Problem 1

Evaluate , where and is the elliptic cylinder oriented away from the -axis and given by , , where , , and are positive constants.

Problem 2

Compute the flux of through the curved surface of the cylinder bounded below by the plane , above by the plane , and oriented away from the -axis.

Problem 3

Let be the closed surface that consists of the hemisphere

and it's base

Let be the electric field defined by . Find the electric flux across . Write the integral over the hemisphere using spherical coordinates, and using the outward pointing normal.

The flux is given by the integral:

Find , , , , and and then evaluate the integral over , , and all of .

Since always points out of the surface and since is parallel to our surface at the flux through is zero.

Problem 4

Suppose is a radial force field, is a sphere of radius 8 centred at the origin, and the flux integral .

Let be the sphere of radius 48 centred at the origin, consider the flux integral .

(a). If the Magnitude of is Inversely Proportional to the Square of the Distance from the Origin, what is the Value of

The flux is independent of radius so

(b). If the Magnitude of is Inversely Proportional to the Cube of the Distance from the Origin, what is the Value of

Problem 5

Set up a double integral for calculating the flux of the vector field through the open-ended circular cylinder of radius 3 and height 4 with its base on the xy-plane and centred about the positive z-axis, oriented away from the z-axis.

The flux is given by the integral:

Find , , , and and the evaluate the integral.

Problem 6

Calculate the flux of the vector field

through the square of side length with one vertex at the origin, one edge along the positive y-axis, one edge in the xz-plane with and , oriented downward with normal .

Problem 7

Consider the vector field and the surface of the upper hemisphere of the sphere .

Setup the Double Integral for Calculating the Flux of over the Surface and then evaluate it.