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When using the formula for the surface area integral and using some change of coordinates, i.e. spherical coordinates, does the area element dA change to $r^2sin(\phi)\partial\phi\partial\theta$ ? Because I noticed in Dr. Paul Lamar's final example here he doesn't use the spherical area element. He simply uses the area element $\partial\phi\partial\theta$.

That's confusing to me. I feel like if he does a spherical change of coordinates, the area element he should be using is the spherical area element.

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Exactly where? The final example (number 4) is a cylinder, so no spherical coordinates there... –  Hans Lundmark Sep 26 '12 at 6:50
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If you are referring to Example 2 (the only example with spherical), then notice that he does conclude the surface element changes is $r^2\sin(\phi) d\phi d\theta$, where he calculates that $\|r_{\theta} \times r_{\phi}\| = 4 \sin{\phi}$. This is precisely what you want, since the radius of the sphere was $r = 2$. In that problem, he uses $dA$ to mean $d\phi d\theta$.

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