# week10 - Math 23 Sections 110-113 B. Dodson Week 10...

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Math 23 Sections 110-113 B. Dodson Week 10 Homework: [due April 15] 16.1, 16.2 vector ﬁelds and line integrals 16.3 Fundamental Theorem for line integrals 16.4 Green’s Formula 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) . Solution:

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2 . By deﬁnition, Z C = Z C 1 + Z C 2 . For
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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 .

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week10 - Math 23 Sections 110-113 B. Dodson Week 10...

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