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265Lesson22Problems(5)

# 265Lesson22Problems(5) - z = x 2 y 2 lying above the disk x...

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MATH 265: MULTIVARIABLE CALCULUS: LESSON 22 PROBLEMS Problems are worth 20 points each. (1) One common parametrization of the sphere of radius 1 centered at the origin is -→ r ( u, v ) = sin u cos v -→ ı + sin u sin v -→ + cos u -→ k . Find formulas for the two normals to the sphere at parameter values ( u, v ). The algebra will look a little bit intimidating, but things actually simplify nicely, par- ticularly if sin u is factored from each component of the normal vectors, and the remaining vector portion is compared to the original -→ r ( u, v ). (2) Compute the flux of the vector field -→ F ( x, y, z ) = z -→ k across the upper hemisphere of radius 1 centered at the origin with outward point normals. You should be able to make some use of the result of problem (1). (3) Evaluate the surface integral of vector field -→ F ( x, y, z ) = x -→ ı + y -→ + ( x + y ) -→ k over the portion S of the paraboloid
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Unformatted text preview: z = x 2 + y 2 lying above the disk x 2 + y 2 ≤ 1. Use outward pointing normals. (4) Evaluate the surface integral of the vector ﬁeld-→ F ( x,y,z ) = h sin y, sin z,yz i over the rectangle 0 ≤ y ≤ 2, 0 ≤ z ≤ 3 in the yz-plane with the normal pointing in the negative x direction. (5) Evaluate ZZ S z-→ k · d-→ S where S is the part paraboloid z = 1-x 2-y 2 above the xy-plane together with the bottom disk x 2 + y 2 ≤ 1 in the xy-plane. Use outward pointing normals. (6) (Bonus question: worth 10 points. Total points for assignment not to exceed 100.) Find the ﬂow rate of water with velocity ﬁeld-→ v ( x,y,z ) = h 2 x,y,xy i meters per second across the part of the cylinder x 2 + y 2 = 9 with x,y ≥ 0, and 0 ≤ z ≤ 4 ( x,y,z measured in meters). 1...
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