# Grad of composite function

1. Aug 18, 2009

### t_n_p

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

http://img21.imageshack.us/img21/8175/46521897.jpg [Broken]

3. The attempt at a solution

I think I have a starting point, but I'm not 100% sure
Basically I thought of just computing grad(f(α(t)) · dα/dt and showing its equal to zero.

Am I on the right track, or shall I try another approach?

Last edited by a moderator: May 4, 2017
2. Aug 19, 2009

### lanedance

hey tnp, i think you might want to start by taking the gradient of both sides of the original equation: grad(f(α(t)) = grad(c)

as the prove is an "if & only if", you might have to think about whether this proves both directions of the theorem, ie. "if" and "only if"

3. Aug 19, 2009

### t_n_p

\cdotok, say I were to take grad of lhs, I need to apply chain rule since f(α(t))

so..

df/dt = (df/dα)(dα/dt)

I recognise (dα/dt) as $$\dot{\alpha}$$, which leads me to

df/dt = (df/dα)($$\dot{\alpha}$$)

Also, since $$\alpha$$ has components $$\alpha$$1, $$\alpha$$2, $$\alpha$$3, ......, $$\alpha$$n+1

df/dα = ($$\partial$$f/d$$\alpha$$1, $$\partial$$f/d$$\alpha$$2, ....,$$\partial$$f/d$$\alpha$$n+1) which I recognise is $$\nabla$$f(α(t)),

this df/dt = $$\nabla$$f(α(t)) $$\cdot$$ $$\dot{\alpha}$$ which is what I wanted.

I hope i'm correct up to here and it isn't too messy to show with the latex...

But as you said before, the question states, if and only if, which means I have to show both ways. Puzzled as to how to do the reverse way...

4. Aug 20, 2009

### t_n_p

can anybody help please? :P

5. Aug 20, 2009

### Дьявол

Could you possibly provide me the name of the textbook?

Regards.

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