I am stuck on a question in my calc III class which is shown in the link below (part a). I completely understand how to find the Jacobian; however I don't understand why the relationship shown is true. I have searched for hours for an answer to this but cannot seem to find one.
In terms of the Jacobian, I got $$J = \frac{1}{2u}$$.
Thanks for your time.
Hint. As regards (a), note that $x(u(x,y),v(x,y))=x$ and $y(u(x,y),v(x,y))=y$. Hence, by taking the derivative with respect to $x$ of both sides of $x(u(x,y),v(x,y))=x$, by the chain rule we get $$\frac{\partial x}{\partial u}\cdot \frac{\partial u}{\partial x}+ \frac{\partial x}{\partial v}\cdot \frac{\partial v}{\partial x}=1.$$ On the other hand, the derivative with respect to $y$ of $x(u(x,y),v(x,y))=x$ yields $$\frac{\partial x}{\partial u}\cdot \frac{\partial u}{\partial y}+ \frac{\partial x}{\partial v}\cdot \frac{\partial v}{\partial y}=0.$$ Can you take it from here?