Integrating Gravitational Attraction in n Dimensions

[Taylor Smith]

Homework Statement

I'm working on a generalization of gravitation to n dimensions. I'm trying to compute gravitational attraction experienced by a point mass y due to a uniform mass distribution throughout a ball of radius a -- B(0, a).

Homework Equations

3. The Attempt at a Solution [/B]

I've determined an integral that expresses this problem, (ignoring the constants outside the integral) but I'm unsure how to evaluate it.

I have $$A = \int_{B(0,a)} \frac{x - y}{||x - y||^n} dvol_n(x)$$
I believe this can be expressed as a function of x_n, thus I've further simplified to
$$A = \int_{B(0,a)} \frac{x_n - r}{||x - re_n||^n} dvol_n(x)$$
where $r$ is the norm of y, and e_n is the unit vector that is 0 in all but the nth position. I'm unsure how to proceed with this integral. I'm trying to express it in terms of only a and r.

 

[mfb]

What is y (and similar r) and why does A do not depend on it?
I would split the integral in three parts:
- radial direction
- angle between x and the nth direction
- all other directions

3 dimensions are the first where these integrals are all meaningful, so it might be useful to study this case first and then generalize this.