 Jacobian matrix

For a differentiable map $F:U\subset {\mathbb{R}}^n\to {\mathbb{R}}^m$, the Jacobian matrix at $a$ represents the Fréchet derivative in the chosen coordinate bases:Rows correspond to output components and columns to input directions. The Jacobian matrix may be rectangular; only for $m=n$ does its Jacobian determinant exist.

For $F(x,y)=(x^2-y^2,2xy)$,The inverse function theorem therefore gives local invertibility away from the origin. The map is still globally two-to-one on the punctured plane because $F(-x,-y)=F(x,y)$. Under composition the chain rule becomes the matrix multiplication identity $J_{G\circ F}(a)=J_G(F(a))J_F(a)$.