Solutions to some problems (Lectures 15-20)
Problem Set 17, problem 3(c)
Decide whether the vector eld could be a gradient eld. Justify your answer. Hint. Find potential functions for the vector elds.
F (x, y, z) = xz 2 i + x2 +z 2 j + x2 +z 2 k
Solutions to some problems (Lectures 27, 28)
Problem Set 27, problem 4
Show that if a is a constant vector and f (x, y, z) is a function, then
div(f a) = (gradf ) a.
Solution. Let a = a1 i + a2 j + a3 k. Then
f (x, y, z)a = f (x, y, z)a1 i + f (x, y, z)a2
Solutions to some problems from Problem Sets 27
Problem set 2
1. Let D be the region inside the unit circle centered at the origin. Decide
without calculation whether the integrals are positive, negative, or zero.
(j) D xex dA. Solution. The region D is s
1. RUse the divergence theorem, which says the flux through S is equal to
R div F dV where R is the region inside the box.
(e + x) +
(2y + cos x) +
(e z) = 1 + 2 1 = 2.
The volume of the box is 33 = 27, so
div F dV = 2