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
2
1.5
1
0.5
c=-1
c=1/2
c=4
y
0
c=0
c=3
-0.5
-1
-1.5
-2
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 denition here, e.g., if : I U (or S ) is a curve with (t0 )
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
dt
(a)
(X + Y) (t) =
d
[X + Y] (t)
dt
= X(t) + Y(t)
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 = 1 and y = 1.
R2
y
N(p) = (p, (0, 1)
.
.
.
.
.
.
.
.
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 certainly other approaches, e.g., someone from class
ha
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 set if c < 0, so f 1 (c) is not an n-surface. Next, si
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.
level sets for f(x,y)=x2y2
2.5
2
1.5
c=1
1
c=0
0.5
c=
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 field: (x,y) -> (x,y),(0,1)
clear
clf
% pick grid large so
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 associative) for any three elements a, b, c G, the val