 
Summary: Extra Problems on Vector Calculus
Charles Martin
October 28, 2009
Note: Problems marked with are more difficult. Try the easier ones first.
1. Write in spherical coordinates: {(x, y, z) : x2
+ y2
+ z2
1 and x, y, z 0}. Integrate x over this region.
2. The plane y = z slices the cylinder x2
+ y2
= 4. Parametrize the resulting curve.
3. Let S be the unit sphere centered at the origin, with unit normal directed outward. Set F = xi + yj + zk.
Find the flux of F across S in three ways:
(a) Directly.
(b) Geometrically.
(c) Using the Divergence Theorem.
4. Write in cylindrical coordinates: {(x, y, z) : x2
+ y2
1 and x2
+ z2
