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

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

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

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

View Full Document
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 ·

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

View Full Document
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
Ask a homework question - tutors are online