Description
1. Let F� : R3 −→ R3 be the vector field given by
F� (x, y, z) = ay2
ˆ j + (by2 2
)ˆ ı + 2y(x + z)ˆ + z k.
(i) For which values of a and b is the vector field F� conservative?
(ii) Find a function f : R3 −→ R such that F� = grad f, for these values.
(iii) Find the equation of a surface S with the property that for every
smooth oriented curve C lying on S,
F� · d�s = 0, C
for these values.
2. Let S be the rectangle with vertices (0, 0, 0), (1, 0, 0), (1, 2, 2) and
(0, 2, 2). Find the flux of the vector field F� : R3 −→ R3, given by
� 2
ˆı − z2ˆ F 2ˆ (x, y, z) = y j + x k,
ˆ through S in the direction of the unit normal vector nˆ, for which nˆ ·k >
0.
3. Let Ca(P) be the circle of radius a centered at P and oriented
counter-clockwise. A smooth rotation free vector field F� is defined on
the whole of R2, except for the points P0 = (0, 0), P1 = (4, 0), and
P3 = (8, 0), and
F� ·d�s = −2, F� ·d�s = 1 and F� ·d�s = 3. C2(P0) C6(P0) C10(P0)
Find the following line integrals.
(a)
�
F� · d�s.
C1(P1)
(b)
�
F� · d�s.
C1(P2)
(c)
�
F� · d�s.
C6(P2)
4. (6.3.16)
5. (7.1.4)
1
6. (7.1.20)
7. (7.2.13)
9. (7.2.17)
10. (7.3.11)
11. (7.3.13)
12. (7.3.16)
13. (7.3.18)
14. (7.3.19)
Just for fun: Let ω be a k-form on Rn. Show that
d(dω) = 0.
This basic fact about d is often expressed in the formula d2 = 0.
2