Explore BrainMass

Jacobian of a 3-component mapping

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.

Solution Preview

The basic approach is to define new variables x, y, z to denote the first, second, and third components, respectively, of the mapping f:

x = x(u, v) = 2u/(1 + u^2 + v^2)
y ...

Solution Summary

First, the Jacobian J of the given mapping is found. Then J is used to find the transpose J^T of the Jacobian. Finally, the product (J^T)J is computed, and a step-by-step derivation is given to prove that (J^T)J does indeed satisfy the stated condition.