Changing notation in differential forms

asked 2013-06-07 05:36:47 -0500

Dox gravatar image

Dear all:

I'm trying to compute the curvature of a Schwarzschild metric using differential forms (Cartan formalism),

sage: reset()
sage: var('t,r,theta,phi')
sage: coords = [t,r,theta, phi]
sage: U = CoordinatePatch((t,r,theta, phi))
sage: Omega = DifferentialForms(U)
sage: X = function('X', r, latex_name=r"\Xi")
sage: f = exp(X)
sage: vi =[]
sage: for i in xrange(len(coords)):
...           vi.append(DifferentialForm(Omega,1))
sage: vi[0][0] = f
sage: vi[1][1] = 1/f
sage: vi[2][2] = r
sage: vi[3][3] = r*sin(theta)
sage: dvi=[]
sage: for i in xrange(len(coords)):
...       dvi.append(diff(vi[i]))
sage: dvi
[-e^X(r)*D[0](X)(r)*dt/\dr, 0, dr/\dtheta, sin(theta)*dr/\dphi + r*cos(theta)*dtheta/\dphi]

I'd like to know if it's possible to manipulate the result in a way that:

  • the term D[0](X)(r) in the last line could be written as X'(r) or just X'.
  • the dvi is expressed in terms of the vi-forms instead of the Omega-basis.

Any help is thanked.

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