s07wk13 - Z C ·· = ZZ D 1 dA Checking the region using y...

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Math 23 B. Dodson Week 13 Homework: [due April 20] 16.3 Fundamental Theorem for line integrals 16.4 Green’s Formula 16.5 Curl and Divergence Text Ex 16.3.4b: Evaluate the line integral Z C F · dr where F = < 3 + 2 xy, x 2 - 3 y 2 >, and C is given by r ( t ) = < e t sin t, e t cos t >, 0 t π. Solution: We check F = < P, Q > for P y = Q x , to verify that the integral is path independent, with starting point r (0) = < 0 , 1 >, ending point r (1) = < 0 , - e π > . We go back to P = 3 + 2 xy = f x (from < P, Q > = < f x , f y > ) and take the partial integral f ( x, y ) = Z (3 + 2 xy ) dx = 3 x + x 2 y + c ( y ) , where the function c ( y ) is the “constant of integration” for ∂x . We now check f y = x 2 + c ( y ) = x 2 - 3 y 2 = Q, and get c ( y ) = - 3 y 2 ; c ( y ) = - y 3 ; f ( x, y ) = 3 x + x 2 y - y 3 . Then Z C = [3 x + x 2 y - y 3 ] (0 , - e π ) (0 , 1) = (0 - 0 - ( - e 3 π )) - (0 + 0 - (1)) = 1 + e 3 π .
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2 Week 13 Homework: 16.4 Green’s Formula Problem 16.4.9: Use Green’s Theorem to evaluate Z C ( y + e x ) dx + (2 x + cos ( y 2 )) dy, where C is the positively oriented boundary of the region D enclosed by the parabolas y = x 2 , x = y 2 . Solution: For Q = 2 x + cos ( y 2 ) we have Q x = 2 , and for P = y + e x , P y = 1 , so Q x - P y = 2 - 1 = 1 , and Green’s Theorem gives
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Unformatted text preview: Z C ··· = ZZ D 1 dA. Checking the region, using y = x 2 in the equation x = y 2 gives x = x 4 , so x 4-x = x ( x-1)( x 2 + x + 1) = 0 , with solutions x = 0 , 1 . Solving x = y 2 for y = √ x gives the iterated integral Z 1 Z √ x x 2 1 dydx = Z 1 ‡ [ y ] √ x x 2 · dx = Z 1 ( √ x-x 2 ) dx = h 2 3 x 3 2-1 3 x 3 i 1 = 2 3-1 3 = 1 3 . Week 13 Homework: 16.5 Curl and Divergence Problem 16.5.5: Find Div( ± F ) and Curl( ± F ) when ± F = < e x sin y, e x cos y, z > . Solution: For ± F = < P, Q, R >, Div( ± F ) = P x + Q y + R z , so Div( ± F ) = e x sin y-e x sin y + 1 = 1 . Using the cross product formula we have Curl( ± F ) = < R y-Q z , R x-P z , Q x-P y > = < , , > . Notice that if ± F = < f x , f y , f z > is the...
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