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Unformatted text preview: Math 23 B. Dodson Week 12 Homework: [due April 13] 16.1, 16.2 vector fields and line integrals Week 13 Homework: [due April 20] 16.3 Fundamental Theorem for line integrals 16.4 Green’s Formula 16.5 Curl and Divergence [Slides for Week 11 Homework attached.] Problem 16.1.26. Find and sketch the gradient field of f ( x, y ) = 1 2 ( x + y ) 2 . Solution. The gradient field→ ∇ f = < f x , f y >, so f x = 1 2 · 2( x + y ) = x + y, and f y = 1 2 · 2( x + y ) = x + y gives→ ∇ f = < x + y, x + y > . For the sketch, we pick a few values→ ∇ f (1 , 2) = < 3 , 3 >,→ ∇ f (1 , 3) = < 4 , 4 >,→ ∇ f (1 , 2) = < 1 , 1 > and draw displacement vectors < 3 , 3 >, < 4 , 4 >, < 1 , 1 > starting at (1 , 2) , (1 , 3) , (1 , 2) respectively. Problem 16.2.5: Evaluate the line integral Z C xy dx + ( x y ) dy, where C is the curve consisting of line segments from (0 , 0) to (2 , 0) and from (2 , 0) to (3 , 2) . 2 Solution: By definition, Z C = Z C 1 + Z C 2 ....
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This note was uploaded on 02/29/2008 for the course MATH 23 taught by Professor Yukich during the Spring '06 term at Lehigh University .
 Spring '06
 YUKICH
 Integrals

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