SKKU-Calculus-Sec-15-5 Curl and Divergence

15.5    Curl and Divergence                             by SGLee-HSKim

1. Find (a) the curl and (b) the divergence of the vector field.

.

(a) curl :

(b) div :

.

(a) curl :  ,  (b) div : .

.

(a) curl :  ,  (b) div : .

.

(a) curl :  ,  (b) div : .

.

(a) curl :  ,  (b) div : .

.

(a) curl : ,

(b) div : .

.

(a) curl :  ,  (b) div : .

.

(a) curl :  ,  (b) div : .

2. Let  be a scalar function and  a vector field. State whether each expression is meaningful. If not, explain why. If so, state whether it is a scalar function or a vector field.

(a) curl

Not meaningful, curl must take a vector field.

Meaningful, vector field

(c) div

Meaningful, scalar function

Meaningful, vector field

Not meaningful

Meaningful, vector field

Meaningful, scalar function

Not meaningful

Not meaningful

(j) div(div)

Not meaningful

Not meaningful

Meaningful, scalar function

3. Is there a vector field  on  such that curl ? Explain.

No, curl.

4. Is there a vector field  on  such that curl ? Explain.

No, curl.

5. (a) Let  be a differentiable vector field with

div. Define a vector field  by

,

and . Prove that .

(b) Let . Find  such that .

6. Evaluate the line integral , where  and  is the curve given by  .

Let . We then observe that

. Note first that direct computations show that , which implies that there exists a scalar function  with .

We then have the following equations:

.

Solving these equations, one can see that , where  is a constant.

Thus, due to the Fundamental Theorem for line integral,

.

8. If  .

a. Prove that the line integral  is independent of the curve  joining two given points  and .

b. Show that there exists a scalar function  such that  and find .

c. Also find the work done in moving an object from  to .

9. Let  and . Then verify the following identities:

(1)                      (2) curl

(3)                (4)