The Jacobian and area differential

[WMDhamnekar]

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 $\lim_{u\to p}\frac {||T(u)-T(p)-J(p)(u-p)||}{||u-p||}=0.$when it exists, J(p) is the total derivative of T(u). What does this symbol || || indicate?

 

[I like Serena]

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 $\lim_{u\to p}\frac {||T(u)-T(p)-J(p)(u-p)||}{||u-p||}=0.$when it exists, J(p) is the total derivative of T(u). What does this symbol || || indicate?

Hi Dhamnekar Winod,

The notation $\|\mathbf x\|$ means the vector norm of $\mathbf x$.
If nothing else is specified it is the usual length of the vector and:
$$\|\mathbf x\| = \sqrt{x^2+y^2}$$