Integrate after Pullback

asked 2018-02-23 20:29:37 +0200

629 ●8 ●27 ●40

How to integrate a differential for obtained by pullback?

Let say I have calculated the pullback of some differential form, the result is (in latex):

$\rho^2\sin(\theta)\,\mathrm{d}\rho\wedge\mathrm{d}\theta\wedge\mathrm{d}\phi$

The triple integral is:

sage: integral(integral(integral(rho^2*sin(theta),rho,0,1),theta,0,pi/7),phi,0,pi/5)

But I want this process to be automatic, i.e. not manually write the integrand in the triple integral by myself. I want to be able to extract the integrand from the pullback and put it in the triple integral. I that possible?

Daniel

answered 2018-02-23 21:53:54 +0200

eric_g
8209 ●10 ●61 ●162 https://luth.obspm.fr/...

Suppose your 3-form is tau; to extract the integrand, simply use tau[1,2,3].expr(), or tau[frame,1,2,3,chart].expr() if the vector frame and the chart are not the default ones on the manifold. Note that expr() returns a symbolic expression, which can be passed directly to integrate. For instance, for the example of question 41228, this would be

sage: tau.display(spherical.frame(), spherical)
tau = rho^2*sin(theta) drho/\dtheta/\dphi
sage: tau[spherical.frame(),1,2,3,spherical].expr()
rho^2*sin(theta)

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thanks Eric.

danielvolinski ( 2018-02-24 09:38:08 +0200 )edit

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