Finw99s - Solutions to Math 20E Final Winter 99 Lindblad 1 Adding the two equations together gives 3 x y 2 = 0 so y = 3 x 2 Inserting this in the

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Unformatted text preview: Solutions to Math 20E Final Winter 99, Lindblad. 1. Adding the two equations together gives 3 x- y + 2 = 0 so y = 3 x + 2. Inserting this in the first equation gives z = 3 y- 2 x +1 = 3(3 x +2)- 2 x +1 = 7 x +7. Hence the equation of the plane is ( x,y,z ) = ( t, 3 t + 2 , 7 t + 7) = (1 , 3 , 7) t + (0 , 2 , 7). 2. (a). F ( x,y,z ) = ∇ φ = y 2 i + 2 xy j . (b). Along the curve we have dx y 2 = dy 2 xy = dz so ydy = 2 xdx and dz = 0. Hence y 2 = 2 x 2 + C so y = ± √ 2 x 2 + C 1 and z = C 2 Since the curve starts at (1 ,- 1 , 1) we must have ( x,y,z ) = ( t,- √ 2 t 2- 1 , 1) 3. (a) φ x = ze x- y , φ y =- ze x- y and φ z = e x- y . Integrating these equations give φ = ze x- y + h 1 ( y,z ), φ y = ze x- y + h 2 ( x,z ) and φ = ze x- y + h 3 ( x,y ) which can all be satisfied if we pick h 1 = h 2 = h 3 = 0 in wchich case φ = ze x- y . (b) The end point of the curve is (1 , , 0) and the initial point is (0 , , 1) so C F · R = φ (1 , , 0)- φ (0 , , 1) =- 1....
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This note was uploaded on 03/16/2010 for the course MATH 223B taught by Professor Staff during the Spring '09 term at University of Arizona- Tucson.

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Finw99s - Solutions to Math 20E Final Winter 99 Lindblad 1 Adding the two equations together gives 3 x y 2 = 0 so y = 3 x 2 Inserting this in the

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