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Unformatted text preview: Math 200, Spring 2010 Handout #22 Section 16-1 Vector Fields Definition of a Vector Field Let D be a subset of 2 . A vector field on 2 is a function F that assigns to each point ( ) , x y in D a two-dimensional vector ( ) , x y F . F can be expressed in terms of its component functions as ( ) ( ) ( ) 1 2 , , , x y F x y F x y = + F i j , where 1 F and 2 F are scalar functions of two variables. Let W be a subset of 3 . A vector field on 3 is a function F that assigns to each point ( ) , , x y z in W a three-dimensional vector ( ) , , x y z F . F can be expressed in terms of its component functions as ( ) ( ) ( ) ( ) 1 2 3 , , , , , , , , x y z F x y z F x y z F x y z = + + F i j k , where 1 F , 2 F and 3 F are scalar functions of three variables. A vector field ( ) ( ) ( ) ( ) 1 2 3 , , , , , , , , , , x y z F x y z F x y z F x y z = F is smooth if its component functions 1 F , 2 F , and 3 F are continuously differentiable . 1. Sketch the vector field: (a) ( ) , x y y = + F i j (b) ( ) , x y y x = F i j (c) ( ) 2 2 , y x x y x y = + i j F (d) ( ) , , x y z = F k Velocity Field A velocity field on 3 is a vector field that assigns a vector to each point ( ) , , x y z in domain W . The speed at any given point in indicated by the length of the vector....
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This note was uploaded on 02/24/2011 for the course MATH 200 taught by Professor Jamesdcampbell during the Spring '10 term at Santa Barbara City.
- Spring '10