Question

Let (u,v)=u^2-v^2,y(u,v)=2uv.

Find the the Jacobian determinant,J(u,v).

Answer 1

Let us first write the Jacobian matrix :

$J = \begin{pmatrix} \partial_u x & \partial_v x \\ \partial _u y & \partial_v y \end{pmatrix}$

Now let us calculate every term in this matrix :

$\begin{cases} \partial_ u x = 2u \\ \partial_v x = -2v \\ \partial _u y = 2v \\ \partial_v y = 2u \end{cases}$

And now we will calculate the determinant :

$\det J = 2u \cdot 2u - 2v\cdot (-2v) = 4(u^2+v^2)$