The two equations $F(x,y,u,v)=0$ and $G(x,y,u,v)=0$ determine $x$ and $y$ implicitly as functions of $u$ and $v$, say $x=X(u,v)$ and $y=Y(u,v)$. Show that $$\frac{\partial X}{\partial u}=\frac{\partial (F,G)/\partial (y,u)}{\partial (F,G)/\partial (x,y)}$$ at points at which the Jacobian $\partial (F,G)/\partial (x,y)\not=0$, and find similar formulas for the partial derivatives ${\partial X/\partial v},{\partial Y/\partial u}$ and ${\partial Y/\partial v}$.
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Here Jacobian means the determinant of the Jacobian matrix as explained in the following link - http://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant.