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# 254q3 - xy i x 2 2 yz j y 2 k Is F conservative If so...

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Quiz 3 of Math 254.01 (Au10) November 23, 2010 (Tuesday) The Ohio State University This quiz consists of two (2) pages and two (2) problems and is worth a total of 15 points. The point value of each problem is indicated. To obtain full credit, you must have the correct answers along with relevant supporting work to justify them. Partial credit will be given based on the work that is shown. However, answers without sufficient support- ing work will receive no credit. Also, you must complete the quiz in 15 minutes. Good luck! Name: Signature: OSU Internet Username: 1. (7) Use Green’s theorem to evaluate R C F · d r , where F ( x, y ) = y 2 cos x i + ( x 2 + 2 y sin x ) j and C is the triangle from (0 , 0) to (2 , 6) to (2 , 0) to (0 , 0) . Solution: The region bounded by the triangle is given by D = { ( x, y ) R 2 | 0 y 3 x, 0 x 2 } . Also, note that the triangle is being traversed clockwise. Hence Green’s theorem implies that Z C F · d r = - Z - C F · d r = - Z - C y 2 cos x dx + ( x 2 + 2 y sin x ) dy = - ZZ D ∂x ( x 2 + 2 y sin x ) - ∂y ( y 2 cos x ) dA = - ZZ D (2 x + 2 y cos x - 2 y cos x ) dA = - Z D 2 x dA = - Z 2 0 Z 3 x 0 2 x dy dx = - Z 2 0 6 x 2 dx = - 16 .

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2. (8) Calculate
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Unformatted text preview: xy i + ( x 2 + 2 yz ) j + y 2 k . Is F conservative? If so, compute the most general potential function f of F . Solution: One has ∇ × F = ± ± ± ± ± ± ± ± i j k ∂ ∂x ∂ ∂y ∂ ∂z 2 xy x 2 + 2 yz y 2 ± ± ± ± ± ± ± ± = (2 y-2 y ) i-(0-0) j + (2 x-2 x ) k = Since, in addition to ∇ × F = , the domain of F is R 3 , which is simply-connected, and the components of F have continuous partial derivatives, F is conservative. If f is a potential function of F , then f x = 2 xy ⇒ f ( x,y,z ) = x 2 y + g ( y,z ) , f y = x 2 + 2 yz ⇒ f ( x,y,z ) = x 2 y + y 2 z + h ( x,z ) , f z = y 2 ⇒ f ( x,y,z ) = y 2 z + k ( x,y ) for some functions g , h , and k of respective variables. By letting g ( y,z ) = y 2 z + C, h ( x,z ) = C, k ( x,y ) = x 2 y + C for an arbitrary constant C , one has f ( x,y,z ) = x 2 y + y 2 z + C. Page 2...
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254q3 - xy i x 2 2 yz j y 2 k Is F conservative If so...

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