Uniform Distribution on a Sphere: Part 3b (The Solution by Calculus)

(This post continues where we left off in Part 2 and gives a more direct solution than Part 3a. However this post uses multivariable/vector calculus, and I do not provide many explanations for readers unfamiliar with that material.)

As we saw in Part 2, the problem with picking our spherical angles \theta and \phi uniformly was that the area on our two-dimensional grid changed when we transformed that surface into the sphere in three dimensions. In particular if we wanted to calculate the probability mass in a region R of our spherical angles, we have the formula from Part 0:

\int_R\!f_{\theta,\phi}(\theta,\phi)\,\mathrm{d}\theta\,\mathrm{d}\phi

where f_{\theta,\phi} is our probability density function. However when we move to the sphere our area elements are no longer little rectangles with area \mathrm{d}\theta\cdot\mathrm{d}\phi; they are little patches on the surface of a sphere. What area do these corresponding patches have?

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