Multivariable calculus, partial derivatives

[Feodalherren]

Homework Statement

Homework Equations

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.

 

[micromass]

Are you familiar with the chain rule?

 

[Feodalherren]

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

 

[micromass]

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?

 

[Feodalherren]

Spoiler

Correct?

 

[micromass]

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

 

[Feodalherren]

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

 

[micromass]

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

$F$ 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)?

 

[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.

 

[Ray Vickson]

Homework Statement

Homework Equations

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.

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.

 

[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?