Chain rule for multivariable functions confusion. General formula

Question

I'm struggling to work out what is what in the formula for the chain rule in general. Say we have $f(x,y)=(u(x,y),v(x,y)).$ And then we have $g(x,y)=(p(x,y),q(x,y))$. So $(f \circ g)(x,y)=u(p(x,y),q(x,y)),v(p(x,y),q(x,y)).$ What is the formula for $\frac{\partial (f \circ g)}{\partial x} $? I can't seem to get it to make any sense.

Answer 1

You have the chain $$ (x,y)\to (p(x,y), q(x,y))\to (u(p(x,y),q(x,y)), v(p(x,y),q(x,y)). $$ and therefore $$ \frac{\partial (f\circ g)}{\partial x}=\left(\frac{\partial u}{\partial s}\cdot \frac{\partial p}{\partial x}+\frac{\partial u}{\partial t}\cdot\frac{\partial q}{\partial x},\frac{\partial v}{\partial s}\cdot \frac{\partial p}{\partial x}+\frac{\partial v}{\partial t}\cdot\frac{\partial q}{\partial x}\right), $$ where the derivatives $\frac{\partial u}{\partial s}$, $\frac{\partial u}{\partial t}$ etc. are evaluated at $(p(x,y), q(x,y))$.

It is easier to use the matrix language and multiply the corresponding Jacobian matrices.