Problem 1

We've seen two calculations of the energy of a system. One equation tells us , while another says .

Consider a force given by for some differentiable function . A particle of mass is being acted on by and no other forces, and its position at time is given by .

True or false: .

True they are both equal they just use different conventions for the potential energy, as conservative vector fields can be interpreted as potential energies so represents some potential energy and so does

Problem 2

For each of the following fields, decide which of the following holds:

• A. The screening test for conservative vector fields tells us is conservative.
• B. The screening test for conservative vector fields tells us is not conservative.
• C. The screening test for conservative vector fields does not tell us whether is conservative or not.

(a).

A. The screening test tells us the field is conservative.

(b).

B. The screening test tells us the field is not conservative.

(c).

C. The screening test is inconclusive as the domain is not simply connected

(d).

B. The screening test tells us the field is not conservative.

Problem 3

Suppose is conservative and let , , and be constants. Find a potential for , OR give a conservative field and constants , , and for which is not conservative.

Since is conservative for some potential function we can then notice that we can then see that giving us our potential function .

Problem 4

Prove, or find a counterexample to, each of the following statements.

(a). If is a Conservative Field and is a Non-conservative Field, then is Non-conservative

For to be conservative and for to not be conservative .

is not conservative as it's gradient is not zero.

(b). If and Are both Non-conservative Fields, then is Non-conservative

Consider and even though both fields are non-conservative their sum is the function which is conservative.

(c). If and Are both Conservative Fields, then is Conservative

Procedural

For and to be conservative and .

is conservative as it's gradient is zero.

Problem 5

Let be the domain consisting of all such that , and let be the vector field

Is conservative on ? Give reasons for your answer.

is conservative on as it's curl is zero and it's well defined on all of .

Problem 6

Find a potential for , or prove none exists.

Problem 7

Find a potential for , or prove none exists.

Problem 8

Find a potential for , or prove none exists.

There does not exist a potential for as it is not conservative.

Problem 9

Find a potential for

or prove none exists.

Problem 10

Determine whether or not each of the following vector fields are conservative. Find the potential if it is.

(a).

The vector field is conservative as it's irrotational.

(b).

The vector field is not conservative as it is not irrotational.

Problem 11

Let .

(a). For what Values of the Constants and is the Vector Field Conservative on ?

(b). If and Have the Values Found in (a), Find a Potential Function for

Problem 12

Find the velocity field for a two-dimensional incompressible fluid when there is a point source of strength at the origin. That is, fluid is emitted from the origin at area rate cm/sec. Show that this velocity field is conservative and find its potential.

The direction of must be facing away from the origin

Problem 13

For some differentiable, real-valued functions , we define

Verify that is conservative.

Problem 14

Describe the region in where the field

has curl .

The field has a vanishing curl on the line .