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:

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.

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

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

$\left[\mathrm{mathematicalnotation}=\mathrm{true}\right]$

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

$\mathrm{*\; Partial\; match\; of\; \text{'}}\mathrm{anticommutativepre}\mathrm{\text{'}\; against\; keyword\; \text{'}}\mathrm{anticommutativeprefix}\text{'}$

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

$\left[\mathrm{anticommutativeprefix}=\left\{\mathrm{\theta }\right\}\right]$

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

$f\left(x\,y\,z\,{\mathrm{\theta }}_1\,{\mathrm{\theta }}_2\,{\mathrm{\theta }}_3\right)$

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

$\frac{{\partial }^6}{\partial x\partial y\partial z\partial {\mathrm{\theta }}_1\partial {\mathrm{\theta }}_2\partial {\mathrm{\theta }}_3}f\left(x\,y\,z\,{\mathrm{\theta }}_1\,{\mathrm{\theta }}_2\,{\mathrm{\theta }}_3\right)$

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

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])

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

${\mathrm{D}}_{1,2,3,4,5,6}\left(f\right)\left(x\,y\,z\,{\mathrm{\theta }}_1\,{\mathrm{\theta }}_2\,{\mathrm{\theta }}_3\right)$

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

$\frac{{\partial }^6}{\partial x\partial y\partial z\partial {\mathrm{\theta }}_1\partial {\mathrm{\theta }}_2\partial {\mathrm{\theta }}_3}f\left(x\,y\,z\,{\mathrm{\theta }}_1\,{\mathrm{\theta }}_2\,{\mathrm{\theta }}_3\right)$