2. Facts from calculus and their d-dimensional analogues
The basic fact from calculus that powers the whole discussion is:
The identity with a = 1 is proved by the trick of calculating the square of theintegral in polar coordinates. The general identity follows by change of variablefrom x to .
This fact generalizes to higher-dimensional integrals. Set v = (v1, ..., vd)and dv = (dv1 ... dvd),and let A be a symmetric dby d matrix.
We use the fact that a symmetric matrix A isdiagonalizable: there exists an orthogonal matrix U(so Ut = U-1)such that UAU-1 is the diagonal matrix Bwhose onlynonzero entries are b11, ... , bddalong the diagonal. Then A = U-1BUand vtAv = vtU-1B Uv =vtUtB Uv =wtBw where w = Uv,using Ut = U-1 and (Uv)t =vtUt. Since Uis orthogonal detU = 1 and the change of variable from v to wdoes notchange the integral:
This follows from Proposition 1 by completion of the square in the exponent and a change of variables.
The generalization to ddimensions replaces awith Aas before and bwiththe vector b = (b1, ... , bd)
This is proven exactly like Proposition 2.If we write this integral as Zbthen the integral of Proposition 2 is Z0and this proposition can be rewritten as