is the curve of intersection of the plane y z 2 and the cylinder x oriented counterclockwise when viewed from above.

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1 The questions below are representative or actual questions that have appeared on final eams in Math from pring 009 to present. The questions below are in no particular order. There are tpicall 10 questions of this kind on an given Math final eam with the subject matter evenl distributed across the eam. The list below ma contain redundanc because some questions are routinel and randoml re-used on final eams. 1. Use the Divergence Theorem to find the flu of F across if 4 3 F(,, z), z, 4 z and is the surface bounded b the clinder 1and the planes z and z 0. Use toke s Theorem to evaluate F d r, where F (,, z),, z and is the curve of intersection of the plane z and the clinder oriented counterclockwise when viewed from above. 3. Find a parametric representation of the top half of the cone 1. is 4. Find an equation of the tangent plane to the surface given b parametric equations u, v, z u v at the point (1, 1, 3) 5. Find the mass of the thin plate that occupies the region bounded b, 0, 1with densit function (, ). 6. Find the mass of the thin wire bent into the shape of (0,0) and (,8) with densit function (, ). 3 between the points 7. et up and evaluate the line integral 3 3 d d where is the closed curve which is a rectangle, positivel oriented, with sides 0, 3,, Find the potential function, if it eists, for F(,, z) e i ( e e z ) j e z k. 9. Use curl to determine whether the vector field F(,, z) e i e j zk is conservative 10. Evaluate r F d rwhere F(, ) i jand is given b 3 ( t) t, 1 t, 0 t 1

2 11. ketch the surface given b the parameterization sin cos, sin sin, z cos where 0 and Find the divergence of the vector field F(,, z) cos( ) isin( z) k 13. Find an equation of the tangent plane to the parametric surface u, v, z uv for u1, v et up and evaluate the surface integral to find the flu of F across if F ze, ze, z and is the part of the plane z 1 0in the first octant with downward orientation. 15. Evaluate 1 d where is the surface with vector equation r( u, v) ucos vi usin vj vk, 0 u 1, 0 v 16. In the homework, ou were asked to find the flu of F across if (,, z) 3z 1, 1, 1. What F i j k and is the cube with vertices aspect of doing this problem was unique, or different, from the other homework problems involving flu? 17. uppose ou evaluate the surface integral of a scalar function, where the surface is a sphere centered at the origin with radius 14 and the scalar function is f (,, z) 1. What would this surface integral represent under these conditions? 18. Use toke s Theorem to evaluate z Fdr, where F (,, z) e, e, e and is the boundar of the part of the plane z in the first octant. 19. Find a parametric representation of the top half of the cone z Use spherical coordinates to ET UP, BUT DO NOT EVALUATE 9 dv z z. and H is the hemisphere 9, 0 H 1. Find the mass of the thin plate that occupies the region bounded b, 0, 1with densit function (, ).. Use toke s Theorem to compute the integral curlf dwhere F(,, z) zi zj k and is the part of the sphere lies inside of the clinder 1 and above the -plane. z 4 that

3 3. Evaluate F d using the Divergence Theorem where (,, ) ( z ) sin( ) F z i e j k and is the surface of the region E bounded b parabolic clinder z z 1 and the planes z 0, 0 and 4. tate whether each of the following is meaningful. If the epression is not meaningful, state the reason(s) wh. If the epression is meaningful, state whether the result is vector or scalar. a. grad ( div( F )) : b. div( curl( F )) : c. grad ( div( curl( F )) : d. grad ( F) curl( F ): e. curl( curl( F) curl( F )) : 5. Use a surface integral to find the mass of the part of the sphere z 4that lies inside the clinder 1and above the - plane if the densit function (,, z) 6. Find the flu of F z,, across the unit sphere r (, ) sin cos,sin sin, cos, 0, 0, oriented outward, given: rr sin r r sin rr sin cos, sin sin, sin cos F is NOT conservative 7. Find the work done b the force field F(, ), in moving an object along three line segments: from the origin along the -ais to the point (1, 0), then along a line segment from (1, 0) to (0, 1), and finall back to the origin along the -ais.

4 8. Find the mass of a thin wire in the shape of the heli sin t, cos t, z 3 t, 0 t if the densit of the wire is a constant k.

5 9. Find the flu of F across where: of the solid bounded b the clinder F(,, z) i j z k and is the surface z 1 and the planes 0 and how that tokes Theorem is true for the vector field F(,, z) i j z k, where is the part of the paraboloid z 1 that lies above the -plane and has upward orientation. Do this b computing the respective integrals below: a. b. F d r: curlf d: 31. Find a parametric representation of the plane that passes through the point 1,, 3 and contains the vectors 1, 1, 1 and 1, 1,1. 3. Identif (name) and describe the surface that has parametric representation r (, ), cos, sin. Answer must be justified 33. Find the gradient vector field of f (,, z) sin z 34. If ou were to plot the graph of some gradient vector field, F, together with the contour map of its corresponding potential function f,what relationship(s), if an, between the vectors and the level curves would ou epect to see? 35. Find the flu of F across where: 3 3 F sin z, zsin, 3z and is the surface of the solid bounded b the hemispheres z 4, z 1 and the plane 0 z. 36. Use a surface integral to evaluate F d rwhere F e, e, e z and is the boundar of the part of the surface which is the plane z in the first octant. This surface is oriented upward. 37. Find the work done b the force field between the points (1, 1) and (4, -). F, in moving an object 38. Find the mass of a thin wire in the shape of a heli given b sin t, cos t, z 3t where 0t if the densit (,, z) is equal to the square of the distance from the point (,, z) to the origin.

6 39. Use the Divergence Theorem to find the flu of F across if F(,, z) z, z, z and is the surface of the tetrahedron with vertices (0,0,0), (1,0,0), (0,1,0), (0,0,1) 40. Use toke s Theorem to evaluate curlf d, is the part of the sphere upward orientation. 41. Find the work done b the force field 3 F (,, z) z, z, z e and z 5 that lies above the plane z 1 with F (, ), in moving an object along three line segments: from the origin along the -ais to the point (1, 0), then along a line segment from (1, 0) to (0, 1), and finall back to the origin along the -ais in a LOED FAHION. 4. Find the curl and divergence of the vector field (,, z) e F i e j zk curl( F ) = div( F ) = 43. Evaluate: 4 d d, where vector field 4 F(, ) i j and is the closed triangular curve consisting of the line segments from (0, 0) to (1, 1), from (1, 1) to (0, 1) and from (0, 1) to (0, 0) with positive orientation. 44. Find a parametric representation for the part of the clinder between the planes 0 and 5. z 16 that lies 45. Find the mass of a thin wire in the shape of the circular heli given b cos t, sin t, z t, 0 t with densit function (,, z) sin z. 46. Find an equation of the tangent plane to the parametric surface r( u, v) v i uvj u k, 0 u 3, 3 v 3 for u1, v. 47. Find the work done b the force field F(, ) i arctan j in moving an 1 object along the curve given b r( t) t i tj, 0 t Evaluate F d rwhere F(, ) e, cos region enclosed b parabolas and and is boundar of the

7 49. Find a parametric representation of the part of the clinder between the planes 0 and 5. ketch this surface. z 16 that lies ketch: 50. Find the surface area of the part of the paraboloid plane z Evaluate z that lies under the F d r for F (,, z) z, z, on the line segment from (1, 0, 0) to (3, 4, ). 5. Evaluate f d for f (,, z) 1 r( u, v) ucos vi usin vj vk, 0 u 1, 0 v where is given b 53. et up but DO NOT EVALUATE a double integral to find the flu of F(,, z) i zj k where is the surface z 4 in the first octant, with orientation toward the origin. 54. Find curl( F) and div( F) if F(,, z) cos zj sin k. 55. What does the direction of the vector curl( F) represent where fluid flow is concerned? 56. What does the magnitude of the vector ( ) curl F represent where fluid flow is concerned?

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