Is the Gradient of a Composite Function Always Zero?

[t_n_p]

Homework Statement

http://img21.imageshack.us/img21/8175/46521897.jpg

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?

 

[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"

 

[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...

 

[t_n_p]

can anybody help please? :P

 

[Дьявол]

Could you possibly provide me the name of the textbook?

Regards.