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DifferentialGeometry

 IntegrateForm
 evaluate a p-fold iterated integral of a differential p-form

 Calling Sequence IntegrateForm(omega, integrationlimits)

Parameters

 omega - a differential p-form on a p-dimensional manifold N integrationlimits - a p-term sequence t1 = a1 .. b1, t2 = a1(t1) .. b1(t1), t3 = a3(t1, t2) .. b3(t1, t2), ..., where t1, t2, t3, ... are coordinates on N, defining a p-dimensional region in N

Description

 • 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.

Examples

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

Example 1.

Integrate the 2-form (x^2 + 3*x*y)*dx &w dy over the triangle T with vertices (0, 0), (1, 0), and (0, 1).

 > $\mathrm{DGsetup}\left(\left[x,y\right],M\right):$
 > $\mathrm{ω}≔\mathrm{evalDG}\left(\left({x}^{2}+3xy\right)\mathrm{dx}&w\mathrm{dy}\right)$
 ${\mathrm{ω}}{≔}\left(\left({{x}}^{{2}}{+}{3}{}{x}{}{y}\right){}{\mathrm{dx}}\right){}{\bigwedge }{}{\mathrm{dy}}$ (1)

To evaluate the double integral over T we note that for a point (x, y) in T the variable x ranges from 0 to 1 and, for a given x value, y ranges from 0 to 1 - x.

 > $\mathrm{IntegrateForm}\left(\mathrm{ω},x=0..1,y=0..1-x\right)$
 $\frac{{5}}{{24}}$ (2)

Example 2.

Compute the line integral of the 1-form omega = y^2*dx + z^2*dy + x*y*z*dz along the curve x = sin(t)*cos(t), y = sin(t)*cos(t), z = exp(t) from t = 0 to t = Pi.

 > $\mathrm{DGsetup}\left(\left[t\right],N\right):$$\mathrm{DGsetup}\left(\left[x,y,z\right],\mathrm{E3}\right):$
 > $\mathrm{ω}≔\mathrm{evalDG}\left({y}^{2}\mathrm{dx}+{z}^{2}\mathrm{dy}+xyz\mathrm{dz}\right)$
 ${\mathrm{ω}}{≔}{{y}}^{{2}}{}{\mathrm{dx}}{+}{{z}}^{{2}}{}{\mathrm{dy}}{+}{z}{}{x}{}{y}{}{\mathrm{dz}}$ (3)
 > $\mathrm{Φ}≔\mathrm{Transformation}\left(N,\mathrm{E3},\left[x=\mathrm{sin}\left(t\right)\mathrm{cos}\left(t\right),y=\mathrm{sin}\left(t\right)\mathrm{cos}\left(t\right),z={ⅇ}^{t}\right]\right)$
 ${\mathrm{Φ}}{≔}\left[{x}{=}{\mathrm{sin}}{}\left({t}\right){}{\mathrm{cos}}{}\left({t}\right){,}{y}{=}{\mathrm{sin}}{}\left({t}\right){}{\mathrm{cos}}{}\left({t}\right){,}{z}{=}{{ⅇ}}^{{t}}\right]$ (4)
 > $\mathrm{IntegrateForm}\left(\mathrm{Pullback}\left(\mathrm{Φ},\mathrm{ω}\right),t=0..\mathrm{π}\right)$
 ${-}\frac{{3}}{{10}}{+}\frac{{3}}{{10}}{}{{ⅇ}}^{{2}{}{\mathrm{π}}}$ (5)

Example 3.

Compute the surface integral of the 1-form omega = y^2*z^2*dx &w dy + x^2*y^2*dy &w dz + x^2*z^2*dx &w dz over the surface of the ellipsoid x^2 + y^2/4 + z^2/9 = 1.

We shall parameterize the surface of the ellipsoid with coordinates (theta, phi) and map x = cos(theta)*sin(phi), y = 2*sin(theta)*sin(phi), z = 3*cos(phi).

 > $\mathrm{DGsetup}\left(\left[\mathrm{θ},\mathrm{φ}\right],N\right):$$\mathrm{DGsetup}\left(\left[x,y,z\right],\mathrm{E3}\right):$
 > $\mathrm{ω}≔\mathrm{evalDG}\left(-3{y}^{2}z\mathrm{dx}&w\mathrm{dy}+x{z}^{2}\mathrm{dy}&w\mathrm{dz}-{x}^{2}y\mathrm{dx}&w\mathrm{dz}\right)$
 ${\mathrm{ω}}{≔}{-}\left(\left({3}{}{{y}}^{{2}}{}{z}{}{\mathrm{dx}}\right){}{\bigwedge }{}{\mathrm{dy}}\right){-}\left(\left({{x}}^{{2}}{}{y}{}{\mathrm{dx}}\right){}{\bigwedge }{}{\mathrm{dz}}\right){+}\left({x}{}{{z}}^{{2}}{}{\mathrm{dy}}\right){}{\bigwedge }{}{\mathrm{dz}}$ (6)
 > $\mathrm{Φ}≔\mathrm{Transformation}\left(N,\mathrm{E3},\left[x=\mathrm{cos}\left(\mathrm{θ}\right)\mathrm{sin}\left(\mathrm{φ}\right),y=2\mathrm{sin}\left(\mathrm{θ}\right)\mathrm{sin}\left(\mathrm{φ}\right),z=3\mathrm{cos}\left(\mathrm{φ}\right)\right]\right)$
 ${\mathrm{Φ}}{≔}\left[{x}{=}{\mathrm{cos}}{}\left({\mathrm{\theta }}\right){}{\mathrm{sin}}{}\left({\mathrm{\phi }}\right){,}{y}{=}{2}{}{\mathrm{sin}}{}\left({\mathrm{\theta }}\right){}{\mathrm{sin}}{}\left({\mathrm{\phi }}\right){,}{z}{=}{3}{}{\mathrm{cos}}{}\left({\mathrm{\phi }}\right)\right]$ (7)
 > $\mathrm{IntegrateForm}\left(\mathrm{Pullback}\left(\mathrm{Φ},\mathrm{ω}\right),\mathrm{θ}=0..2\mathrm{π},\mathrm{φ}=0..\mathrm{π}\right)$
 $\frac{{16}}{{5}}{}{\mathrm{π}}$ (8)
 N >