Can someone pls help me solve this Jacobian. Let x^3 + y^3 = u+v, x^2+y^2 = u^3+v^3, prove that J ((x,y)/(u,v)) = 1/2((y^2-x^2)/2uv (u-v)).?

Let F = x^3 + y^3 - u - v = 0
and G = x^2 + y^2 - u^3 - v^3 = 0.

Then, ∂(x,y)/∂(u, v) = -[∂(F, G)/∂(u, v)] / [∂(F, G)/∂(x, y)].
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Computing the individual jacobians:
∂(F, G)/∂(u, v) =

|∂F/∂u ∂F/∂v|
|∂G/∂u ∂G/∂v| =

|....-1......-1...|
|-3u^2 -3v^2| = 3v^2 - 3u^2.
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∂(F, G)/∂(x, y) =

|∂F/∂x ∂F/∂y|
|∂G/∂x ∂G/∂y| =

|3x^2 3y^2|
|...2x.....2y..| = 6x^2 y - 6xy^2.
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So, ∂(x,y)/∂(u, v)
= -(3v^2 - 3u^2)/(6x^2 y - 6xy^2)
= (-v^2 + u^2)/(2x^2 y - 2xy^2).

I hope this helps!

This is simple. This is the determinant value

j = | dx/du dy/dv|
|dx/dv dy/du|