Chapter 1 Solutions
Exercise 1.1 f (x, y ) = x: a level curve f (x, y ) =constant is a vertical line x = c. Curves for c = 1, 0, 1/2, 3 and
4 are graphed below:
Level sets x=c for c=-1,0,1/2,3, and 4
Chapter 9 Hints
Exercise 9.1 Routine calculations.
Exercise 9.2
vf
=
f (p) v for any vector v Rn+1 .
p
Exercise 9.3 Routine calculations.
Exercise 9.4 Routine calculations.
Exercise 9.8 Apply the deni
Chapter 8: Hints and (partial) Solutions
Exercise 8.1 For this problem (prime) denotes a covariant derivative and
follow the covariant derivative denition on pg 45:
(dot) or
d
a usual derivative. Just
Chapter 6: Hints and (partial) Solutions
Exercise 6.1
(a) [n = 1]: Let f (x, y ) = y 2 . Then S is the surface in R2 that is the graph of the equation y 2 = 1. So S consists
of two horizontal lines: y
Chapter 5: Solutions
Exercise 5.1 Choose any distinct points p and q on the n-sphere Sn (n > 0). So p and q are unit vectors in Rn+1 .
Well use a great circle on Sn to connect these points. There are
Chapter 4: Hints
Exercise 4.1 All the functions are smooth. So try and nd points where
level sets of f contain these points.
f vanishes and then gure out which
(a) First note that f 1 (c) is the empty
Chapter 3 (partial) Solutions
Exercise 3.1
(a) When n = 1, f (x, y ) = x2 y 2 and so level sets are described by the equation: x2 y 2 = c. The following plot
contains curves for c = 1, c = 0 and c = 1
Chapter 2 (partial) Solutions
Exercise 2.1
(a) X (x, y ) = (0, 1), a constant vector eld.
vector field: (x,y,0,1)
10
y
5
0
-5
-10
-10
-5
0
x
The m-le below was used to generate the gure:
% vector fiel
Hints for Final Exam Problems
Exercise 8.7
If you know what a group is then skip the box below.
A group (G, ) is a nonempty set G together with a function (binary operation) : G G G such that
(a) ( is