### Jacobian of a 3-component mapping with two independent variables

I have the map f(u,v) = (2u/(1 + u^2 + v^2), 2v/(1 + u^2 + v^2), (1 - u^2 - v^2)/(1 + u^2 + v^2)), and I need help showing that the Jacobian J of this map satisfies the condition that the ijth entry of (J^T)J is given by 4/[(1 + u^2 + v^2)^2] D_{ij} where D_{ij} is the Kronecker delta.