apply the homotopy operator for the de Rham complex on R^n to a differential form
a differential form
equations of the form integrationlimits = list, initialpoint = list, path="straightline", path="zigzag", variableorder = list, cylinderconstruction = var
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.
First define a manifold M with coordinates [x, y, z, w].
frame name: M
Define a differential 3-form omega1 and check that it is closed, that is, that the exterior derivative of omega1 is 0.
ω1 ≔ evalDG⁡y⁢dx &w dz &w dw+x⁢dy &w dz &w dw
To write omega1 as the exterior derivative of a 1-form, we apply the DeRhamHomotopy to omega1.
η1 ≔ DeRhamHomotopy⁡ω1
Check this result.
ExteriorDerivative⁡η1 &minus ω1
Here are a few simple examples which illustrate the use of the options to control the values returned by the DeRhamHomotopy command.
ω2 ≔ evalDG⁡x⁢dx &w dy
eta2a ≔ DeRhamHomotopy⁡ω2
Change the starting point for the straight line defining the integration path.
eta2b ≔ DeRhamHomotopy⁡ω2,initialpoint=x=1,y=2
Use a sequence of coordinate lines starting from the origin. Integrate along the x-axis, y-axis, z-axis, and w-axis.
eta2c ≔ DeRhamHomotopy⁡ω2,path=zigzag
Use a sequence of coordinate lines starting from the origin. Integrate along the y-axis, x-axis, w-axis, and z-axis.
eta2d ≔ DeRhamHomotopy⁡ω2,path=zigzag,variableorder=y,x,w,z
Use a sequence of coordinate lines starting from the point [x=1, y=2, z=0, w=0]. Integrate along the y-axis, x-axis, z-axis, and w-axis.
eta2e ≔ DeRhamHomotopy⁡ω2,path=zigzag,variableorder=y,x,z,w,initialpoint=x=1,y=2
The exterior derivative of each the forms eta2a ... eta2e returns the correct result, namely omega2.
Here are a few examples which illustrate the use of DeRhamHomotopy operator for forms with singularities.
ω3 ≔ evalDG⁡1⁢dx &w dyx3
η3 ≔ DeRhamHomotopy⁡ω3,integrationlimits=1,∞
ω4 ≔ evalDG⁡1⁢dx &w dyx
eta4a ≔ DeRhamHomotopy⁡ω4,path=zigzag
eta4b ≔ DeRhamHomotopy⁡ω4,path=zigzag,initialpoint=x=1,variableorder=y,x,w,z
Here is a simple example of the cylinder construction.
frame name: P
ω ≔ evalDG⁡f⁡x,y,t⁢dx &w dt+g⁡x,y,t⁢dx &w dy
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