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Unformatted text preview: Z c F · ds and use it to explain the eﬀects of a reparametrization of the path c . 6. Consider the vectorvalued function c ( θ ) = (sin 2 θ, cos2 θ,θ ). (a) Sketch the curve traced out by c . (b) Find the length of the curve from the points (0 ,1 ,π/ 2) to (0 ,1 , 3 π/ 2). 7. Let F ( x,y,z ) = ± y 2 2 + yz,xy + xz,xy ¶ . (a) Compute ∇ × F . (b) Show that F is a gradient ﬁeld and ﬁnd F ’s potential function f . (c) The path r ( θ ) = ˆ1 + 4 √ 2 3 π θ cos θ, 2 + 4 √ 2 3 π θ sin θ, ! as 0 ≤ θ ≤ 9 π/ 4 is a parametrization of the spiral that starts at the point (1 , 2 , 0) and ends at the point (2 , 5 , 0). Evaluate the integral Z r F · d s ....
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This note was uploaded on 02/15/2011 for the course MATH 380 taught by Professor Staff during the Summer '08 term at University of Illinois, Urbana Champaign.
 Summer '08
 Staff
 Math, Calculus

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