Exam 16: Vector Calculus

Find a vector representation for the surface. The plane that passes through the point and contains the vectors and ..

A thin wire is bent into the shape of a semicircle If the linear density is , find the exact mass of the wire.

Find a function f such that , and use it to evaluate along the given curve C.

Evaluate . ; S is the part of the torus with vector representation , , .

Match the vector field with one of the plots shown below.

Use the Divergence Theorem to find the flux of F across S; that is, calculate . ; S is the sphere

Suppose that where g is a function of one variable such that . Evaluate where S is the sphere

Find parametric equations for C, if C is the curve of intersection of the hyperbolic paraboloid and the cylinder oriented counterclockwise as viewed from above.

Evaluate the line integral over the given curve C. , where C is the line segment joining (-2, -1) to (4, 5)

Find the curl of the vector field.

Find the curl of the vector field.

Let

A fluid with density flows with velocity Find the rate of flow upward through the paraboloid

Find the area of the surface. The part of the paraboloid ; ,

Evaluate the surface integral. S is the part of the cylinder between the planes and in the first octant.

Find the area of the surface S where S is the part of the plane that lies above the triangular region with vertices , and

Find a function f such that , and use it to evaluate along the given curve C.

Determine whether or not F is a conservative vector field. If it is, find a function f such that

Use Stoke's theorem to evaluate where and C is the boundary of the part of the plane in the first octant.

Evaluate the surface integral. Round your answer to four decimal places. S is surface