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difforms

 d
 exterior differentiation

 Calling Sequence d(expr) d(expr, forms)

Parameters

 expr - expression or list of expressions forms - (optional) list of 1-forms, i.e. wdegree=one

Description

 • The function d computes the exterior derivative of an expression. If the expression is a list, then d is applied to each element in the list.
 • When d is called with an expression and a list of 1-forms, any name of type scalar in the expression will be expanded in these 1-forms.  It is assumed that the 1-forms are independent.  For each 1-form, a new scalar is created to be the component for that 1-form.
 • The command with(difforms,d) allows the use of the abbreviated form of this command.

Examples

 > $\mathrm{with}\left(\mathrm{difforms}\right):$
 > $\mathrm{defform}\left(f=0,\mathrm{w1}=1,\mathrm{w2}=1,\mathrm{w3}=1,v=1,x=0,y=0,z=0\right)$
 > $d\left({x}^{2}y\right)$
 ${2}{}{x}{}{y}{}{d}{}\left({x}\right){+}{{x}}^{{2}}{}{d}{}\left({y}\right)$ (1)
 > $d\left(f\left(x,y,z\right)\right)$
 $\left(\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}{,}{y}{,}{z}\right)\right){}{d}{}\left({x}\right){+}\left(\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}{,}{y}{,}{z}\right)\right){}{d}{}\left({y}\right){+}\left(\frac{{\partial }}{{\partial }{z}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({x}{,}{y}{,}{z}\right)\right){}{d}{}\left({z}\right)$ (2)
 > $d\left(f,\left[\mathrm{w1},\mathrm{w2},\mathrm{w3}\right]\right)$
 ${\mathrm{fw1}}{}{\mathrm{w1}}{+}{\mathrm{fw2}}{}{\mathrm{w2}}{+}{\mathrm{fw3}}{}{\mathrm{w3}}$ (3)
 > $d\left(f,\left[d\left(x\right),d\left(y\left[1\right]\right),v\left[1\right]\right]\right)$
 ${\mathrm{fx}}{}{d}{}\left({x}\right){+}{{\mathrm{fy}}}_{{1}}{}{d}{}\left({{y}}_{{1}}\right){+}{{\mathrm{fv}}}_{{1}}{}{{v}}_{{1}}$ (4)
 > $d\left(f\mathrm{w1}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&ˆ\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{w2}\right)$
 ${\mathrm{&^}}{}\left({d}{}\left({f}\right){,}{\mathrm{w1}}{,}{\mathrm{w2}}\right){+}{f}{}{d}{}\left({\mathrm{w1}}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&ˆ}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\mathrm{w2}}{-}{f}{}{\mathrm{w1}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&ˆ}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{d}{}\left({\mathrm{w2}}\right)$ (5)
 > $d\left(\left[xy,{y}^{2},\mathrm{w2}\right]\right)$
 $\left[{y}{}{d}{}\left({x}\right){+}{x}{}{d}{}\left({y}\right){,}{2}{}{y}{}{d}{}\left({y}\right){,}{d}{}\left({\mathrm{w2}}\right)\right]$ (6)