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DifferentialGeometry

  

IntegrateForm

  

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

 

Calling Sequence

Parameters

Description

Examples

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

withDifferentialGeometry:

 

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).

DGsetupx,y,M:

ωevalDGx2+3xydx &w dy

ωx2+3xydxdy

(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.

IntegrateFormω,x=0..1,y=0..1x

524

(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.

DGsetupt,N:DGsetupx,y,z,E3:

ωevalDGy2dx+z2dy+xyzdz

ωy2dx+z2dy+zxydz

(3)

ΦTransformationN,E3,x=sintcost,y=sintcost,z=ⅇt

Φx=sintcost,y=sintcost,z=ⅇt

(4)

IntegrateFormPullbackΦ,ω,t=0..π

310+310ⅇ2π

(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).

DGsetupθ,φ,N:DGsetupx,y,z,E3:

ωevalDG3y2zdx &w dy+xz2dy &w dzx2ydx &w dz

ω3y2zdxdyx2ydxdz+xz2dydz

(6)

ΦTransformationN,E3,x=cosθsinφ,y=2sinθsinφ,z=3cosφ

Φx=cosθsinφ,y=2sinθsinφ,z=3cosφ

(7)

IntegrateFormPullbackΦ,ω,θ=0..2π,φ=0..π

165π

(8)
N > 

See Also

DifferentialGeometry

Pullback

Transformation

VectorCalculus[int]

 


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