Web Work 9 (MATH 317)

Problem 1

The region lies between the spheres and and within the cone with ; its boundary is the closed surface, , oriented outward. Find the flux of out of .

Problem 2

Let and

(a). Find the Flux of out of the Rectangular Solid

(b). For what Values of is the Flux Largest?

(c). What is that Largest Flux?

Problem 3

Use the Divergence Theorem to calculate the flux of the vector field and through the surface given by the sphere of radius centred at the origin with outwards orientation. Be sure that you are able to explain your answers geometrically.

with giving the interior of a sphere

Problem 4

Consider the flux integral . is the surface of the box with faces , , , , , and closed and oriented outward, and .

(a). Evaluate the Flux Integral Using the Divergence Theorem

(b). Directly Evaluate the Flux Integral

Problem 5

(a). Let . Calculate where is the Sphere of Radius 1 Centred at the Origin

(b) Let Be the Sphere of Radius 4 Centred at the Origin; Let Be the Sphere of Radius 7 Centred at the Origin; Let Be the Box of Side 8 Centred at the Origin with Edges Parallel to the Axes. without Calculating Them, Arrange the following Integrals in Increasing order

Problem 6

Compute the flux of the vector field through the surface , which is a closed cylinder of radius 1, centred on the -axis, with , and oriented outward.

Problem 7

Let . Evaluate for each of the following regions :

(a).

(b).

(c).

Problem 8

Use the divergence theorem to find the outward flux of the vector field across the boundary of the rectangular prism: , , and .

Problem 9

Suppose . Let be the solid bounded by the paraboloid and the plane . Let be the closed boundary of oriented outward.

(a). Use the Divergence Theorem to Find the Flux of through

(b). Find the Flux of out the Bottom of (the Truncated paraboloid) and the top of (the disk)

Problem 10

Use the divergence theorem to calculate the flux of the vector field through the sphere of radius 5 centred at the origin and oriented outward.

The gradient is odd over the region so it must sum to 0

Problem 11

Use the divergence theorem to calculate the flux of the vector field out of the closed, outward-oriented surface bounding the solid and .

Problem 12

Let . Use the divergence theorem to evaluate where is the top half of the sphere oriented upwards.

Problem 13

Use the Divergence Theorem to calculate the flux of across , where and is the surface of the tetrahedron enclosed by the coordinate planes and the plane

Problem 14

Compute the outward flux of the vector field across the boundary of the right cylinder with radius 7 with bottom edge at height and upper edge at .

Note: The vectors in this field point outwards from the origin, so we would expect the flux across each face of the cylinder to be positive.

Compute the flux using the following methods