# Multivariable calculus, partial derivatives

1. Mar 4, 2014

### Feodalherren

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution
Umm can somebody explain to me what just happened. None of that makes any sense to me what so ever.

2. Mar 4, 2014

### micromass

Staff Emeritus
Are you familiar with the chain rule?

3. Mar 4, 2014

### Feodalherren

Of course. The notation in this problem is so confusing that I can't follow what's happening.

4. Mar 4, 2014

### micromass

Staff Emeritus
Let's use the notation $G(s,t) = (u(s,t), v(s,t))$.

What you need to do is to compute

$$\frac{\partial}{\partial s} F\circ G$$

What will that be according to the chain rule?

5. Mar 4, 2014

Correct?

6. Mar 4, 2014

### micromass

Staff Emeritus
What happened to $F$? I can only see $G$ showing up.

7. Mar 4, 2014

### Feodalherren

Ok I have no idea.. I can't remember what FoG means :/. You didn't even define F as anything?

8. Mar 4, 2014

### micromass

Staff Emeritus
$F$ is just an arbitrary differentiable map (as shown in the problem statement).

What $F\circ G$ means is that it is the map which sends $(s,t)$ to $F(G(s,t))$.

Can you show me the chain rule you've learned (and perhaps the variations)?

9. Mar 4, 2014

### Feodalherren

I've only learned the one that I demonstrated. You draw one of those trees and then just differentiate however many times you need to in order to get to the variable that you need.

I'm totally lost on the notation yet again.

10. Mar 4, 2014

### Ray Vickson

Is the "subscript" notation throwing you off? If so, forget it and use a more exact nomenclature: $W = F(u,v)$ gives
$$\frac{\partial W}{\partial s} = \frac{\partial F}{\partial u} \frac{\partial u}{\partial s} + \frac{\partial F}{\partial v} \frac{\partial v}{\partial s}\: \longleftarrow \text{ chain rule}\\ \text{ }\\ \text{or}\\ \text{ }\\ W_s = F_u u_s + F_v v_s$$
etc.

11. Mar 4, 2014

### Feodalherren

Ok that helps Ray. Now for the evaluation.

If I know Ws (1,0) then
s=1, t=0

Then I get:
u (1,0) and s (1,0).

Where do I evaluate the first therm dF/du?