Midterm2A.pdf - Midterm 2 Solutions(Version A Problem 1...

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Midterm 2 Solutions (Version A) Problem 1 Consider the vector field F ( x, y, z ) = e y i + ( xe y + e z ) j + ye z k . (a) Show that F is conservative. (b) Calculate the work done by F in moving a particle along the straight line segment C from (0 , 2 , 0) to (4 , 0 , 3) . Solution (a) Since F and its first partial derivatives are continuous everywhere, curl F = 0 will imply F is conservative. We have curl F = i j k ∂x ∂y ∂z e y ( xe y + e z ) ye z = h e z - e z , 0 - 0 , e y - e y i = 0 Thus since curl F = 0 we conclude F is conservative. (b) Since F is conservative, we can find a potential function f so that F = f = h f x , f y , f z i . Then by the Fundamental Theorem of Line Integrals the work done is computed by W = R C F · d r = f (4 , 0 , 3) - f (0 , 2 , 0). We must solve the system e y = f x (1) xe y + e z = f y (2) ye z = f z (3) Integating (1) with respect to x gives f ( x, y, z ) = xe y + c 1 ( y, z ). Now differentiating this expession with respect to y and setting it equal to (2) shows xe y + ∂y c 1 ( y, z ) = xe y + e z . This equation implies c 1 ( y, z ) = ye z + c 2 ( z ), for some function c 2 depending only on z . So at this point, plugging in for
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  • Spring '08
  • wong
  • Vector Calculus, Line integral, Vector field, Gradient

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