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# samplefinal - = h z,x,y i Compute the line integral R C F.d...

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MAC 2313 Final Exam (40 points) Calculators: Not allowed. Time: Two hours. Each problem is worth 8 points. Read the questions carefully. No credit will be given for answering questions which were not asked. Please show all working and sketch pictures whenever you can. 1. Determine whether the vector field F ( x, y, z ) = h cos x + 2 yz, sin y + 2 xz, z + 2 xy i is conservative. If so, find a potential function. Let C be the curve r ( t ) = ( t, t, t 5 ), 0 t 1. Compute R C F .d r . 2. Let F ( x, y ) = h y, x i and let C be the circle x 2 + y 2 = 1, with positive orientation. Evaluate the line integral Z C F .d r either directly or using Green’s Theorem. 3. Let f ( x, y, z ) and g ( x, y, z ) be functions. Assuming that the appropriate partial derivatives exist and are continuous, prove the identity div( f × ∇ g ) = 0 . 4. Let C be the triangular curve with vertices (1 , 0 , 0), (0

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Unformatted text preview: ) = h z,x,y i . Compute the line integral R C F .d r (for whichever orientation you like). 5. Let F ( x,y,z ) = h xz,yz,xy i . Let S be the part of the sphere x 2 + y 2 + z 2 = 4 that is inside the cylinder x 2 + y 2 = 1 and above the xy-plane. Compute ZZ S Curl F .d S . 1 2 Formulae These are included only to jog your memory - you are supposed to know what they mean. (a) SA = ZZ R s 1 + ± ∂f ∂x ² 2 + ± ∂f ∂y ² 2 dA. (b) Z C F .d r = Z C P dx + Q dy = ZZ R ∂Q ∂x-∂P ∂y dA. (c) ∇ f = ± ∂f ∂x , ∂f ∂y , ∂f ∂z ² . Div F = ∂M ∂x + ∂N ∂y + ∂P ∂z . Curl F = ³ ³ ³ ³ ³ ³ i j k ∂ ∂x ∂ ∂y ∂ ∂z M N P ³ ³ ³ ³ ³ ³ . (d) ZZ S Curl F .d S = Z C F .d r ....
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samplefinal - = h z,x,y i Compute the line integral R C F.d...

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