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 : 
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.
(b) grad
Meaningful, vector field
(c) div
Meaningful, scalar function
(d) curl(grad
)
Meaningful, vector field
(e) grad
Not meaningful
(f) grad(div
)
Meaningful, vector field
(g) div(grad
)
Meaningful, scalar function
(h) grad(div
)
Not meaningful
(i) curl(grad
)
Not meaningful
(j) div(div
)
Not meaningful
(k) (grad
)
(div
)
Not meaningful
(l) div(curl(grad
))
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) 
7. Find 
.