Problem 1

Let be the cube , let be the square , and let have continuous first partial derivatives.

(a). Use the Fundamental Theorem of Calculus to Show that $$\iiint_V \frac{\partial f}{\partial z}(x,y,z),dx,dy,dz = \iint_R f(x,y,1),dx,dy - \iint_R f(x,y,0),dx,dy.$$

(b). Use the Divergence Theorem to Show that $$\iiint_V \frac{\partial f}{\partial z}(x,y,z),dx,dy,dz = \iint_R f(x,y,1),dx,dy - \iint_R f(x,y,0),dx,dy.$$

Problem 2

(a). By Applying the Divergence Theorem to , where is an Arbitrary Constant Vector, Show that $$\iiint_V \nabla\varphi,dV = \iint_{\partial V} \varphi,\hat{n},dS.$$

(b). Show that the Centroid of a Solid with Volume is given by $$(\bar{x}, \bar{y}, \bar{z}) = \frac{1}{2|V|}\iint_{\partial V}(x^2+y^2+z^2),\hat{n},dS.$$

Problem 3

Let be the unit sphere centred at the origin and oriented by the outward pointing normal. If , evaluate the flux of through :

(a). Directly

(b). By Applying the Divergence Theorem

Problem 4

Evaluate, by two methods, the integral , where , is the surface , and is the outward pointing unit normal to .

(a). First, by Direct Computation of the Surface Integral

(b). Second, by Using the Divergence Theorem

Problem 5

Let , let be the solid defined by , let be the bottom surface of (namely , ), and let be the curved portion of the boundary of (namely , ). Denote by the volume of and compute, in terms of :

(a). With Pointing downward

(b).

(c). With Pointing Outward. Use the Divergence Theorem to Answer at Least One of Parts (a), (b), and (c)

Problem 6

Evaluate the integral , where , is the surface for , and is the upward pointing normal, by two methods:

(a). First, by Direct Computation of the Surface Integral

(b). Second, by Using the Divergence Theorem

Problem 7

(a). Find the Divergence of the Vector Field

(b). Find the Flux of the Vector Field of (a) outward through the Sphere of Radius Centred at the Origin in

Problem 8

The sides of a grain silo are described by the portion of the cylinder with . The top of the silo is given by the portion of the sphere lying within the cylinder and above the -plane. Find the flux of the vector field

out of the silo.

Problem 9

Let be the ball of volume centred at the point , and let be the sphere that is the boundary of . Find the flux of outward (from ) through .

Problem 10

Let . Let be the portion of the surface which is above the -plane. What is the flux of downward through ?

Problem 11

Use the divergence theorem to find the flux of through the part of the ellipsoid

with . (Note: the ellipsoid has volume .)

Problem 12

Let where and .

(a). Find

(b). Find the Flux of outwards through the Spherical Surface

(c). Do the Results of (a) and (b) Contradict the Divergence Theorem? Explain Your Answer

No they don't as the field is undefined at which changes the value.

(d). Let Be the Solid Region Bounded by the Surfaces , , and . Let Be the Bounding Surface of . Determine the Flux of outwards through

(e). Let Be the Solid Region Bounded by the Surfaces , , and . Let Be the Bounding Surface of . Determine the Flux of outwards through

Problem 13

Consider the ellipsoid given by with the unit normal pointing outward.

(a). Parametrise

(b). Compute the Flux of the Vector Field

(c). Verify Your Answer in (b) Using the Divergence Theorem

Problem 14

Evaluate the flux integral , where

and is the surface of the solid region bounded by the cylinder and the planes and . The surface is positively oriented (its unit normal points outward).

Problem 15

Find the flux of the vector field through the cylindrical surface whose equation is , extending from to . (Only the curved part of the cylinder is included, not the two disks bounding it.) The orientation of the surface is outward, pointing away from the -axis.

Problem 16

The surface is the part above the -plane of the surface obtained by revolving the graph of around the -axis. The surface is oriented such that the normal vector has positive -component. The circle with radius and centre at the origin in the -plane is the boundary of .

Find the flux of the divergenceless vector field through .

Problem 17

Let be the part of the paraboloid contained in the cone and oriented in the upward direction. Let

Evaluate the flux integral .

Problem 18

Evaluate the surface integral where

and is the boundary of the solid region enclosed by the paraboloid and the plane , with outward pointing normal.

Problem 19

Let be the part of the sphere between the planes and , oriented away from the origin. Let

Compute the flux integral .

Problem 20

Let be the boundary of the solid region

Find the flux of the vector field

outward through .

Problem 21

Let be the hemisphere , , oriented with pointing away from the origin. Evaluate the flux integral where

Problem 22

Let be the solid region between the plane and the paraboloid . Let

(a). Compute the Flux of outward through the Boundary of

(b). Let Be the part of the Paraboloid Lying below the Plane, Oriented with a Normal Vector that Has a Positive Component. Compute the Flux of through

Problem 23

Consider the vector field .

(a). Compute

(b). Let Be the Sphere , Oriented Outwards. Compute

(c). Let Be the Sphere , Oriented Outwards. Compute

(d). Are Your Answers to (b) and (c) the Same or Different? Give a Mathematical Explanation of Your Answer

Problem 24

Let be the vector field defined by

Calculate the flux integral where is the boundary surface of the solid region

with outer normal.

Problem 25

Consider the vector field . Let the surface be the part of the sphere that lies above the plane , oriented downwards.

(a). Find the Divergence of

(b). Compute the Flux Integral

Problem 26

Let be the sphere , oriented inward. Compute the flux integral where

Problem 27

Consider the vector field .

(a). Calculate

(b). Find the Flux of through the Surface Defined by , , Using the outward Normal to

Problem 28

Let be the portion of the sphere that lies above the -plane. Find the flux of outward across .

Problem 29

Find the flux of outward through the hemispherical surface

Problem 30

Let be the cylinder , . Calculate the flux of the vector field

outward through the curved part of the surface of .

Problem 31

Find the flux of upward through the first octant part of the sphere .

Problem 32

Let and let

• be the portion of the cylinder that lies inside the sphere ,
• be the portion of the sphere that lies outside the cylinder , and
• be the solid bounded by and .

Compute:

(a). With Pointing inward

(b).

(c). With Pointing Outward. Use the Divergence Theorem to Answer at Least One of Parts (a), (b), and (c)

Problem 33

Let be the electric field due to a charge configuration that has density . Gauss' law states that, if is any solid in with surface , then the electric flux

is the total charge in . Here, as usual, is the outward pointing unit normal to . Show that for all in . This is one of Maxwell's equations. Assume that and are well-defined and continuous everywhere.

Problem 34

Let be a solid in with surface . Show that

where and, as usual, is the outer normal to . See if you can explain this result geometrically.

Problem 35

Let be the sphere of radius , centred at the origin and with outward orientation. Given the vector field :

(a). Calculate (using the definition) the Flux of through , that Is, Compute the Flux by Evaluating the Surface Integral Directly

(b). Calculate the Same Flux Using the Divergence Theorem

Problem 36

Consider the cube of side length that lies entirely in the first octant (, , ) with one corner at the origin and another corner at . Denote as the open surface that consists of the union of the 5 faces of the cube that do not lie in the plane . The surface is oriented so that the unit normal vectors point outwards. Determine the value of

where .

Problem 37

Let be the solid region between the plane and the paraboloid . Let

(a). Compute the Flux of outward through the Boundary of

(b). Let Be the part of the Paraboloid Lying below the Plane, Oriented with a Normal Vector that Has a Positive Component. Compute the Flux of through

Problem 38

(a). Find an upward Pointing Unit Normal Vector to the Surface at the point

(b). Now Consider the part of the Surface Which Lies within the Cylinder , Called . Compute the upward Flux of through

(c). Find the Flux of through the Cylindrical Surface in between and . the Orientation is Outward, away from the -axis

Problem 39

(a). Find the Divergence of the Vector Field

(b). Find the Flux of through the Upper Hemisphere , , Oriented in the Positive -direction

(c). Specify an Oriented Closed Surface such that the Flux is Equal to

Problem 40

Evaluate the surface integrals. (Use any method you like.)

(a). , Where is the part of the Cone where and

(b). , Where and is the Rectangle with Vertices , , , , Oriented so that the Normal Vector Points upward

(c). , Where and is the Boundary Surface of the Box , , , with the Normal Vector Pointing outward

Problem 41

Let be the open surface , ; let be the open surface , ; and let be the planar surface , . Let

where , , , and are constants.

(a). Find the Flux of upwards across

(b). Find All Values of the Constants , , , and so that the Flux of outwards across the Closed Surface is Zero

(c). Find All Values of the Constants , , , and so that the Flux of outwards across the Closed Surface is Zero

Problem 42

Let be the ellipsoid and its outward unit normal.

(a). Find if

(b). Find if

Problem 43

Let be a smoothly bounded domain, with boundary and outer unit normal . Prove that for any vector field which is continuously differentiable in ,

Problem 44

Recall that if is a smooth closed surface with outer normal field , then for any smooth function on ,

where is the solid bounded by . Show that as a consequence, the total force exerted on the surface of a solid body contained in a gas of constant pressure is zero. (Recall that the pressure acts in the direction normal to the surface.)

Problem 45

Let be a smooth 3-dimensional vector field such that the flux of out of the sphere is equal to for every . Calculate .

Problem 46

Let and let be the part of the surface having , oriented with normal pointing away from the origin. Here is a constant. Compute the flux of through .

Problem 47

Let be a solution of Laplace's equation

in . Let be a smooth solid in .

(a). Prove that the Total Flux of out through the Boundary of is Zero

(b). Prove that the Total Flux of out through the Boundary of Equals $$\iiint_\mathcal{R}\left[\left(\frac{\partial u}{\partial x}\right)^2 + \left(\frac{\partial u}{\partial y}\right)^2 + \left(\frac{\partial u}{\partial z}\right)^2\right]dV.$$

Problem 48

Let be the part of the solid cylinder satisfying , and let be the boundary of . Given :

(a). Find the Total Flux of outward through

(b). Find the Total Flux of outward through the (vertical) Cylindrical Sides of . (Hint: for )

Problem 49

A smooth surface lies above the plane and has as its boundary the circle in the plane . This circle also bounds a disk in that plane. The volume of the 3-dimensional region bounded by and is 10 cubic units. Find the flux of

through in the direction outward from .