Exam 18: Fundamental Theorems of Vector Analysis

Compute where c is the curve shown in the figure.

Compute where and S is the upper half of the sphere of radius 2; that is, with upward-pointing normal.

Let where , is oriented with normal pointing to the origin, and is oriented in the opposite direction. Let be the vector field . Compute

Let , and w be a region in whose boundary is a closed piecewise smooth surface S. The integral is equal to which of the following?

Compute the surface integral where S is the half sphere , , oriented with outward pointing normal, and

Let be the boundary of the region enclosed by the surfaces and oriented with outward-pointing normal. Evaluate if

Let where and , oriented upward. I is equal to which of the following?

Compute where S is the portion of the surface of the sphere with radius and center that is above the plane oriented upward, and F is the vector field .

Use Stokes' Theorem to compute where S is the part of the surface , oriented outward, and .

Compute where and is the boundary of the tetrahedron formed by the planes oriented with outward-pointing normal.

Compute where C is the curve consisting of the line segment : on the x-axis together with the curve : in the positive direction.

Let S be a closed and smooth surface with outward-pointing normal which is the boundary of a solid V in Let be a vector field whose components have continuous partial derivatives. A) Compute . B) What is ? Explain.

Use Stokes' Theorem to find the line integral of the vector field around the curve which is the intersection of the plane with the cylinder , oriented counterclockwise as viewed from above.

Compute where is the intersection line of the surfaces and

Compute where c is the curve starting at the origin and ending at .

Use Stokes' Theorem to compute the flux of through the surface S which is the part of the paraboloid below the plane , oriented upward. The vector field is given by .

Evaluate where is the boundary of the region enclosed by the surfaces and oriented with inward-pointing normal, and .

Compute where c is the curve of intersection between the sphere and the plane . The integration on c is counterclockwise when viewing from the point .

Find the flux of the vector field through the boundary of the region enclosed by and oriented with inward-pointing normal.

Use the Divergence Theorem to compute the surface integral , where S is the surface , oriented outward, and F is the vector field .