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Date: 06/03/97 at 17:22:08
From: Fresco

I understand the mathematical way of finding the gradient of a three-
dimensional graph, but I think geometrically.  Since the gradient is a
vector, I know there is some geometrical significance of it.  Please
explain in geometric terms what a gradient is.

Date: 06/03/97 at 20:14:28
From: Doctor Tom

Hello Fresco,

Actually, there's a very nice geometric interpretation of the
gradient.  I assume you're talking about a surface over the x-y plane,
where the height of the surface is given by z = f(x,y).

The gradient vector is (df/dx, df/dy), where the "d" should really be
the symbol for partial derivative.

If you evaluate this vector at a given point (x,y), it points in the
direction of steepest climb up the surface.  The size of the vector is
proportional to the slope in that direction.

a surface to find local maxima (or you can go in the opposite
direction to find the minimun).

If the surface represents the strengh of an electric field, then the
gradient represents the direction and magnitude of force that a
charged particle in that field will experience.  (Or the negative of
the force if the particle has a negative charge.)

-Doctor Tom,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/

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