Partial Derivatives and GradientDate: 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/ |
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