MVC
Quiz #4
Instructions: SHOW ALL WORK !!
No Calculator allowed.

1. (7 pts) Let F ( x, y, z )   3x 2  y  iˆ   y  z  ˆj  

NAME:
zˆ
k
y
a) Calculate the curl of F .
b)
2.
MVC
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.
(5 pts) Determine whether or not x(t )  (4e2t , e2t , t 2 ) is a flow line for the vector field
x


F( x, y, z )   x  4 y, ,ln( y )  . Explain work carefully.
2


1

3. (8 pts) Find the flow line for the vector field F ( x, y, z )   , 2, z  3  with x(0)  (1,5, 2) .
x

4.(5 pts) If f ( x, y, z ) and F ( x, y, z ) are suitably differentiable, show: div( fF )  f F  f div( F )
MVC
5. (8 pts) Let x (t )   et cos t , et sin t , et  ,0  t  4 .
a. Find the arc-length parameter s (t ) .
b. Find the unit tangent T , the unit normal N , and the curvature  .
MVC
6. Let F :
2

2
be defined by F ( x, y )  ( y, x) .
a. (2 pts) Is F conservative? Explain.
b. (1 pt) Is F incompressible? Explain.
c. (2 pts) Find a flow line for F through the point (1,0).
d. (1 pts) Find a flow line for F through the point (a,b) where a 2  b 2 .
MVC