Question about the gradient of a function

Hello everyone,

This might be a bit of a silly question. Just looking at the definition of a gradient of a scalar field in wikipedia:

http://en.wikipedia.org/wiki/Gradient" [Broken]

So, the gradient points in the direction of the greatest increase in scalar field.

From the definition with the partial derivatives, it is not clear to me why that vector should point in the direction of greatest increase though. I understand how one computes the gradient but from that definition how can one conclude that this will point in the direction of greatest change?

Many thanks for any help you can give me.

Cheers,
Luc

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Answers and Replies

LCKurtz
Science Advisor
Homework Helper
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Think about the directional derivative of f at a point P in the direction of a unit vector u:

$$D_{\hat u}f(P) = \nabla f (P)\cdot \hat u = |\nabla f(P)| |\hat u|\cos(\theta)= |\nabla f(P)| \cos(\theta)$$

where $\theta$ is the angle between u and the gradient. Notice that the only variable in this equation is the direction of u. The directional derivative will be max when $\cos\theta$ = 1, which is when $\theta= 0$ which says u is pointed in the direction of the gradient. So the gradient must point in the direction of max rate of change at P.

Ah! That makes sense. Many thanks for the explanation. Much appreciated.

Luc