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9 Nov 2019
Calculus III Problems:
One common parametrization of the sphere of radius 1 centered at the origin is rrightarrow(u, v) = sin u cos v irightarrow + sin u sin v jrightarrow + cos u krightarrow. 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, particularly if sin u is factored from each component of the normal vectors, and the remaining vector portion is compared to the original rrightarrow(u, v). Compute the flux of the vector field Frightarrow(x, y, z) = z krightarrow 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). Evaluate the surface integral of vector field Frightarrow (x, y, z) = x irightarrow + y jrightarrow + (x + y) krightarrow over the portion S of the paraboloid z = x2 + y2 lying above the disk x2 + y2 1. Use outward pointing normals. Evaluate the surface integral of the vector field Frightarrow (x, y, z) = sin y, sin z, yz over the rectangle 0 y 2, 0 z 3 in the yz- plane with the normal pointing in the negative x direction. Evaluate int intSzkrightarrow  d Srightarrow where S is the part paraboloid z = 1 - x2 - y2 above the xy-plane together with the bottom disk x2 + y2 1 in the xy-plane. Use outward pointing normals.
Calculus III Problems:
One common parametrization of the sphere of radius 1 centered at the origin is rrightarrow(u, v) = sin u cos v irightarrow + sin u sin v jrightarrow + cos u krightarrow. 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, particularly if sin u is factored from each component of the normal vectors, and the remaining vector portion is compared to the original rrightarrow(u, v). Compute the flux of the vector field Frightarrow(x, y, z) = z krightarrow 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). Evaluate the surface integral of vector field Frightarrow (x, y, z) = x irightarrow + y jrightarrow + (x + y) krightarrow over the portion S of the paraboloid z = x2 + y2 lying above the disk x2 + y2 1. Use outward pointing normals. Evaluate the surface integral of the vector field Frightarrow (x, y, z) = sin y, sin z, yz over the rectangle 0 y 2, 0 z 3 in the yz- plane with the normal pointing in the negative x direction. Evaluate int intSzkrightarrow  d Srightarrow where S is the part paraboloid z = 1 - x2 - y2 above the xy-plane together with the bottom disk x2 + y2 1 in the xy-plane. Use outward pointing normals.