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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 r rightarrow (u, v). Compute the flux of the vector field F rightarrow (x, y, z) = z k rightarrow 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 F rightarrow (x, y, z) = x i rightarrow + y j rightarrow + (x + y)k rightarrow over the portion S of the paraboloid z = x2 + y2 lying above the disk x2 + y2 le 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 le y le 2, 0 le z le 3in the yz-plane with the normal pointing in the negative x direction. Evaluate S z krightarrow Â· d S rightarrow where S is the part paraboloid z = 1 - x2 - y2 above the xy-plane together with the bottom disk x2 + y2 le 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 r rightarrow (u, v). Compute the flux of the vector field F rightarrow (x, y, z) = z k rightarrow 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 F rightarrow (x, y, z) = x i rightarrow + y j rightarrow + (x + y)k rightarrow over the portion S of the paraboloid z = x2 + y2 lying above the disk x2 + y2 le 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 le y le 2, 0 le z le 3in the yz-plane with the normal pointing in the negative x direction. Evaluate S z krightarrow Â· d S rightarrow where S is the part paraboloid z = 1 - x2 - y2 above the xy-plane together with the bottom disk x2 + y2 le 1 in the xy-plane. Use outward pointing normals.