If x=u(1+v) y=v(1+u), find the Jacobian of x,y with respect to u, v.

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Solution

$J = ⎡ ⎢ ⎣ \begin{matrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{matrix} ⎤ ⎥ ⎦ = [\begin{matrix} 1 + v & u v & 1 + u \end{matrix}] | J | = (1 + v) (1 + u) - u v = 1 + u + v$

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