I guess the origins of the pdf are not that mysterious. It's just a generalization of what happens in the unidimensional case.
For some hints on how to remember the pdf in the multivariate case :
- If you ou replace the square root of the standard deviation by its equivalent : the square root of the determinant of the covariance matrix, because it has to be a unidimensional value and it kind of summarizes what happens in this covariance matrix.
- appears every time there is a normal distribution, so you will multiply as many times as there "are" normal distributions in the Gaussian vector.
- Usually, we have . If you have studied a bit quadratic forms, you will know that the square is a "special case" of the product of transposes.
As for a (more) formal approach (mine obviously isn't ), you may want to have a look at the book mentioned here : http://www.gbv.de/dms/goettingen/229762905.pdf (section 18.1), although I couldn't find it elsewhere than amazon.com.
And this : Multivariate normal distribution: Encyclopedia - Multivariate normal distribution looks nice, but I haven't seen any further because there are many links to read...