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s07wk12 - Math 23 B Dodson Week 12 Homework[due April 13...

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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) .
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2 Solution: By definition, Z C = Z C 1 + Z C 2 . For C 1 we pick the simplest parameterization x = t, y = 0 for 0 t 2 . Then substituting the parameterization, xy = 0 , so Z C 1 xy dx = 0; and, y = 0 gives dy = 0 · dt, so Z C 1 ( x - y ) dy = 0 .
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