Chain rule for Jacobian matrices

Remarks

This is the chain rule in several variables, written with matrices: the matrix of a composition of linear maps is the product of their matrices. For $k = m = n$, taking determinants gives $\det J_{f \circ g}(a) = \det J_f(g(a)) \cdot \det J_g(a)$. For example, let $g(r, \theta) = (r \cos \theta, r \sin \theta)$ and $f(x, y) = x^2 + y^2$. Then $J_f(g(r, \theta)) J_g(r, \theta) = (2r \cos \theta, \, 2r \sin \theta) J_g(r, \theta) = (2r, \, 0)$, which is the Jacobian matrix of $(f \circ g)(r, \theta) = r^2$.