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
0
c=0 c=3
y
-0.5
-1
-1.5
-2

Linear Algebra Basics A vector space (or, linear space) is an algebraic structure that often provides a home for solutions of mathematical models. Linear algebra is a study of vector spaces and a class of special functions (called linear transformations)

Hints Chapter 5
December 12, 2004
Math 5010/6010
#5-5 Prove that a k-dimensional (vector) subspace of Rn is a k-dimensional manifold. hint: Give the supspace a name and note that its equal to the span of a set A consisting of k linear independent vectors

You'll be asked to choose and do four problems from a list of six problems. The six problems will be taken from the following list of exercises from the text: 4-3, 4-9 (but not all of these), 4-11, 4-14, 4-16, 4-19, 4-25, 4-26

The dot product in Rn .
1. Dot product in Rn : If x = (x1 , . . . , xn ) and y = (y1 , . . . , yn ) are points in Rn then their dot product is dened
n
by x y = k=1 xk yk . This is a natural extension to higher dimensions of the familiar dot product from
e

Some Properties of the Determinant Function
The determinant of a square matrix A, det A, is a real number. However, rather than thinking of the determinant as a
function of the entire matrix A its nature is better revealed by regarding it as a function of

Chapter 8: Hints and (partial) Solutions
Exercise 8.1
(a) For this problem
(prime) denotes a covariant derivative and
derivative denition (pg 45):
(X + Y ) =
d
[X + Y] (t)
dt
= X(t) + Y(t)
(dot) or
d
a usual derivative. Use covariant
dt
d
[X + Y] (t) N(

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: Hints and (partial) Solutions Exercise 5.1 Start with distinct points p and q on the n-sphere Sn . So p and q are unit vectors in Rn+1 . There are three cases (well. only two really, the rst is a special case of the third): (a) If p q (i.e., p

Chapter 4: Hints and Solutions
Exercise 4.1 All the functions are smooth. So try and nd points where f vanishes and then gure out which
level sets of f contain these points.
(a) First note that f 1 (c) is the empty set if c < 0, so f 1 (c) is not an n-sur

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
5
y
0
-5
-10 -10 -5 0
x
5
10
The m-le below was used to generate the gure:
% vector field: (x,y) -> (x,y),(0,1) clear clf % pick grid larg

Notation and Terminology
1. R: The symbol R denotes the set of real numbers together with usual real number arithmetic. The distance between two numbers x and y is the absolute value of their dierence: |x y |. 2. Rn : For each positive integer n the set R