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.

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

$\mathrm{defform}\left(f=0\,\mathrm{w1}=1\,\mathrm{w2}=1\,\mathrm{w3}=1\,v=1\,x=0\,y=0\,z=0\right)$

$d\left(x^{2}y\right)$

$2xyd\left(x\right)+x^{2}d\left(y\right)$

$d\left(f\left(x\,y\,z\right)\right)$

$\left(\frac{\partial }{\partial x}f\left(x\,y\,z\right)\right)d\left(x\right)+\left(\frac{\partial }{\partial y}f\left(x\,y\,z\right)\right)d\left(y\right)+\left(\frac{\partial }{\partial z}f\left(x\,y\,z\right)\right)d\left(z\right)$

$d\left(f\,\left[\mathrm{w1}\,\mathrm{w2}\,\mathrm{w3}\right]\right)$

$\mathrm{fw1}\mathrm{w1}+\mathrm{fw2}\mathrm{w2}+\mathrm{fw3}\mathrm{w3}$

$d\left(f\,\left[d\left(x\right)\,d\left(y\left[1\right]\right)\,v\left[1\right]\right]\right)$

$\mathrm{fx}d\left(x\right)+{\mathrm{fy}}_{1}d\left(y_1\right)+{\mathrm{fv}}_1{v}_1$

$d\left(f\mathrm{w1}\&ˆ\mathrm{w2}\right)$

$\mathrm{\&^}\left(d\left(f\right)\,\mathrm{w1}\,\mathrm{w2}\right)+fd\left(\mathrm{w1}\right)\&ˆ\mathrm{w2}-f\mathrm{w1}\&ˆd\left(\mathrm{w2}\right)$

$d\left(\left[xy\,y^2\,\mathrm{w2}\right]\right)$

$\left[yd\left(x\right)+xd\left(y\right)\,2yd\left(y\right)\,d\left(\mathrm{w2}\right)\right]$