I don't understand the following definition. If we let $u=\langle u,v \rangle$ , $p=\langle p,q\rangle,$ $x=\langle x,y \rangle$,then (x,y)=T(u,v) is given in vector notation by
x=T(u). A coordinate transformation T(u) is differentiable at a point p , if there exists a matrix J(p) for which...