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Surface integral with differential forms 
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#1
Jan1109, 08:04 PM

P: 6

Hi, I'm trying to solve a problem in David Bachman's Geometric Approach to Differential Forms (teaching myself.) The problem is to integrate the scalar function f(x,y,z) = z^2 over the top half of the unit sphere centered at the origin, parameterized by [tex]\phi(r,\theta) = (rcos\theta, rsin\theta, \sqrt{1  r^2}), 0 \leq r \leq 1, 0 \leq \theta \leq 2\pi[/tex]. I think we can evaluate this surface integral using the formula [tex]\int\int_{S}f(x,y,z)dS = \int\int_{D}f(\phi(r, \theta))\phi_{r}\times\phi_{\theta}drd\theta[/tex] yielding:
[tex] \int^{2\pi}_{0}\int^{1}_{0}(1r^2)\partial\phi/\partial r \times \partial\phi/\partial\theta dr d\theta [/tex] [tex] \int^{2\pi}_{0}\int^{1}_{0}(1r^2)<r^2cos\theta/\sqrt{1  r^2}, r^2sin\theta/\sqrt{1  r^2}, r> dr d\theta [/tex] [tex] \int^{2\pi}_{0}\int^{1}_{0}(1r^2)\sqrt{(r^4cos^2\theta + r^4sin^2\theta)/(1  r^2) + r^2} dr d\theta [/tex] [tex] \int^{2\pi}_{0}\int^{1}_{0}(1r^2)\sqrt{(r^2/(1  r^2)} dr d\theta [/tex] [tex] \int^{2\pi}_{0}d\theta\int^{1}_{0}r\sqrt{1  r^2} dr = 2\pi/3 [/tex] However, using differential forms, if we let [tex]\omega = z^2 dx \wedge dy[/tex] and use the same parameterization to integrate [tex]\omega[/tex] over the mentioned manifold, we get [tex]\int_{M}\omega = \int_{D}(1  r^2)\cdot(\partial\phi/\partial r, \partial\phi/\partial\theta)dx \wedge dy[/tex] (Here [tex]\partial\phi/\partial r, \partial\phi/\partial\theta[/tex] are the tangent vectors being acted on by [tex]dx \wedge dy[/tex]) [tex]\int_{D}(1  r^2) det[cos\theta \; rsin\theta , \; \; sin\theta \; rcos\theta]drd\theta[/tex] (the matrix rows are separated by the comma.) [tex]\int^{2\pi}_{0}\int^{1}_{0}(1  r^2)(r)drd\theta = \pi/2[/tex] Am I doing something wrong? If anyone can help I'd appreciate it, thanks! 


#2
Jan1209, 02:31 AM

P: 1

Hi!
In the first case you are doing nothing more than integrating the form [tex]f(\phi(r,\theta))\:\phi^*[dr\wedge d\theta]=f(\phi(r,\theta)) dS[/tex] acting on the vectors [tex]\partial_r\phi,\:\partial_{\theta}\phi[/tex] (here [tex] \phi^{*}[/tex] represents the pushforward of a form). This corresponds to projecting the tangent vectors [tex]\partial_r\phi,\:\partial_{\theta}\phi[/tex] to the plane (x,y) and then integrating on the disk [tex]0 \leq r \leq 1, 0 \leq \theta \leq 2\pi[/tex], with the factor [tex]\partial\phi/\partial r \times \partial\phi/\partial\theta[/tex] accounting for the deformation due to projection to the horizontal plane. So this is the right result. In the second case you are integrating [tex]f\;dx\wedge dy[/tex], but [tex]dx\wedge dy [/tex] is not the surface element [tex]dS[/tex]. Thus the result is different. 


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