Exam 18: Fundamental Theorems of Vector Analysis

Let and S be the part of the sphere between the planes and , oriented outward. The integral is equal to which of the following?

Compute where and S is the boundary of the pyramid determined by the planes .

Use Stokes' Theorem to compute , where and S is the part of the surface satisfying S is oriented with outward-pointing normals.

Compute where and , oriented outward.

Use Green's Theorem to evaluate the line integral along the contour of the triangle ABD with vertices

Compute where S is the ellipsoid with outward-pointing normal and .

Compute where c is the curve of intersection of the surfaces and , oriented counterclockwise.

Use Green's Theorem to evaluate where c is the piecewise linear path starting at (0,- 2) and then traveling to (2,4), (- 2,2), and (0,- 2), in that order.

Evaluate where is the ellipsoid oriented outward, and

Evaluate where c is the circle oriented counterclockwise and .

Evaluate where and S is the sphere oriented with outward-pointing normal.

Compute where is the intersection line of the surfaces and

Let S be the part of the paraboloid which is above the plane oriented upwards, and let . A) Explain why where is the disc of radius 3 in the xy-plane oriented upward. B) Compute the surface integral

Compute where and is the boundary of the tetrahedron with vertices and , oriented with outward-pointing normal.

Compute where and is the boundary of the tetrahedron with vertices , oriented with outward-pointing normal.

Use Green's Theorem to evaluate where is the closed curve shown in the following figure.

Let be the vector field , w be a region in containing the origin in its interior , and S be the boundary of w that is a closed surface with outward-pointing normal. Compute

Let , and let S be the surface , together with the two vertical sides. Compute where

Evaluate where is the curve , oriented clockwise.

Let S be the boundary of the region V defined by oriented with outward-pointing normal. Let . The surface integral is equal to which of the following?