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Page 1 of 6 Name ________________________ Calculus 3 Test #04: Vector Calculus Assigned: Thursday 5/01/2008 Due: Monday 5/05/2008 You must show your work to get full credit. Score: __ / 130 1. ( __ / 20 ) Compute the work done by the force field F y , x x y x y2 (1, 0, 1) along the helix of radius 1 that completes 1 turn over a height of 1. Previous page score: ________ This page score: ________ 2 2 2 , 0 from (1, 0, 0) to Running score: ________ Page 2 of 6 2. Consider the line integral Name ________________________ y 2 2 x dx x 2 dy where C is the square from (0, 0) to (1, 0) to (1, 1) to C (0, 1) to (0, 0) as shown below. a. ( __ / 8 ) Evaluate the line integral directly. b. ( __ / 7 ) Evaluate the line integral using Green’s Theorem. c. ( __ / 3 ) If this integral represents the line integral of a vector field along C, state the vector field being integrated. d. ( __ / 2 ) State whether or not the vector field is conservative and why. Previous page score: ________ This page score: ________ Running score: ________ Page 3 of 6 Name ________________________ 3. Consider the vector field F sin x , 2 y 2 , z a. ( __ / 7 ) Compute the divergence of this vector field. b. ( __ / 11 ) Compute the curl of this vector field. c. ( __ / 2 ) State whether or not this vector field is irrotational and why. 4. ( __ / 10 ) If A is a constant vector and r x, y, z , prove that A r 2A Previous page score: ________ This page score: ________ Running score: ________ Page 4 of 6 Name ________________________ 5. ( __ / 20 ) Evaluate the surface integral zdS , where S is the hemisphere z 9 x2 y 2 . S Previous page score: ________ This page score: ________ Running score: ________ Page 5 of 6 Name ________________________ 6. ( __ / 20 ) Verify the divergence theorem by computing both integrals for F x 2 , 2 y, x 2 , where Q is the region bounded by x 2 y z 4 and the coordinate planes. Previous page score: ________ This page score: ________ Running score: ________ Page 6 of 6 Name ________________________ 7. ( __ / 20 ) Verify Stokes’s theorem by computing both integrals for F 2 x, z 2 x, xz 2 , where S is the portion of z 1 x 2 y 2 above the xy plane. Previous page score: ________ This page score: ________ Running score: ________