ex2 - Math 251 Sections 1-3 Practice Problems

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Unformatted text preview: Math 251 Sections 1-3 Practice Problems ———————————————————————————————————— (i) Find a potential for the vector field F ( x,y,z ) = y i + ( x + z ) j + y k . (ii) Evaluate the line integral Z C sin xdx where C is the arc of the curve x = y 4 from (1 ,- 1) to (1 , 1). We can take c ( t ) = ( t 4 ,t ) ,- 1 ≤ t ≤ 1. So, x = t 4 ,dx = 4 t 3 dt So we get Z C sin xdx = Z 1- 1 sin( t 4 )4 t 3 dt. Caution: what we did above was integrating the vector field < sin x, > along the given curve. (iii) Let S be the part of the cone z = p x 2 + y 2 beneath the plane z = 1 with downward normal. a)Evaluate the surface integral of the vector field x i + y j + z 4 k over S . We can take the parametrization Φ( x,y ) = ( x,y, p x 2 + y 2 ). We have 0 ≤ z ≤ 1. Since z = p x 2 + y 2 = r , this becomes 0 ≤ r ≤ 1. Φ x ( x,y ) = < 1 , ,x/r > , Φ y ( x,y ) = < , 1 ,y/r > , n = Φ x × Φ y = <- x/r,- y/r, 1 > . This normal points upward because 1 > 0. So the integral is- Z 2 π Z 1 < x,y, ( p x 2 + y 2 ) 4 > · <- x/r,- y/r, 1 > ( rdrdθ ) and you should replace x,y with their polar equivalents and take the dot product....
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This note was uploaded on 02/08/2011 for the course ECONOMICS 101 taught by Professor June during the Spring '08 term at Rutgers.

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ex2 - Math 251 Sections 1-3 Practice Problems

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