WORKSHEET XVI
Vector Fields, Flow lines, divergence
1. Sketch each of the following vector fields. When possible, find isoclines to facilitate the
sketching process.
(a)
F(x, y) = (3, 3)
(b)
F(x, y) = (4, 0)
(c)
F(x, y) = (x, 0)
(d)
F(x, y) = (0, x)
(e)
F(x, y) = (x, y)
(f)
F(x, y) = (3x, 3y)
(g)
F(x, y) = (-x, -y)
(h)
F(x, y) = (-x, y)
(i)
F(x, y) = (y, x)
(j)
F(x, y) = (y, -2x)
(k)
F(x, y) = (x, -y)
(l)
F(x, y) = (y, -x)
(m)
F(x, y) = (x, x2)
(n)
F(x, y, z) = (y, -x, 0)
(o)
F(x, y, z) = (0, x, y)
2. Describe the differences (direction, norm, etc.) among the three vector fields V1(x,y) = y i + x j, V2(x,y) = y i – x j, V3 ( x, y ) 
V 4 ( x, y ) 
y
x2  y2
i
x
x2  y2
j and
y
x
i 2
j
2
x y
x  y2
2
3. Sketch each of the following gradient vector fields. (Use appropriate contour diagrams
to aid you in your sketch.)
4.
(a)
F(x, y) = xy
(b)
F(x, y) = (3x + y)
(c)
F(x, y) = (x2 + y2)
(d)
F(x, y) = (x2 – y)
(e)
F(x, y, z) = (x + 3y + 5z)
(f)
F(x, y, z) = (x2 + y2 + z2)
(a) Show that F(x, y) = (-y, x) is not a gradient vector field.
(b) Show that F(x, y) = (y, x) is a gradient vector field.
5.
Show that (t) = (cos t, sin t) is a flow line of the vector field F(x, y) = -y i + x j. Can
you find other flow lines?
6.
Solve a pair of differential equations to find the equations of the flow lines of
F(x, y) = (2x, 3).
7.
Solve a pair of differential equations to find the equations of the flow lines of
F(x, y) = (x, -y).
8.
Sketch a few flow lines for the vector fields given in question 1.
9.
Define divergence of a vector field F (denoted div F or F).
10.
Compute the divergence of the vector field F(x, y, z) = x2y i + z j + xyz k.
11.
Let V(x, y, z) = x i be the velocity vector field of a fluid in space. Relate the sign of
the divergence with the rate of change of volume under the flow.
12.
Sketch a few flow lines for F(x, y) = y i. Calculate div F and explain why this
answer is consistent with your sketch.
Sketch a few flow lines for F(x, y) = -3x i – yj. Calculate div F and explain why
13.
this answer is consistent with your sketch.
(Marsden) Let (t) be a flow line of a gradient field F (x, y, z) = -g. Prove that
14.
g((t)) is a decreasing function of t.
15.
Matching: (1) F(x, y) = (-x, -y), (2) F(x,y) = (y, -2x), (3) F(x, y) = (4y, -x),
(4) F(x, y) = (x, x2) (5) F(x, y) = (y, x)
5
5
0
0
5
5
5
0
5
5
5
0
5
5
0
5
0
5
5
0
5
5
0
5
1.0
0.5
0.0
0.5
1.0
1.0
0.5
0.0
0.5
1.0
As far as the laws of mathematics refer to reality, they are not certain; as far as
they are certain, they do not refer to reality.
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