Conversions between diff, D, and Physics[diff]  convert derivatives between the diff and D notations

Calling Sequence


convert(expr, diff)
convert(expr, D)


Parameters


expr



any valid Maple object





Description


•

The Physics package provides a framework for computing with commutative, anticommutative, and noncommutative objects at the same time. Accordingly, it is possible to differentiate with respect to anticommutative variables; the command used to perform these derivatives is the diff command of the Physics package. (herein referred to as diff).

•

convert/D and convert/diff are converter routines between the D and diff formats for representing derivatives. The equivalence for anticommutative high order derivatives written in the D format and diff format of the Physics package is as in:

$\frac{{\partial}^{2}}{\partial {{\mathrm{\theta}}}_{1}\partial {{\mathrm{\theta}}}_{2}}f\left({{\mathrm{\theta}}}_{1}\,{{\mathrm{\theta}}}_{2}\right)={\mathrm{D}}_{1,2}\left(f\right)\left({{\mathrm{\theta}}}_{1}\,{{\mathrm{\theta}}}_{2}\right)$

where the derivative above should be interpreted as: first differentiate with respect to ${\mathrm{\theta}}_{1}$, then with respect to ${\mathrm{\theta}}_{2}$ (or the opposite times $\mathrm{1}$); and the right hand side is not interpreted as a commutative higher order derivative.



Examples


Load the Physics package and set a prefix to identify anticommutative variables (see Setup for more information).
>

$\mathrm{with}\left(\mathrm{Physics}\right)\:$

>

$\mathrm{Setup}\left(\mathrm{mathematicalnotation}=\mathrm{true}\right)$

$\left[{\mathrm{mathematicalnotation}}{=}{\mathrm{true}}\right]$
 (1) 
>

$\mathrm{Setup}\left(\mathrm{anticommutativepre}=\mathrm{\theta}\right)$

$\mathrm{*\; Partial\; match\; of\; \text{'}}{}\mathrm{anticommutativepre}{}\mathrm{\text{'}\; against\; keyword\; \text{'}}{}\mathrm{anticommutativeprefix}{}\text{'}$
 
$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$
 
$\left[{\mathrm{anticommutativeprefix}}{=}\left\{{\mathrm{\theta}}\right\}\right]$
 (2) 
Consider a commutative function depending on commutative and anticommutative variables, and one higher order derivative of it.
>

$f\left(x\,y\,z\,\mathrm{\theta}\left[1\right]\,\mathrm{\theta}\left[2\right]\,\mathrm{\theta}\left[3\right]\right)$

${f}{}\left({x}{\,}{y}{\,}{z}{\,}{{\mathrm{\theta}}}_{{1}}{\,}{{\mathrm{\theta}}}_{{2}}{\,}{{\mathrm{\theta}}}_{{3}}\right)$
 (3) 
>

$\mathrm{diff}\left(\,x\,\mathrm{\theta}\left[3\right]\,y\,\mathrm{\theta}\left[1\right]\,z\,\mathrm{\theta}\left[2\right]\right)$

$\frac{{{\partial}}^{{6}}}{{\partial}{x}{\partial}{y}{\partial}{z}{\partial}{{\mathrm{\theta}}}_{{1}}{\partial}{{\mathrm{\theta}}}_{{2}}{\partial}{{\mathrm{\theta}}}_{{3}}}\phantom{\rule[0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}{\,}{y}{\,}{z}{\,}{{\mathrm{\theta}}}_{{1}}{\,}{{\mathrm{\theta}}}_{{2}}{\,}{{\mathrm{\theta}}}_{{3}}\right)$
 (4) 
Note in the above that the commutative differentiation variables are collected as a group to be applied first, then the anticommutative ones.
>

$\mathrm{lprint}\left(\right)$

Physics:diff(Physics:diff(Physics:diff(diff(diff(diff(f(x,y,z,theta[1],theta
[2],theta[3]),x),y),z),theta[1]),theta[2]),theta[3])
 
Rewrite this expression in D notation, then convert back to diff notation.
>

$\mathrm{convert}\left(\,\mathrm{D}\right)$

${{\mathrm{D}}}_{{1}{,}{2}{,}{3}{,}{4}{,}{5}{,}{6}}{}\left({f}\right){}\left({x}{\,}{y}{\,}{z}{\,}{{\mathrm{\theta}}}_{{1}}{\,}{{\mathrm{\theta}}}_{{2}}{\,}{{\mathrm{\theta}}}_{{3}}\right)$
 (5) 
>

$\mathrm{convert}\left(\,\mathrm{diff}\right)$

$\frac{{{\partial}}^{{6}}}{{\partial}{x}{\partial}{y}{\partial}{z}{\partial}{{\mathrm{\theta}}}_{{1}}{\partial}{{\mathrm{\theta}}}_{{2}}{\partial}{{\mathrm{\theta}}}_{{3}}}\phantom{\rule[0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}{\,}{y}{\,}{z}{\,}{{\mathrm{\theta}}}_{{1}}{\,}{{\mathrm{\theta}}}_{{2}}{\,}{{\mathrm{\theta}}}_{{3}}\right)$
 (6) 


See Also


convert/D, convert/diff, D, diff, Physics, Physics conventions, Physics examples, Physics Updates, Tensors  a complete guide, MiniCourse Computer Algebra for Physicists, Physics,diff, Physics,diff,anticommutative, Setup

