Math 2210-1
Homework 8
Due Wednesday July 28
Show all work. Please box
√ your answers. Be sure to write in complete sentences when appropriate. Also,
I prefer exact answers like 2 instead of 1.414. Note that a symbol indicates that graph paper might be
useful for that problem.
Vector Fields
1.
Sketch the graphs of the following vector fields.
(a) F~ (x, y) = 2~i + 3~j
(b) F~ (x, y) = y~i
(c) F~ (~r) = 2~r
(d) F~ (x, y) = (x + y)~i + (x − y)~j
2. For each of the vector fields below, calculate div F~ and sketch F~ .
~r
(a) F~ (~r) = 3 (in 3-space), ~r 6= ~0.
r
(b) F~ (x, y, z) = z~k.
3. Find the divergence of the given vector field.
(a) F~ (x, y) = (x2 − y 2 )~i + 2xy~j
~r
(b) F~ (~r) = (in 3-space), ~r 6= ~0
r
~
(c) F (x, y) = −x~i + y~j
(d) F~ (x, y) = −y~i + x~j
(e) F~ (x, y, z) = (−x + y)~i + (y + z)~j + (−z + x)~k
4. Compute the curl of the following vector fields:
(a) F~ (x, y, z) = (x2 − y 2 )~i + 2xy~j
~r
(b) F~ (~r) = (in 3-space), ~r 6= ~0
r
(c) F~ (x, y, z) = x2~i + y 3~j + z 4~k
2
(d) F~ (x, y, z) = ex~i + cos y~j + ez ~k
(e) F~ (x, y, z) = 2yz~i + 3xz~j + 7xy~k
(f) F~ (x, y, z) = (−x + y)~i + (y + z)~j + (−z + x)~k
(g) F~ (x, y, z) = (x + yz)~i + (y 2 + xyz)~j + (zx3 y 2 + x7 y 6 )~k
5. If F~ is any vector field whose components have continuous second partial derivatives, show that
div curl F~ = 0.
6. For any constant vector field ~c, and any vector field F~ , show that div (F~ × ~c) = ~c · curl F~ .
7. For ~c a constant vector field and F~ any vector field, show that curl (F~ + ~c) = curl F~ .
8. For φ a scalar function and F~ a vector field, show that curl (φF~ ) = φ curl F~ + (∇φ) × F~ .
9. Show that if φ is a harmonic function, then ∇φ is both curl free and divergence free.
Line Integrals
10. Consider Rthe vectorRfield F~ shown Rbelow, together with the paths C1 , C2 and C3 . Arrange the line
integrals C1 F~ · d~r, C2 F~ · d~r, and C3 F~ · d~r in ascending order.
111111111111111111111111111100
0000000000000000000000000000
000000000000000000000000000011
1111111111111111111111111111
0000000000000000000000000000
1111111111111111111111111111
0000000000000000000000000000
1111111111111111111111111111
0000000000000000000000000000
1111111111111111111111111111
0000000000000000000000000000
1111111111111111111111111111
000000000000000
111111111111111
0000000000000000000000000000
1111111111111111111111111111
000000000000000
111111111111111
0000000000000000000000000000
1111111111111111111111111111
000000000000000
111111111111111
C2
0000000000000000000000000000
1111111111111111111111111111
000000000000000
111111111111111
0000000000000000000000000000
1111111111111111111111111111
000000000000000
111111111111111
0000000000000000000000000000
1111111111111111111111111111
000000000000000
111111111111111
0000000000000000000000000000
1111111111111111111111111111
000000000000000
111111111111111
0000000000000000000000000000
111111111111111111111111111111
00
11.
11
00
C1
11
00
000000000000000000000000000000
11111111111111111100
000000000000000000
11111111111111111111111111111111
00
11
C3
For each of the following figures, say whether the line integral of the pictured vector field over the
given curve is positive, negative, or zero.
12.
(a)
1111111111111111111111111111111111111111111111111111111111111111111111111111111111
0000000000000000000000000000000000000000000000000000000000000000000000000000000000
(b)
1111111111111111111111111111111111111111111111111111111111111111111111111111111111
0000000000000000000000000000000000000000000000000000000000000000000000000000000000
(c)
1111111111111111111111111111111111111111111111111111111111111111111111111111111111
0000000000000000000000000000000000000000000000000000000000000000000000000000000000
(d)
1111111111111111111111111111111111111111111111111111111111111111111111111111111111
0000000000000000000000000000000000000000000000000000000000000000000000000000000000
Compute the line integral of the given vector field along the given path.
(a) F~ (x, y) = x2~i + y 2~j and C is the line from the point (1, 2) to the point (3, 4).
(b) F~ (x, y) = ln y~i + ln x~j and C is the curve given parametrically by ~r(t) = 2t~i + t3~j for 2 ≤ t ≤ 4.
(c) F~ (x, y) = ex~i + ey~j and C is the part of the ellips x2 + 4y 2 = 4 joining the point (0, 1) to the
point (2, 0) in the clockwise direction.
(d) F~ (x, y) = xy~i + (x − y)~j and C is the triangle joining the points (1, 0), (0, 1) and (−1, 0) in the
clockwise direction.
(e) F~ (x, y, z) = x~i + 2zy~j + x~k and C is given by ~r(t) = t~i + t2~j + t3~k for 1 ≤ t ≤ 2.
13.
Consider the vector field F~ = −y~i + x~j. Let C be the unit circle oriented counterclockwise.
(a) Show that F~ has constant magnitude of 1 on the circle C.
(b) Show that F~ is always tangent to the circle C.
R
(c) Show that C F~ · d~r = Length of C.
(d) Explain why this is.
14. Evaluate the line integral
Z
C
(2x + 9z) ds, where C is the curve x = t, y = t2 , z = t3 , 0 ≤ t ≤ 1.
15. Find the work done by the force field F~ (x, y) = ex~ı − e−y~ moving a particle along the curve C given
by x = 3 ln t, y = ln 2t, 1 ≤ t ≤ 5.
Independence of Path
16. Decide whether or not the following vector fields could be gradient fields. Justify your answer.
(a) F~ (x, y) = x~i
z
~k
~j + √ x
~i + √ y
(b) F~ (x, y, z) = − √
2
2
2
2
x +z
x2 + z 2
x +z
~
(c) G(x,
y) = (x2 − y 2 )~i − 2xy~j
1
(d) F~ (r) = 3 ~r, where ~r = x~i + y~j + z~k.
r
17. The statement below is FALSE. Explain why or give a counterexample.
R
“If C F~ · d~r = 0 for one particular closed path C, then F~ is conservative.”
2
2
18. Suppose that ∇f = 2xex sin y~i + ex cos y~j. Find the change in f between (0, 0) and (1, π2 ) in two
ways:
(a) By computing a line integral.
(b) By computing f .
19. Suppose a particle subject to a force F~ (x, y) = y~i − x~j moves along the arc of the unit circle, centered
at the origin, that begins at (−1, 0) and ends at (0, 1) (i.e. clockwise).
(a) Find the work done by F~ . Explain the sign of your answer.
(b) Is F~ conservative? Explain.
20. The line integral of F~ = (x + y)~i + x~j along each of the following paths is 3/2.
(i) The path (t, t2 ), with 0 ≤ t ≤ 1.
(ii) The path (t2 , t), with 0 ≤ t ≤ 1.
(iii) The path (t, tn ), with n > 0 and 0 ≤ t ≤ 1.
Verify this in two ways:
(a) Using a parameterization to compute the line integral.
(b) Using the Fundamental Theorem of Calculus for Line Integrals.
Green’s Theorem in the Plane
21. Suppose F~ (x, y) = x~. Show that the line integral of F~ around a closed curve in the xy-plane measures
the area of the region enclosed by the curve.
22.
Use Green’s Theorem to exaluate the following. Sketch the region.
(a)
(b)
I
IC
(2x + y 2 ) dx + (x2 + 2y) dy, where C is the closed curve formed by y = 0, x = 2, and y =
x3
.
4
(e3x + 2y) dx + (x2 + sin y) dy, where C is the rectangle with vertices (2,1), (6,1), (6,4), and
C
(2,4).
23. Let F~ =
x
y
~ı − 2
~ = F~1~ı + F~2~
x2 + y 2
x + y2
(a) Show that
∂F2
∂F1
=
.
∂x
∂y
(b) Show, by using the parametrization x = cos t, y = sin t, that
the unit circle.
I
C
F1 dx + F2 dy = −2π, where C is
(c) Why doesn’t this contradict Green’s Theorem?
24. If F~ = (x2 + y 2 )~ı + 2xy~, find the flux of F~ across the boundary C of the unit square with vertices
(0,0), (1,0), (1,1), and (0,1).