Let $R(s,t) = G(u(s,t), v(s,t))$, where $G$, $u$, and $v$ are differentiable. What is $R_s(1,2)$ and $R_t(1,2)$?

Question

Here's everything that's given:

$u(1,2)$ = $5$

$u_s(1,2)$ = $4$

$u_t(1,2)$ = $-3$

$v(1,2)$ = $7$

$v_s(1,2)$ = $2$

$v_t(1,2)$ = $6$

$G_u(5,7)$ = $9$

$G_v(5,7)$ = $-2$

I would post my attempt, but I have no idea where to start. I know I have to split $R_s$ and $R_t$ into a bunch of partial derivatives involving the other variables, but I haven't found anything that works, nor do I have an answer in my book to check for an answer.

Answer 1

Since $G$ is a function of $u$ and $v$, and $u$ and $v$ are each functions of $s$ and $t$, we have: $$R_s=\frac{\partial G}{\partial u}\cdot\frac{\partial u}{\partial s} + \frac{\partial G}{\partial v}\cdot\frac{\partial v}{\partial s}, \quad R_t=\frac{\partial G}{\partial u}\cdot\frac{\partial u}{\partial t} + \frac{\partial G}{\partial v}\cdot\frac{\partial v}{\partial t}.$$

Now, at the point with coordinates $s=1,t=2$, we obtain: $$R_s(1,2)=G_u(5,7)\cdot u_s(1,2)+ G_v(5,7)\cdot v_s(1,2)=9\cdot4-2\cdot2=32.$$

$$R_t(1,2)=G_u(5,7)\cdot u_t(1,2)+ G_v(5,7)\cdot v_t(1,2)=9\cdot(-3)-2\cdot6=-39.$$