Question related to inverse map of 2 variables and chain rule.

Question

Let $(x,y)=F(u,v)$, and $(u,v)=G(x,y)$, (assume continuous etc).

Then $D(G\circ F)(u,v)=\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ by the chain rule.

If I use the chain rule I get the matrix $D(G\circ F)(u,v)=\begin{bmatrix} 1 & \frac{\partial{u}}{\partial{v}} \\ \frac{\partial{v}}{\partial{u}} & 1 \end{bmatrix}$.

So $\frac{\partial{u}}{\partial{v}}$ and $\frac{\partial{v}}{\partial{u}}$ must be equal to zero.

My problem is that I need help understanding why these partials are zero, I've thought about it for a while but can't explain why. Thanks.

Answer 1

It's because $u$ and $v$ are independent variables. Therefore, when $u$ alone changes, $v$ remains the same and therefore $\frac{\partial v}{\partial u}=0$. For the same reason, $\frac{\partial u}{\partial v}=0$.