Vector Fields
Reading assignment: 3.3, 3.4
Recommended exercises: 3.3: 1, 3, 17, 19, 21, 23, 25, 27
3.4: 1 – 13, 15, 17,19,21, 25
1
1) Show that the path x (t )  (cos(t  1), t 3  1,  2) is tangent to the surface
t
3
3
3
x  y  z  xyz  0 when t = 1.
2) Sketch the following vector fields:
a) F ( x, y )   x , y 
b) f , where f ( x, y ) 
x
y
3)
Verify that the path is a flow line of the indicated vector field.
3
2

a. x (t )   2sin t ,3cos t , 2t  ; F ( x, y, z )   y,  x, 2  .
2
3

b. x (t )   e 2t , e5t  ; F ( x, y)   2 x,5 y 
4)
Calculate the flow line x (t ) of the given vector field F that passes through the given point.
a. F ( x, y)    x, y  ; x (0)   2,3
b. F ( x, y, z )  1, y, z 2  ; x (0)  1,1,1
Calculate the curl of each vector field.
5) F(x, y) = cos x i + sin y j
6)
F(x, y, z) = (3xz)i + (zex + y)j + (y sin x)k
7)
F(x, y, z) = r = xi + yj + zk
8) Let F(x, y) = 3x2yi + (x3 + y3)j.
a) Verify that the curl of this vector field is zero.
b) By integrating, find a function f so that F  f . Don’t forget your “constants” of integration!
 x
9) Let f ( x, y, z )  e x cos   . Verify that curl(f )  0 .
z
10) For each vector field in 4 given below, either find a function for which it is the gradient, or
explain why no such function exists. Variables are in the order x, y, z, w.
a) (sin y + w cos x + 1)e1 + (sinz + x cos y + 1)e2 + (sin w + y cos z)e3 + (sin x + z cos w)e4.
b) yz e1 + xz e2 + xy e3 + yw e4.
  
11) By direct calculation, verify the product rule  ( gF )  g (curl F )  g  F