Is there a closed form expression for the covariance of a uniform distribution in a n-ball? I would like to develop a test for vector sums of points sampled from a uniform distribution in a n-ball. I need the covariance of the distribution to relate this to the CLT. I've taken a stab at it but I am not sure if I am correct.

Any pointers to statistics of uniform distributions in a n-ball would be very useful. Thanks.