Given two diffeomorphisms A and B, each of which is also a vector field, that is,

each entry of A or B is a vector. For example,

A = $\displaystyle \begin{bmatrix} (a,b) & (c,d) & (e,f) \\ (g,h) & (i,j) & (k,l) \\ (m,n) & (o,p) & (q,r) \end{bmatrix}$

Does anybody know how to calculate the product of A and B?

Thanks!