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math226fall07pf

math226fall07pf - PRACTICE PROBLEMS Below are some type of...

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PRACTICE PROBLEMS Below are some type of problems related to the material covered after the 2nd midterm (note that final exam is comprehensive). The following formulas you may need will be provided: Spherical coordinates: x = ρ sin v cos u, y = ρ sin v sin u, z = ρ cos v, ρ 0 , 0 u 2 π, 0 v π ; the absolute value of the Jacobian | ( x, y, z ) /∂ ( u, v, ρ ) | = ρ 2 sin v. The volume of the ball of radius K : V = 4 πK 3 / 3; The area of a sphere of radius K : A = 4 πK 2 . Parametrization of a sphere of radius K : x = K sin v cos u, y = K sin v sin u, z = K cos v, 0 u 2 π, 0 v π ; and its normal vector ~ N ( u, v ) . Often it is convenient to start integration in multiple integrals with param- eter u or θ [0 , 2 π ] and to know that R 2 π 0 sin θ dθ = R 2 π 0 cos θ dθ = 0: 1 . a) Given ~ F ( x, y ) = h P, Q i = - y/ ( x 2 + y 2 ) , x/ ( x 2 + y 2 ) , check whether Q x = P y and find R C ~ F · d~ r for C : x 2 + y 2 = 1; b) Given ~ F ( x, y ) = h P, Q i = x/ ( x 2 + y 2 ) , y/ ( x 2 + y 2 ) , check whether Q x = P y and find R C ~ F · d~ r for C : x 2 + y 2 = 1 .
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