Jacobian Calculation for Transformation (x, y) to (u, v)

[franky2727]

calculate the jacobian d(x,y)/d(u,v) of the transformation u=x²+y²
v=x+y

for this do i first have to calculate the jacobian d(u,v)/d(x,y) then do 1over the answer? because i would assume the matrix to be det|{(dudx,dudy)(dvdx,dvdy)} but with (u,v) on top i cannot get this

 

[HallsofIvy]

calculate the jacobian d(x,y)/d(u,v) of the transformation u=x²+y²
v=x+y

for this do i first have to calculate the jacobian d(u,v)/d(x,y) then do 1over the answer? because i would assume the matrix to be det|{(dudx,dudy)(dvdx,dvdy)} but with (u,v) on top i cannot get this

You can do any of three things:
1) Solve for x and y as functions of u and v.
2) Use implicit differentiation to find $\partial x/\partial u$, $\partial y/\partial u$, $\partial x/\partial v$, and $\partial y/\partial v$.
3) Take the reciprocal of d(u,v)/d(x,y).