MVC
Quiz #4
Instructions: SHOW ALL WORK !!
Calculator allowed.
45 minutes
1.
z
Let F ( x, y)   3x 2  y  iˆ   y  z  ˆj     ˆj
 y
a)
Calculate the curl of F .
b)
2.
MVC
NAME:
Is there a scalar function f such that F  f ? If so, find f. If no such function
exists, explain how you know this to be the case.
whether x(t )  (4e2t , e2t , t 2 ) is a flow
x


F( x, y, z )   x  4 y, ,ln( y )  . Explain work carefully.
2


Determine
line
for
the
vector
field
2

3. Find the flow line for the vector field F ( x, y, z )   3 x, y 3 ,  with x(0)  (5, 2, 2) .
z

2

Let x (t )   x(t ), y(t ), z (t )  . The x(t )  F ( x (t ))   x(t ), y(t ), z(t )    3 x, y 3 ,  .
z

3t
3t
x  3x  x  Ce . Since x(0)  5, x(t )  5e .
dy
1
1
4
2
y  y 3  3  dt  2  t  C. Since y (0)  2,  C  y 2 
y
y
2y
8
1  8t
1  8t
2
z2
z   zdz  2dt 
 2t  C. Since z (0)  2,2  C  z 2  4t  2  z   4t  2
z
2
2


Therefore, x (t )   5e3t ,
,  4t  2 
1  8t


4. If f ( x, y, z ) and F ( x, y, z ) are suitably differentiable, show: div( fF )  f F  fdiv( F )
MVC
Let x (t )   et cos t , et sin t , et  ,0  t  4 .
MVC
a.
Find the arclength parameter s (t ) .
b.
Find the unit tangent T , the unit normal N , and the curvature  .
5. Find an integral expression for the length of the curve in which the cylinders x 2  z 2  1
b
and y  z  1 . Your answer should be of the form
2
2
 f (t )dt .
a
MVC