Directional Derivatives and Gradient Vectors

Directional Derivatives and Gradient Vectors - ~ b = | ~a...

Info iconThis preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon
12.5 Directional Derivatives and Gradient Vec- tors We have calculated derivatives in the + x -direction ( f x ) and the + y -direction ( f y ). What about any other direction? Let P ( a, b ) be a point and f ( x, y ) a function. Suppose we want to find the derivative of f at ( a, b ) in the direction of a unit vector ~u = < u 1 , u 2 > . Parameterize the line through ( a, b ) in the direction of ~u . By the chain rule, Defn. Gradient Vector The gradient vector of f ( x, y ) at a point P ( a, b ) is Defn. Directional Derivative The directional derivative of f in the direction of any unit vector ~u is So, evaluated at ( x, y ): 1
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Example 1 Find the directional derivative of f ( x, y ) = 2 + 1 2 x - y at P (2 , 0) in the direction of ~v = < 1 , 2 > . Example 2 f ( x, y, z ) = xy + ln z , P (2 , - 1 , 1), ~v = < 1 , 0 , 4 > . Note From 1224: ~a ·
Background image of page 2
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 4
Background image of page 5
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ~ b = | ~a || ~ b | cos θ So, 1. 2 2. 3. Example 3 Suppose T ( x, y ) = 70 + xy represents level curves around a heat source. Let P (2 ,-1) be a point where you are sitting. 1. How hot is it? 2. If you move in the direction of ~ A = <-1 , 1 > is it getting hotter or cooler? 3 3. Find the direction of maximum temperature. .. increase? decrease? What is the maximum increase? 4. Find the direction to stay at 68 ◦ . 4 Example 4 Suppose at P (1 , 0) the derivative in the direction of <-4 , 2 > is 1 and in the direction of < 2 , 1 > it’s -2. Find the derivative at P (1 , 0) in the direction of < 1 ,-1 > . 5...
View Full Document

This note was uploaded on 03/03/2010 for the course MATH 2224 taught by Professor Mecothren during the Spring '03 term at Virginia Tech.

Page1 / 5

Directional Derivatives and Gradient Vectors - ~ b = | ~a...

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online