Multivariate calculus

This note covers some practical issues when dealing with function of several variables: differentiation, integration on vector spaces. The usual textbook approach is to use partial derivative, which I find highly unsatisfactory for several reasons. First, differentiating non linear functional of vectors and matrices is difficult to do in the context of partial derivative: for example, differentiating formula involving inverse matrices or determinants, and those often appear in statistics (think maximum likelihood estimator of multivariate Gaussian).

Also, when integrating over vector spaces or matrix spaces, it is often required to do a transform on the integrand for easier computation. For example, integrating over the space of definite positive matrices is often easier when the matrices are triangularized: this requires finding the Jacobian of some matrix transform, which is easier to do when the Jacobian is understood as the matrix representation of the differential.

multivariate_calculus.pdf

Note: This is really incomplete for now. More recent sources may be found here (bzr branch)