18 . 325 : Finite Random Matrix Theory Volumes and Integration

because as we saw in Handout #3, the Jacobian emerges when we write the exterior product of the dy’s in terms of the dx’s. We will only concern ourselves with integration of n-forms on manifolds of dimension n. In fact, most of our manifolds will be flat (subsets of R), or surfaces only slightly more complicated than spheres. For example, the Stiefel manifold Vm,n of n by p orthogonal matrices Q (Q Q = Im) which we shall introduce shortly. Exterior products will give us the correct volume element for integration. If the xi are Cartesian coordinates in n-dimensional Euclidean space, then (dx) ≡ dx1 ∧ dx2 ∧ . . . dxn is the correct volume element. For simplicity, this may be written as dx1dx2 . . . dxn so as to correspond to the Lebesgue measure. Let qi be the ith component of a unit vector q ∈ R. Evidently, n parameters is one too many for specifying points on the sphere. Unless qn = 0, we may use q1 through qn−1 as local coordinates on the sphere, and then dqn may be thought of as a linear combination of the dqi for i < n. ( ∑