The Gradient

Date: 06/03/97 at 17:22:08
From: Fresco
Subject: high school:calculus:gradient

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.  

Thanks for your time.

Date: 06/03/97 at 20:14:28
From: Doctor Tom
Subject: Re: high school:calculus:gradient

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.

There are lots of mathematical techniques to "follow the gradient" on 
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/