}_{{1}{,}{2}{,}{2}{,}{2}}$ (2.4)

Here is a general formula for the total derivative of a function with dependencies on the 2-jet of $u.$

 E > $\mathrm{vars}≔x,y,{u}_{[]},{u}_{1},{u}_{2},{u}_{1,1},{u}_{1,2},{u}_{2,2}:$
 E > $\mathrm{PDEtools}[\mathrm{declare}]\left(F\left(\mathrm{vars}\right)\right)$
 ${F}{}\left({x}{,}{y}{,}{{u}}_{\left[\right]}{,}{{u}}_{{1}}{,}{{u}}_{{2}}{,}{{u}}_{{1}{,}{1}}{,}{{u}}_{{1}{,}{2}}{,}{{u}}_{{2}{,}{2}}\right){}{\mathrm{will now be displayed as}}{}{F}$ (2.5)
 E > $\mathrm{TotalDiff}\left(F\left(\mathrm{vars}\right),x\right)$
 ${{F}}_{{{u}}_{\left[\right]}}{}{{u}}_{{1}}{+}{{F}}_{{{u}}_{{1}}}{}{{u}}_{{1}{,}{1}}{+}{{F}}_{{{u}}_{{2}}}{}{{u}}_{{1}{,}{2}}{+}{{F}}_{{{u}}_{{1}{,}{1}}}{}{{u}}_{{1}{,}{1}{,}{1}}{+}{{F}}_{{{u}}_{{1}{,}{2}}}{}{{u}}_{{1}{,}{1}{,}{2}}{+}{{F}}_{{{u}}_{{2}{,}{2}}}{}{{u}}_{{1}{,}{2}{,}{2}}{+}{{F}}_{{x}}$ (2.6)
 E > $\mathrm{TotalDiff}\left(F\left(\mathrm{vars}\right),y\right)$
 ${{F}}_{{{u}}_{\left[\right]}}{}{{u}}_{{2}}{+}{{F}}_{{{u}}_{{1}}}{}{{u}}_{{1}{,}{2}}{+}{{F}}_{{{u}}_{{2}}}{}{{u}}_{{2}{,}{2}}{+}{{F}}_{{{u}}_{{1}{,}{1}}}{}{{u}}_{{1}{,}{1}{,}{2}}{+}{{F}}_{{{u}}_{{1}{,}{2}}}{}{{u}}_{{1}{,}{2}{,}{2}}{+}{{F}}_{{{u}}_{{2}{,}{2}}}{}{{u}}_{{2}{,}{2}{,}{2}}{+}{{F}}_{{y}}$ (2.7)

The total derivative satisfies the usual rules of differentiation.

 E > $\mathrm{TotalDiff}\left(y{u}_{1,0}{v}_{0,1},y\right)$
 ${y}{}{{u}}_{{1}{,}{0}}{}{{v}}_{{0}{,}{1}{,}{2}}{+}{y}{}{{v}}_{{0}{,}{1}}{}{{u}}_{{0}{,}{1}{,}{2}}{+}{{u}}_{{1}{,}{0}}{}{{v}}_{{0}{,}{1}}$ (2.8)
 E > $f≔\mathrm{simplify}\left(\mathrm{TotalDiff}\left(\mathrm{arctan}\left(\frac{{u}_{0,3}}{{v}_{2,0}}\right),x\right)\right)$
 ${f}{≔}{-}\frac{{{u}}_{{0}{,}{3}}{}{{v}}_{{0}{,}{1}{,}{2}}{-}{{u}}_{{0}{,}{1}{,}{3}}{}{{v}}_{{2}{,}{0}}}{{{u}}_{{0}{,}{3}}^{{2}}{+}{{v}}_{{2}{,}{0}}^{{2}}}$ (2.9)

Multiple total derivatives can also be calculated by using TotalDiff. We differentiate 2 times with respect to $x$ and 3 times with respect to $y$ to get $u\left[1,1,2,2,2\right]$.

 E > $\mathrm{TotalDiff}\left({u}_{2},\left[1,1,2,2,2\right]\right)$
 ${{u}}_{{1}{,}{1}{,}{2}{,}{2}{,}{2}{,}{2}}$ (2.10)
 E > $\mathrm{TotalDiff}\left({u}_{[]}{v}_{[]},\left[1,1,1\right]\right)$
 ${{u}}_{\left[\right]}{}{{v}}_{{1}{,}{1}{,}{1}}{+}{3}{}{{v}}_{{1}{,}{1}}{}{{u}}_{{1}}{+}{3}{}{{v}}_{{1}}{}{{u}}_{{1}{,}{1}}{+}{{v}}_{\left[\right]}{}{{u}}_{{1}{,}{1}{,}{1}}$ (2.11)

Example 2.

Total differentiation extends to differential forms and contact forms on jet spaces.

 E > $\mathrm{TotalDiff}\left({\mathrm{du}}_{2},x\right)$
 ${{\mathrm{du}}}_{{1}{,}{2}}$ (2.12)
 E > $\mathrm{TotalDiff}\left(\left({\mathrm{du}}_{2}\right)&w\left({\mathrm{dv}}_{1}\right),y\right)$
 ${{\mathrm{du}}}_{{2}}{}{\bigwedge }{}{{\mathrm{dv}}}_{{1}{,}{2}}{-}{{\mathrm{dv}}}_{{1}}{}{\bigwedge }{}{{\mathrm{du}}}_{{2}{,}{2}}$ (2.13)
 E > $\mathrm{TotalDiff}\left({\mathrm{Cu}}_{2},x\right)$
 ${{\mathrm{Cu}}}_{{1}{,}{2}}$ (2.14)
 E > $\mathrm{TotalDiff}\left({\mathrm{du}}_{2},\left[1,1,2\right]\right)$
 ${{\mathrm{du}}}_{{1}{,}{1}{,}{2}{,}{2}}$ (2.15)
 E > $\mathrm{TotalDiff}\left(\left({\mathrm{du}}_{2}\right)&w\left({\mathrm{dv}}_{1}\right),\left[1,2\right]\right)$
 ${{\mathrm{du}}}_{{2}}{}{\bigwedge }{}{{\mathrm{dv}}}_{{1}{,}{1}{,}{2}}{-}{{\mathrm{dv}}}_{{1}}{}{\bigwedge }{}{{\mathrm{du}}}_{{1}{,}{2}{,}{2}}{+}{{\mathrm{du}}}_{{1}{,}{2}}{}{\bigwedge }{}{{\mathrm{dv}}}_{{1}{,}{2}}{+}{{\mathrm{du}}}_{{2}{,}{2}}{}{\bigwedge }{}{{\mathrm{dv}}}_{{1}{,}{1}}$ (2.16)

Example 3.

The DifferentialGeometry package supports an alternative jet notation. For example, if there are 2 independent variables $\left(x,y\right)$, then would now represent the 3rd mixed partial derivative of once with respect to and twice with respect to $y.$

 E > $\mathrm{Preferences}\left("JetNotation","JetNotation2"\right)$
 ${"JetNotation1"}$ (2.17)
 E > $\mathrm{DGsetup}\left(\left[x,y\right],\left[u,v\right],J,3\right):$
 J > $\mathrm{TotalDiff}\left({u}_{1,2},x\right)$
 ${{u}}_{{2}{,}{2}}$ (2.18)
 J > $\mathrm{TotalDiff}\left({u}_{1,2},y\right)$
 ${{u}}_{{1}{,}{3}}$ (2.19)
 E > $\mathrm{TotalDiff}\left({u}_{1,2},\left[2,3\right]\right)$
 ${{u}}_{{3}{,}{5}}$ (2.20)

Revert to the default jet notation.

 J > $\mathrm{Preferences}\left("JetNotation","JetNotation1"\right)$
 ${"JetNotation2"}$ (2.21)