Thread: gradient field

how can I prove that if s(t) is a flux line of a gradient field F=-1*nabla(U)
tehn U(s(t)) is a decreasing function of t

always points in the direction of greatest increase of the function U. Therefore, points in the direction of greatest decrease.

that I was gessing that was going to be the starting point. I just can't get any further...how does that prove what I want to prove?

Originally Posted by Mppl

how can I prove that if s(t) is a flux line of a gradient field F=-1*nabla(U)
tehn U(s(t)) is a decreasing function of t

Since is the direction of greatest decrease (as said above), then any would be a decreasing function of t.

Basically it isn't asking you to prove that s(t) is a flux line, it's asking that IF s(t) is a flux line with the gradient field of F, then is U(s(t)) a decreasing function.

Well, let me see! You're saying that s(t) is a decreasing function imagine it is... Why Does ir imply that U(s(t)) is a decreasing function aswell? Can't get there. Bur I would lime you to tell me aswell Why the hell is s(t) a decreasing function. Isn't ir just a curve?

No one has said that s(t) is a decreasing function! In fact, if U is a function of two or three variables, s(t) maps R into or which are not ordered so "increasing" and "decreasing" make no sense for s(t). I am beginning to think you are completely confused as to what this question is asking.

(By the way, you will have to take "decreasing" in the more general sense of "non-increasing". For example, if U is a constant function the U(s(t)) will be constant for any s.)

So you're stating simply that s(t) is decreasing, so any function, U(s(t)) is decreasing.

I gess I've not commited a mistake saying someone told it was decreasing...someone said that s(t) is decreasing I'm not mad, not yet...

And why would that statement be right? imagine U(s(t))=-s(t) then if s(t) is increasing U(s(t)) is decreasing. Maybe you're not explaning things clearly...or maybe (and most probably, I'm missing something) doesn't it have a formal proof?

Originally Posted by moderata

Since is the direction of greatest decrease (as said above), then any would be a decreasing function of t.

Basically it isn't asking you to prove that s(t) is a flux line, it's asking that IF s(t) is a flux line with the gradient field of F, then is U(s(t)) a decreasing function.

So you're stating simply that s(t) is decreasing, so any function, U(s(t)) is decreasing.

Okay, I didn't see this last sentence. Of course, it isn't correct. As I said, it, in general, makes no sense to say that s(t) is decreasing and, even if it were, it does not follow from that that U(s(t)) is decreasing.

The crucial point is, as I said before that [tex]\nabla U[tex] points in the direction of fastest increase and so points in the direction of fastest decrease.

Originally Posted by Mppl

............... doesn't it have a formal proof?

Here goes one:
F = -grad(U), s(t) is a curve such that its velocity vector( derivative) s'(t) = F. Let's compute the derivative of U(s(t)) as a function of t.
d( U(s(t)) )/dt (using the chain rule, use <,> to denote inner product) = <grad(U), s'(t)> = <-F, F> = -||F||^2 < 0. DONE.