This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: The derivative of f at P o (x o ,y o ) in the direction of the unit vector u=u 1 i+u 2 j is the number df ds u,Po = lim sf 1 2 ( , ) ( , ) o o o o f x su y su f x y s + + provided the limit exits. The gradient vector (gradient) of f(x,y) at a point P o (x o ,y o ) is the vector df df f dx dy = + i j obtained by evaluating the partial derivatives of f at P o . If the partial derivatives of f(x,y) are defined at P o (x o ,y o ), then , ( ) , o p u Po df f ds = u the scalar product of the gradient f a P o and u. Properties of the Directional Derivative D u f= cos f f = u 1. The function f increases most rapidly when cos =1, or when u is the direction of h f. That is, at each point P in its domain, f increases most rapidly in the direction of the gradient vector f f at P. The derivatives in this direction is D u f = ( 29 cos 0 . f f = 2. Similarly, f decreases most rapidly in the direction of h f. The derivative in this direction is D u f = ( cos( )) . f f =  3. Any direction u orthogonal to the gradient is a direction of a zero change in f because then equals / 2 and D u f= ( 29 cos / 2 f f = = At every point (x o ,y o ) in the domain f(x,y), the gradient of f is normal to the level curve through (x o ,y o ). The tangent plane at the point P o (x o ,y o ,z o ) on the level surface f(x,y,z)=c is the plane through P o normal to o P f f . The normal line of the surface at P o is the line through P o is the line through P o parallel to o p f f ....
View
Full
Document
This note was uploaded on 09/23/2007 for the course MATH 1920 taught by Professor Pantano during the Spring '06 term at Cornell University (Engineering School).
 Spring '06
 PANTANO
 Math, Derivative, Multivariable Calculus, Cos, Duf, cos cos

Click to edit the document details