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plugin-FExRevProbSolutions

plugin-FExRevProbSolutions - Math 255 Winter 2010 Review...

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Math 255 Winter 2010 Review problems for Final Exam 1. The electrostatic potential at (0 , 0 , - a ) of a charge of constant density σ on the hemisphere S : x 2 + y 2 + z 2 = a 2 , z 0 is U = S σ x 2 + y 2 + ( z + a ) 2 dS. Show that U = 2 πσ a (2 - 2). Solution The surface is a graph z = a 2 - x 2 - y 2 . We therefore project the integral onto the xy plane and convert to polar coordinates to find that we need to calculate 2 π 0 a 0 σ 2 a 2 + 2 a a 2 - r 2 a a 2 - r 2 r d r d θ . After integrating with respect to θ we make the substitution w = a 2 - r 2 to find that 2 a πσ a 2 0 d( a + w ) a + w = 2 πσ a (2 - 2) . 1

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2. Find the value of the line integral C yzdx + xzdy + xydz where C is the curve with initial point P 0 = (1 , 0 , 0) given by. a) x = cos t , y = sin t , z = t 2 with t [0 , π ]. b) The line segment between P 0 and ( - 1 , 0 , - π ). c) The ellipse x 2 + 4 y 2 = 1 and z = 0 parametrized clockwise. d) The circle x 2 + y 2 = 9 and z = 0 parametrized counterclockwise as viewed from above. Use the fact that the path C is the boundary of the region x 2 + y 2 9 with z = 0. Use Green’s Theorem to compute value of the line integral. e) The same path as in part d ), but now using Stokes Theorem. Use the fact that the path C in this case is the boundary of the hemisphere x 2 + y 2 + z 2 = 9 with z 0. f) Can you use the fundamental theorem of line integrals to compute the value of any of the above line integrals? Explain.
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plugin-FExRevProbSolutions - Math 255 Winter 2010 Review...

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