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That equation is rather unnecessary showing off, I don't see it being used later on, and the author doesn't bother trying to explain it. It looks like (going by the GM/r) that it might be the gravitational potential for a mass distribution described by a bunch of spherical harmonics.

Yes, especially as one of the key advantages of using a projection is that you don't have to worry about this stuff. You can just work with a Cartesian coordinate system and the results will probably be right.

The equation is barely even showing off: if you decode it, that's just a bunch of algebraic manipulations. There's not even any calculus in there.

Calculus is basically algebra with a bigger vocabulary of operations.

Depends. Calculus deals with limits and integration. Algebra is more interested in how groups, rings and fields and other structures work.

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