Partial Derivatives and Gradient

Date: 12/12/98 at 02:27:21
From: alfredo
Subject: Partial and directional derivative, gradient

I don't understand the definitions of partial and directional 
derivatives and gradients. Could you explain the meaning and relations 
between them?

Thank you.

Date: 12/12/98 at 07:39:54
From: Doctor Jerry
Subject: Re: Partial and directional derivative, gradient

Hi Alfredo,

I'll assume that you know the meaning and definitions of partial 
derivatives, which are directional derivatives in the {1,0} and {0,1} 
directions, for functions of two variables. The directional derivative 
D_u f(a) at a point a = {a_1, a_2} and in the u = {u_1, u_2} direction 
(u_1 means u sub 1, etc) is the rate of change of f in the u-direction, 
that is, the limit as h->0 of:

   [f(a+h*u) - f(a)]/h

where u = {u_1, u_2} is a unit vector, and h is a number.

If u = {1,0}, then D_u f(a) is the partial derivative f_x of f with 
respect to x.

Under mild hypotheses, D_u f(a) = grad(f) dot u, where 
grad(f) = {f_x, f_y}. The gradient direction is the direction in which 
the directional derivative is a maximum.

For more information on the gradient, see the archives:

   The Gradient
   http://mathforum.org/dr.math/problems/fresco6.3.97.html   

- Doctor Jerry, The Math Forum
  http://mathforum.org/dr.math/