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# Math 232 Midterm II Name_______________________________ (Please Print) Last, First Answer each question as accurately as possible. Please explain...

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Math 232 Midterm II Name_______________________________ (Please Print) Last, First Answer each question as accurately as possible. Please explain everything clearly. Show all your work for full credit . In particular, you must show the steps for integration. If you need to use any of the theorems, make sure that you check all the hypotheses. If you have any questions, please see me. Staple this sheet together with your work to turn in. Please make sure that your solutions are still readable after stapling. 1. (10 pts) Consider the transformation ( ) 2 : , 1 T x v y u v = = + . Analyze and describe clearly the image D in the xy -plane of R in the uv -plane under T , where ( ) { } , 0 1, 0 1 R u v u v = . 2. (10 pts) Consider the rectangle with vertices ( ) ( ) ( ) ( ) 1, 1 , 1, 1 , 1,2 , and 1,2 - - - - . Compute the work done by the force field ( ) 2 2 , sin , y x y x xy e x = + + F in moving an object along C , where C is the path that goes around the boundary of the rectangle clockwise twice starting at the vertex ( ) 1, 1 - - . 3. (10 pts) Consider the vector field ( ) , , 2 ,2 ,2 x y z x y z = F . If C is any path from ( ) 0,0,0 to ( ) 1 2 3 , , a a a and 1 2 3 , , a a a = a , show that C d = ⋅ F r a a .
Math 232 Midterm II Name_______________________________ (Please Print) Last, First 4. (10 pts) Evaluate the line integral, 2 2 C x z ds + , where C is the circle ( ) 0, cos , sin t a t a t = r , 0 2 t π ≤ ≤ . You may assume that 0 a > . 5. (10 pts) Evaluate ( ) ( ) 2,2,2 2 2 1,1,1 1 1 x y dx dy dz y z y z + - - . Justify your answer. 6. Let K be an arbitrary smooth simple closed curve in the plane that does not pass through the origin ( ) 0,0 . Let ( ) ( ) 2 2 , ln f x y x y = + . a. (10 pts) Evaluate K f ds ∇ ⋅ n i given that ( ) 0,0 lies outside K. b. (10 pts) Evaluate K f ds ∇ ⋅ n i given that ( ) 0,0 lies inside K . 7. (10 pts) A thin spherical shell of radius 5 centered at the origin has a hole of radius 5 3 2 removed from the top. Find the center of mass of this shell assuming that the density is proportional to the square of the distance from the z -axis. 8. (10 pts) Find an equation of the tangent plane to the given parametric surface, 2 x u = , 2 y v = , z uv = , at the point where 1 u = and 1 v = . 9. (10 pts) Evaluate the surface integral S d ∫∫ F S , where ( ) 4 , , , , x y z x y z = F and S is the part of the cone 2 2 z x y = + beneath the plane 1 z = with downward orientation.

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