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ma441sfinal

# ma441sfinal - of the component of F in the direction of...

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Dr. H. Khanal Spring 2008 MA 441.01 Final Exam (SAMPLE) Name: ID#: ! Read each problem carefully before attempting a solution. Show all work to justify your answers. Answer alone carries no credit. Partial credit only for significant progress towards a correct solution. Box or underline final answers. 1. Complex Integration (1) Integrate Z C ¯ z dz , where C is the upper half of the unit circle | z | = 1. (2) Integrate I C e z z 2 - 9 dz , where C is the circle | z - 3 | = 3. 2. Vector Integration (1) Let F = [ ze xz , 1 , xe xz ]. Is the line integral R C F · d r path independent? If path independent, evaluate the integral between the points A (0 , 0 , 1) and B (1 , 2 , 1). If not, evaluate the integral along a straight line path between the points A (0 , 0 , 1) and B (1 , 2 , 1). (2) Let T be the solid defined by the inequalities x 2 + y 2 9 , 0 z 5, and let S be the surface, or boundary of T . Let n be the unit normal vector field on S that points out of T . Define F ( x, y, z ) = ( x + yz ) i + ( x 4 - y ) j + (2 z + 3 - x ) k . Find Z Z S F · n dA (the flux of F across S , which is the integral of with respect to surface area
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Unformatted text preview: of the component of F in the direction of normal to S ). ( Hint: This is easiest to do using the Divergence Theorem.) (3) Let F ( x,y,z ) = [ e y ,-z, ye x ], S is the hemisphere x 2 + y 2 + z 2 = 1 , x ≤ 0 and n is the unit normal vector ﬁeld to S that points toward the origin. Deﬁne the vector ﬁeld F by (i) Let C denote the boundary of S . Find a vector function r that parametrizes C . (ii) Invoke the Stokes’s Theorem to ﬁnd a line integral whose value is equal to Z Z S ∇ × F · n dA. (iii) Evaluate the line integral from part (ii). 3. Fourier Series (1) Expand the function f ( x ) = x-3 (0 < x < 3) in a Fourier cosine series, and (b) Fourier sine series. (2) Find the complex Fourier series of f ( x ) = x 2 if-π < x < π and f ( x + 2 π ) = f ( x ). Convert the series to real form 1...
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