Let Omega^p(R^n) denote the space of differential p-forms defined on the Euclidean space R^n.  A homotopy operator for the de Rham complex on R^n is a map H: Omega^p(R^n) -> Omega^(p-1)(R^n) such that omega = H(d(omega)) + d(H(omega) ) for any omega in Omega^p(R^n).

Most differential geometry texts give a standard formula for a homotopy operator which involves an integration of the coefficients of the differential from omega along the straight line [tx, ty, tz, ...] from the origin (t = 0) to the general point [x, y, z, ...].  The optional arguments to DeRhamHomotopy allow the user to modify the standard homotopy operator by changing the path of integration.

The option initialpoint = [x = a, y = b, z = c, ...] changes the path of integration to the straight line starting at the point with coordinates [a, b, c, ...]. This option is useful when the differential form omega is singular at the origin.

The option integrationlimits = [infinity, 1] integrates the coefficients of the differential form from a point at infinity to the general point [x, y, z, ...]. This option may be used for differential forms which vanish at infinity.  More precisely, if A(x^1, x^2, x^3, ...) is any coefficient of the differential form omega, then the function a(t) = t*A(tx^1, tx^2, tx^3, ...) must have a finite integral over the interval t = 1 .. infinity.

In many situations, the DeRhamHomotopy command will return a simpler answer by using the path="zigzag" option.  In this case the 'radial' integration from the origin to the point [x, y, z, ...] is replaced by a sequence of straight line integrations along the coordinate lines, first from [0, 0, 0, ...] to [x, 0, 0, ...], then from [x, 0, 0, ...] to' [x, y, 0, ...] and so on.  The option variableorder = list specifies the order in which the integrations along the different coordinate lines are performed.

The option cylinderconstruction = var implements the more general homotopy operator described in the books by Boothby, page 277 and Flanders, page 27.  We consider a manifold M and set N = I x M, where I is an interval with coordinate (say) t.  Let omega be a p-form on N. Then eta = DeRhamHomotopy(omega, cylinderconstruction = t) returns a (p-1)-form on M.  If omega = A(t, x) dx &w dy ... (no dt differentials) then eta = 0.  If omega = A(t, x) dt &w dx &w dy ..., then eta = int(A(t, x) t = 0..1) dx &w dy.  With the cylinderconstruction option, you can easily create customized homotopy operators for the de Rham complex.

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

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

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

$\mathrm{frame\; name:\; M}$

$\mathrm{\ω1}≔\mathrm{evalDG}\left(y\left(\mathrm{dx}\&w\mathrm{dz}\right)\&w\mathrm{dw}+x\left(\mathrm{dy}\&w\mathrm{dz}\right)\&w\mathrm{dw}\right)$

$\mathrm{\ω1}≔y\mathrm{dx}\bigwedge \mathrm{dz}\bigwedge \mathrm{dw}+x\mathrm{dy}\bigwedge \mathrm{dz}\bigwedge \mathrm{dw}$

$\mathrm{ExteriorDerivative}\left(\mathrm{\ω1}\right)$

$0\mathrm{dx}\bigwedge \mathrm{dy}\bigwedge \mathrm{dz}\bigwedge \mathrm{dw}$

$\mathrm{\η1}≔\mathrm{DeRhamHomotopy}\left(\mathrm{\ω1}\right)$

$\mathrm{\η1}≔\frac{yw\mathrm{dx}}{4}\bigwedge \mathrm{dz}-\frac{yz\mathrm{dx}}{4}\bigwedge \mathrm{dw}+\frac{xw\mathrm{dy}}{4}\bigwedge \mathrm{dz}-\frac{xz\mathrm{dy}}{4}\bigwedge \mathrm{dw}+\frac{yx\mathrm{dz}}{2}\bigwedge \mathrm{dw}$

$\mathrm{ExteriorDerivative}\left(\mathrm{\η1}\right)\&minus\mathrm{\ω1}$

$0\mathrm{dx}\bigwedge \mathrm{dy}\bigwedge \mathrm{dz}$

$\mathrm{\ω2}≔\mathrm{evalDG}\left(x\mathrm{dx}\&w\mathrm{dy}\right)$

$\mathrm{\ω2}≔x\mathrm{dx}\bigwedge \mathrm{dy}$

$\mathrm{eta2a}≔\mathrm{DeRhamHomotopy}\left(\mathrm{\ω2}\right)$

$\mathrm{eta2a}≔-\frac{yx\mathrm{dx}}{3}+\frac{x^2\mathrm{dy}}{3}$

$\mathrm{eta2b}≔\mathrm{DeRhamHomotopy}\left(\mathrm{\ω2}\,\mathrm{initialpoint}=\left[x=1\,y=2\right]\right)$

$\mathrm{eta2b}≔-\left(\frac{1}{3}yx-\frac{2}{3}x+\frac{1}{6}y-\frac{1}{3}\right)\mathrm{dx}+\left(\frac{1}{3}{x}^2-\frac{1}{6}x-\frac{1}{6}\right)\mathrm{dy}$

$\mathrm{eta2c}≔\mathrm{DeRhamHomotopy}\left(\mathrm{\ω2}\,\mathrm{path}=''zigzag''\right)$

$\mathrm{eta2c}≔-yx\mathrm{dx}$

$\mathrm{eta2d}≔\mathrm{DeRhamHomotopy}\left(\mathrm{\ω2}\,\mathrm{path}=''zigzag''\,\mathrm{variableorder}=\left[y\,x\,w\,z\right]\right)$

$\mathrm{eta2d}≔\frac{x^2\mathrm{dy}}{2}$

$\mathrm{eta2e}≔\mathrm{DeRhamHomotopy}\left(\mathrm{\ω2}\,\mathrm{path}=''zigzag''\,\mathrm{variableorder}=\left[y\,x\,z\,w\right]\,\mathrm{initialpoint}=\left[x=1\,y=2\right]\right)$

$\mathrm{eta2e}≔\left(\frac{x^2}{2}-\frac{1}{2}\right)\mathrm{dy}$

$\mathrm{ExteriorDerivative}\left(\left[\mathrm{eta2a}\,\mathrm{eta2b}\,\mathrm{eta2c}\,\mathrm{eta2d}\,\mathrm{eta2e}\right]\right)$

$\left[x\mathrm{dx}\bigwedge \mathrm{dy}\,x\mathrm{dx}\bigwedge \mathrm{dy}\,x\mathrm{dx}\bigwedge \mathrm{dy}\,x\mathrm{dx}\bigwedge \mathrm{dy}\,x\mathrm{dx}\bigwedge \mathrm{dy}\right]$

$\mathrm{\ω3}≔\mathrm{evalDG}\left(\frac{1}{x^3}\mathrm{dx}\&w\mathrm{dy}\right)$

$\mathrm{\ω3}≔\frac{\mathrm{dx}}{x^3}\bigwedge \mathrm{dy}$

$\mathrm{\η3}≔\mathrm{DeRhamHomotopy}\left(\mathrm{\ω3}\,\mathrm{integrationlimits}=\left[1\,\mathrm{\infty }\right]\right)$

$\mathrm{\η3}≔-\frac{y\mathrm{dx}}{x^3}+\frac{\mathrm{dy}}{x^2}$

$\mathrm{\ω4}≔\mathrm{evalDG}\left(\frac{1}{x}\mathrm{dx}\&w\mathrm{dy}\right)$

$\mathrm{\ω4}≔\frac{\mathrm{dx}}{x}\bigwedge \mathrm{dy}$

$\mathrm{eta4a}≔\mathrm{DeRhamHomotopy}\left(\mathrm{\ω4}\,\mathrm{path}=''zigzag''\right)$

$\mathrm{eta4a}≔-\frac{y\mathrm{dx}}{x}$

$\mathrm{eta4b}≔\mathrm{DeRhamHomotopy}\left(\mathrm{\ω4}\,\mathrm{path}=''zigzag''\,\mathrm{initialpoint}=\left[x=1\right]\,\mathrm{variableorder}=\left[y\,x\,w\,z\right]\right)$

$\mathrm{eta4b}≔\ln \left(x\right)\mathrm{dy}$

$\mathrm{DGsetup}\left(\left[x\,y\,t\right]\,P\right)$

$\mathrm{frame\; name:\; P}$

$\mathrm{\omega }≔\mathrm{evalDG}\left(f\left(x\,y\,t\right)\mathrm{dx}\&w\mathrm{dt}+g\left(x\,y\,t\right)\mathrm{dx}\&w\mathrm{dy}\right)$

$\mathrm{\omega }≔g\left(x\,y\,t\right)\mathrm{dx}\bigwedge \mathrm{dy}+f\left(x\,y\,t\right)\mathrm{dx}\bigwedge \mathrm{dt}$

$\mathrm{DeRhamHomotopy}\left(\mathrm{\omega }\,\mathrm{cylinderconstruction}=t\right)$

$-\left({\int }_0^{1}f\left(x\,y\,t\right)\ⅆt\right)\mathrm{dx}$