Weekly Assignment for Dec 4 - Dec 8
Do problems on pages 300-301/4,6,8,10 and the following eight problems:

1) In each case, compute the line integral of the given vector field F about the boundary curve, g, of the triangle in the
plane x + y + z = 1 where x, y and z are all positive. In each case, traverse the curve g clockwise as
viewed from z = 100 on the z -axis. Do this by first parametrizing each of the three segments of g.
Then, do the calculation via Stokes’ theorem.
a) F = (z, x, y)
b) F = (x², y , z)
c) F = (xy, z, 0)

2) In each case, Find the flux of the indicated vector field F through the boundary of the cube in R³ where
0 £ x £ 1, 0 £ y £ 1 and 0 £ z £ 1. In each case, do the calculation first by computing the flux through
each face and then use the divergence theorem.
a) F = (1, 1, 1)
b) F = (x, 0, 0)

3) Suppose that F is a vector field with curl F = (1, 1, 1). Find all planes in R³ with the property that the
line integral of F about any closed curve in the plane is zero.

4) Let F be a vector field on R³ and let g be a function on R³. Prove the following:
a) div(gF) = g div(F) + Ñg×F
b) curl(gF) = g curl(F) + Ñg ´ F.

5) Suppose that F is a vector field on R³ and ||F|| £ 1 at all points inside the ball V where x² + y² + z² £ 1.
Explain why -4p £ £ 4p . (Hint: Use the Divergence Theorem to rewrite this integral and then
think about the size of the resulting integrand.)

6) Take F as in the previous problem and let R denote the disc x² + y² £ 1 in the z = 0 plane
and let n = (0, 0, 1) be its normal. Explain why - 2p £ £ 2p.

7) Compute the volume of the tetrahedron V (four sided prism) in R³ given by x ³ 0, y ³ 0, z ³ 0
and x + y + z £ 1. (Thus, the boundary of V has four triangular faces, one each in the planes
x = 0, y = 0, z = 0 and x + y + z = 1.) Do the computation first as an iterated integral, and then via
the Divergence Theorem as a flux integral through the boundary of V of the vector field F = (x, 0, 0).

8) For each angle q between 0 and 2p, let D_q denote the disk in R³ whose radius is 1, center is the
origin in R³ and which lies in the plane cos(q) x + sin(q) z = 0. Let I(q) denote the absolute value of
the line integral over the boundary of D_q of the vector field F = (0, 0, y). What is the maximum value
of I(q) and what are the angles q which have this value? (Hint: The problem is easier if you use
Stokes’ theorem and think about the size of curl(F)×n depends on the angle q.)