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The two equations $F(x,y,u,v)=0, G(x,y,u,v)=0$ determine $x$ and $y$ implicitly as the functions of $u$ and $v$, say $x = X(u,v)$ and $y = Y(uv)$. Show that

$$\dfrac {\partial X} {\partial u} = \dfrac {\partial (F,G)/\partial (y,u)}{\partial(F,G)/\partial (x,y)}$$

Attempt: We have the following equations :

$F[X(u,v), Y(u,v), u,v ] = 0 = G[X(u,v), Y(u,v), u,v ] ~~~........(1)$

Now, the LHS of the problem asks for $X$ to be partially differentiated with respect to $u$. But, I tried differentiating $(1)$ to get nothing?

Could anyone tell me a way to proceed in this problem?

Thank you for helping!

Shreya Taneja
  • 427
  • Use that $\frac{dF}{du} = \frac{\partial F}{\partial u} + \frac{\partial F}{\partial x}\frac{\partial X}{\partial u} + \frac{\partial F}{\partial y} \frac{\partial Y}{\partial u}$ and similary for $G$. Since $F(X,Y,u,v) = 0 = G(X,Y,u,v)$ the total derivative is zero. Now solve the linear system for $\frac{\partial X}{\partial u}$ (eliminate $\frac{\partial Y}{\partial u}$) and use the definition for the determinant of the two Jacobi matrices to get the result. – Winther Feb 04 '15 at 01:13
  • I got it. Thank you for helping me point out my mistake – Shreya Taneja Feb 04 '15 at 01:15

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