Spring 2006 Final Review - Math 53 - Final Exam Review GSI:...

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Unformatted text preview: Math 53 - Final Exam Review GSI: Santiago Canez http://math.berkeley.edu/∼scanez/courses/math53/spring06/ 1. Compute the line integral of F(x, y ) = (2x2 − 3y 2 )i + (2x + 3y 2 )j over the triangle with vertices (−2, 0), (2, 0), and (0, 2) oriented counterclockwise. 2. (a) Define the curl and divergence of a vector field. (b) Determine whether or not the vector field F = x2 y i + zyex j − 2xyz k is the curl of another vector field. 3. Let S be the sphere of radius ρ. (a) Compute the surface area of S . ￿￿ (b) Compute the surface integral S z dS . 4. Let F(x, y, z ) = xi + y j + z k. Determine whether the flux of F across S is negative, positive, or zero for each surface S below. (a) the cone z 2 = x2 + y 2 oriented outward (b) the sphere x2 + y 2 + z 2 = 1 oriented inward (c) the plane y = 2 oriented with normals having positive j component 5. Describe two different methods for computing the flux of F = 2yxi − yex j + (2 − z )k through the upwardly oriented piece of the plane x + y + 2z = 1 in the first octant. 6. (a) Using Stokes’ Theorem, compute the surface integral of curl F over the upper unit hemisphere where F(x, y, z ) = xi − y j + exyz k with orientation given by normals with positive k component. (b) Using Stokes’ Theorem, compute the circulation of F(x, y, z ) = (z − 2y )i + (3x − 4y )j + (z + 3y )k around the circle x2 + y 2 = 4, z = 1 oriented counterclockwise. 7. Compute the flux of F = x3 i + y 3 j + z 3 k through the sphere of radius 3, positively oriented. 8. Let S be the right half of the unit sphere, with inward orientation. Compute the flux of F(x, y, z ) = zey i + (x2 + z 2 )j + sin−1 (exy )k across S . 9. Let S1 and S2 be two upwardly oriented surfaces with the same boundary and let F be a vector field. Date : May 11, 2006 1 (a) Use Stokes’ theorem to show that the flux of curl F through S1 is the same as the flux through S2 . (b) Compute the flux of curl F where F = −y i + xj + z k through the pyramid (without a base!) with vertices at (0, 0, 1), (±1, 0, 0), (0, ±1, 0), upwardly oriented. 10. Let C be the diamond with vertices at (±3, 0) and (0, ±3), oriented clockwise. Compute the line integral of y i − xj F= 2 x + y2 around C . Hint: Replace C (with justification!) by a simpler curve. 2 ...
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