Exam 18: Fundamental Theorems of Vector Analysis

Use Stokes' Theorem to compute the line integral of counterclockwise (as viewed from above) around the triangle with vertices and .

Calculate the circulation of the vector field around the circle , traversed in a counterclockwise direction.

Evaluate where is the boundary of the unit square oriented clockwise.

Evaluate where is the triangle with vertices and , traversed in a counterclockwise direction.

Evaluate where is the circle , traversed in a counterclockwise direction.

Use Stokes' Theorem to compute the line integral where and c is the path made up of the sequence of three line segments: 0 to A, A to B, and B to C. (See the figure.)

Compute the area of the region bounded by the astroid

Evaluate where c is the closed curve traversed in a counterclockwise direction.

Let S be the upper half of the hemisphere including the bottom . S is oriented with outward-pointing normal, and F is the vector field Compute .

Compute where and S is the upper half of the sphere of radius 3, that is, with upward-pointing normal.

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

Let be the vector field Compute the flux of through the surface if the normal to the surface satisfies

Let where C is the ellipse oriented counterclockwise. The value of I is which of the following?

Use the Divergence Theorem to evaluate where and S is the surface , oriented outward.

Compute where and S is the surface defined by , oriented with outward pointing normal.

Let be the surface area of the box in the figure with dimensions , and let W be the interior of the box. Let F be the vector field Which of the following integrals are equal? A) B) C) D) E)

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

Evaluate where is the boundary of the upper hemisphere and Assume is oriented with outward-pointing normal.

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

Compute the area of the shaded region whose boundary consists of the line segment on the x axis and the curve , .