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MATH503 Exam 1 Solns

# MATH503 Exam 1 Solns - Exam#1(Math 503 STUDENT NAME...

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Exam#1 (Math 503) October 25 2010 STUDENT NAME: Instructions: The duration of the test is 50 minutes. The test consists of 3 questions and the marks are specified next to each question. Total mark = 50. Detail your calculations. No aid sheet or calculator allowed. 1) [17] (a) Verify Stokes’ theorem for the vector field F = y i + z j + x k where the open surface S is the ‘upside-down’ paraboloid defined by z = 1 - ( x 2 + y 2 ), z 0, with upward normal. (b) What condition should F satisfy to be conservative? Solution: (a) Here f = 1 - ( x 2 + y 2 ) and G = ∇ × F = ( - 1 , - 1 , - 1), so ZZ S ( ∇ × F ) · n dS = ZZ R - G x ∂f ∂x - G y ∂f ∂y + G z dxdy = - ZZ R (2 x + 2 y + 1) dxdy = - Z 2 π 0 Z 1 0 (2 r cos θ + 2 r sin θ + 1) rdrdθ = - Z 2 π 0 h 2 3 (cos θ + sin θ ) + 1 2 i = - π Z C F · t ds = Z 2 π 0 F · d r = Z 2 π 0 sin θ 0 cos θ · - sin θ cos θ 0 = - Z 2 π 0 sin 2 ( θ ) = - 1 2 Z 2 π 0 h 1 + cos(2 θ ) i = - π (b) F is conservative if ∇ × F = 0 .

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2) [17] (a) Use the divergence theorem to compute the flux RR S F · n dS for the vector field F = z 2 x i + 1 3 y 3 j
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MATH503 Exam 1 Solns - Exam#1(Math 503 STUDENT NAME...

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