With respect to the coordinates t1, t2, t3, ... on N, the p-form omega can be written as omega = f(t1, t2, t3, ...) dt1 &w dt2 &w dt3 ....  The command IntegrateForm integrates the function f(t1, t2, t3, ...) over the p-dimensional region defined by t1 = a1 .. b1, t2 = a1(t1) ... b1(t1), t3 = a3(t1, t2) ... b3(t1, t2), ....

In many cases one is interested in integrating a p-form omega on a manifold M over an imbedding submanifold phi : N -> M.  This is done in the DifferentialGeometry package by first computing the pullback  theta = Phi^*(omega) with the Pullback command and then integrating the resulting p-form theta over N with the IntegrateForm command.

In many cases a more efficient alternative to the IntegrateForm command is provided by the VectorCalculus[int] command.

This command is part of the DifferentialGeometry package, and so can be used in the form IntegrateForm(...) only after executing the command with(DifferentialGeometry).  It can always be used in the long form DifferentialGeometry:-IntegrateForm.

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

$\mathrm{DGsetup}\left(\left[x\,y\right]\,M\right)\:$

$\mathrm{\omega }≔\mathrm{evalDG}\left(\left(x^2+3xy\right)\mathrm{dx}\&w\mathrm{dy}\right)$

$\mathrm{\omega }≔\left(x^2+3xy\right)\mathrm{dx}\bigwedge \mathrm{dy}$

$\mathrm{IntegrateForm}\left(\mathrm{\omega }\,x=0..1\,y=0..1-x\right)$

$\frac{5}{24}$

$\mathrm{DGsetup}\left(\left[t\right]\,N\right)\:$$\mathrm{DGsetup}\left(\left[x\,y\,z\right]\,\mathrm{E3}\right)\:$

$\mathrm{\omega }≔\mathrm{evalDG}\left(y^2\mathrm{dx}+z^2\mathrm{dy}+xyz\mathrm{dz}\right)$

$\mathrm{\omega }≔y^2\mathrm{dx}+z^2\mathrm{dy}+zxy\mathrm{dz}$

$\mathrm{\Phi }≔\mathrm{Transformation}\left(N\,\mathrm{E3}\,\left[x=\sin \left(t\right)\cos \left(t\right)\,y=\sin \left(t\right)\cos \left(t\right)\,z=\exp \left(t\right)\right]\right)$

$\mathrm{\Phi }≔\left[x=\sin \left(t\right)\cos \left(t\right)\,y=\sin \left(t\right)\cos \left(t\right)\,z={\ⅇ}^t\right]$

$\mathrm{IntegrateForm}\left(\mathrm{Pullback}\left(\mathrm{\Phi }\,\mathrm{\omega }\right)\,t=0..\mathrm{\pi }\right)$

$-\frac{3}{10}+\frac{3{\ⅇ}^{2\mathrm{\pi }}}{10}$

$\mathrm{DGsetup}\left(\left[\mathrm{\theta }\,\mathrm{\phi }\right]\,N\right)\:$$\mathrm{DGsetup}\left(\left[x\,y\,z\right]\,\mathrm{E3}\right)\:$

$\mathrm{\omega }≔\mathrm{evalDG}\left(-3y^{2}z\mathrm{dx}\&w\mathrm{dy}+x{z}^2\mathrm{dy}\&w\mathrm{dz}-x^{2}y\mathrm{dx}\&w\mathrm{dz}\right)$

$\mathrm{\omega }≔-3y^{2}z\mathrm{dx}\bigwedge \mathrm{dy}-x^{2}y\mathrm{dx}\bigwedge \mathrm{dz}+x{z}^2\mathrm{dy}\bigwedge \mathrm{dz}$

$\mathrm{\Phi }≔\mathrm{Transformation}\left(N\,\mathrm{E3}\,\left[x=\cos \left(\mathrm{\theta }\right)\sin \left(\mathrm{\phi }\right)\,y=2\sin \left(\mathrm{\theta }\right)\sin \left(\mathrm{\phi }\right)\,z=3\cos \left(\mathrm{\phi }\right)\right]\right)$

$\mathrm{\Phi }≔\left[x=\cos \left(\mathrm{\theta }\right)\sin \left(\mathrm{\phi }\right)\,y=2\sin \left(\mathrm{\theta }\right)\sin \left(\mathrm{\phi }\right)\,z=3\cos \left(\mathrm{\phi }\right)\right]$

$\mathrm{IntegrateForm}\left(\mathrm{Pullback}\left(\mathrm{\Phi }\,\mathrm{\omega }\right)\,\mathrm{\theta }=0..2\mathrm{\pi }\,\mathrm{\phi }=0..\mathrm{\pi }\right)$

$\frac{16\mathrm{\pi }}{5}$