The University of Sydney
School of Mathematics and Statistics
Tutorial 10 (Week 11)
MATH2061: Vector Calculus
Semester 1
Preparatory question (attempt before the tutorial)
1. Find F in each of the following:
(a)
F = 2i + 3j + 4k
F = x2 yz i + xy 2 z j + x
The University of Sydney
MATH1002 Linear Algebra
Semester 1
Exercises for Week 4 (beginning 24 March)
2014
Preparatory exercises should be attempted before coming to the tutorial. Questions labelled with
an asterisk are suitable for students aiming for a
5 Vectors in Space, nVectors
In vector calculus classes, you encountered threedimensional vectors. Now
we will develop the notion of nvectors and learn some of their properties.
6 Overview
We begin by looking at the space R", which we can think of as the
196 Chapter 4 Infinite Sequences and Series
Since a subsequence fsnk g is again a sequence (with respect to k), we may ask whether
fsnk g converges.
Example 4.2.2 The sequence fsn g defined by
1
sn D . 1/ 1 C
n
n
does not converge, but fsn g has subsequen
114 Chapter 3 Integral Calculus of Functions of One Variable
3.1 DEFINITION OF THE INTEGRAL
The integral that you studied in calculus is the Riemann integral, named after the German
mathematician Bernhard Riemann, who provided a rigorous formulation to re
Section 3.2 Existence of the Integral
129
Adding these inequalities and taking account of cancellations yields
0 S.P .0/ /
S.P .r / / 2M.kP .0/ k C kP .1/k C C kP .r
Since P .0/ D P , P .r / D P 0 , and kP .k/ k kP .k
1/
k for 1 k r
1/
k/:
(5)
1, (5) impl
Section 1.2 Mathematical Induction
11
(A) N is nonempty.
(B) Associated with each natural number n there is a unique natural number n0 called
the successor of n.
(C) There is a natural number n that is not the successor of any natural number.
(D) Distinct
54 Chapter 2 Differential Calculus of Functions of One Variable
The definitions of
f .x0 / D lim f .x/;
x!x0
f .x0 C/ D lim f .x/;
x!x0 C
and
lim f .x/
x!x0
do not involve f .x0 / or even require that it be defined. However, the case where f .x0 / is
defi
Section 4.3 Infinite Series of Constants
is an infinite series, and an is the nth term of the series. We say that
the sum A, and write
1
X
an D A;
P1
nDk
201
an converges to
nDk
if the sequence
fAn g1
k
defined by
An D ak C akC1 C C an ;
n k;
P
1
converge
CHAPTER 5
Real-Valued Functions
of Several Variables
IN THIS CHAPTER we consider real-valued function of n variables, where n > 1.
SECTION 5.1 deals with the structure of Rn , the space of ordered n-tuples of real numbers,
which we call vectors. We define
20 Chapter 1 The Real Numbers
(d) S strictly contains T if S contains T but T does not contain S ; that is, if every
member of T is also in S , but at least one member of S is not in T (Figure 1.3.1).
(e) The complement of S , denoted by S c , is the set
Section 2.4 LHospitals Rule
89
or
lim f .x/ D 1
and suppose that
lim g.x/ D 1;
and
x!b
x!b
f 0 .x/
DL
g0 .x/
lim
x!b
Then
lim
x!b
.finite or 1/:
f .x/
D L:
g.x/
(2)
(3)
(4)
Proof We prove the theorem for finite L and leave the case where L D 1 to you
(Exe
Section 2.1 Functions and Limits
31
attains or assumes the value y0 . The set of values attained by f is the range of f . A realvalued function of a real variable is a function whose domain and range are both subsets
of the reals. Although we are concerne
302 Chapter 5 Real-Valued Functions of Several Variables
27.
Let D1 and D2 be compact subsets of Rn . Show that
D D .X; Y/ X 2 D1 ; Y 2 D2
is a compact subset of R2n .
28.
Prove: If S is open and S D A [ B where A \ B D A \ B D ;, then A and B are
open.
Section 2.5 Taylors Theorem
99
where
lim E.x/ D 0:
x!x0
To generalize this result, we first restate it: the polynomial
T1 .x/ D f .x0 / C f 0 .x0 /.x
x0 /;
which is of degree 1 and satisfies
T10 .x0 / D f 0 .x0 /;
T1 .x0 / D f .x0 /;
approximates f so wel
4
Solution Sets for Systems of Linear Equa-
tions
For a system of equations with 7 equations and k: unknowns, one can have a
number of different outcomes. For example, consider the case of 7" equations
in three variables. Each of these equations is the eq
2 Gaussian Elimination
2.1 Notation for Linear Systems
In Lecture 1 we studied the linear system
:r + y = 27
2x y = 0
and found that
:1: = 9
y = 18
We learned to write the linear system using a matrix and two vectors like so:
<3 2) (:> = (2:)
Likewise, we
3 Elementary Row Operations
Our goal is to begin with an arbitrary matrix and apply operations that
respect row equivalence until we have a matrix in Reduced Row Echelon
Form (RREF). The three elementary row operations are:
0 (Row Swap) Exchange any two r
The University of Sydney
MATH1002 Linear Algebra
Semester 1
Exercises for Week 11 (beginning 19 May)
2014
Questions labelled with an asterisk are suitable for students aiming for a credit or higher.
Important Ideas and Useful Facts:
(i) The determinant of
The University of Sydney
MATH1002 Linear Algebra
Semester 1
Exercises for Week 9 (beginning 5 May)
2014
Preparatory exercises should be attempted before coming to the tutorial. Questions labelled with
an asterisk are suitable for students aiming for a cre
The University of Sydney
MATH1002 Linear Algebra
Semester 1
Exercises for Week 12 (beginning 26 May)
2014
Preparatory exercises should be attempted before coming to the tutorial. Questions labelled with
an asterisk are suitable for students aiming for a c
The University of Sydney
MATH1002 Linear Algebra
Semester 1
Exercises for Week 7 (beginning 14 April)
2014
Preparatory exercises should be attempted before coming to the tutorial. Questions labelled with
an asterisk are suitable for students aiming for a
The University of Sydney
MATH1002 Linear Algebra
Semester 1
Exercises for Week 13 (beginning 2 June)
2014
Preparatory exercises should be attempted before coming to the tutorial. Questions labelled with
an asterisk are suitable for students aiming for a c
The University of Sydney
MATH1002 Linear Algebra
Semester 1
Exercises for Week 8 (beginning 28 April)
2014
Preparatory exercises should be attempted before coming to the tutorial. Questions labelled with
an asterisk are suitable for students aiming for a
Solutions to MATH1002 Assignment 2
Semester 1, 2017
1. (a) To find a Cartesian equation for P 0 we may first find a vector equation for P 0 , then turn
the vector equation into a Cartesian equation. A vector equation for P 0 has the form
(r r0 ) n = 0,
wh
THE UNIVERSITY OF SYDNEY
Semester 1, 2014
Information Sheet for MATH1002 Linear Algebra
Websites:
It is important that you check both the MATH1002 website and the Junior Mathematics
website regularly. Both sites may be accessed through Blackboard, or dire
Writing proofs
Tim Hsu, San Jose State University
Revised January 2014
Contents
I
Fundamentals
5
1 Definitions and theorems
5
2 What is a proof ?
5
3 A word about definitions
6
II
The structure of proofs
8
4 Assumptions and conclusions
8
5 The if-then met
7 Linear Transformations
Recall that the key properties of vector spaces are vector addition and scalar
multiplication. Now suppose we have two vector spaces V and W and a map
L between them:
L: V > W
Now, both V and W have notions of vector addition and
6 Vector Spaces
Thus far we have thought of vectors as lists of numbers in R". As it turns
out, the notion of a vector applies to a much more general class of structures
than this. The main idea is to dene vectors based on their most important
properties.
Faculty of Science
School of Mathematics and Statistics
MATH1002: Linear Algebra
Semester 1 , 2016 | 3 Credit Points | Coordinator: Dr Sharon Stephen
([email protected])
1 Introduction
MATH1002 is designed to provide a thorough preparation for
MATHEMATICS PAST EXAM REVISION
1) Underline important terms and numbers in the
question
2) Have a go at the question
3) If you do not know how to do the question look at
your notebook
4) If it isnt in the notebook highlight/mark the question
and move on
5
484 Chapter 7 Integrals of Functions of Several Variables
19.
Let R D a1 ; b2 a2 ; b2 an ; bn . Evaluate
R
R
(a) R .x1 C x2 C C xn / d X
(b) R .x12 C x22 C C xn2 / d X
(c)
20.
R
R
x1 x2 ; xn d X
Assuming that f is continuous, express
Z
Z p
1 y2
1
dy
21.
2
CHAPTER 7
Integrals of Functions
of Several Variables
IN THIS CHAPTER we study the integral calculus of real-valued functions of several
variables.
SECTION 7.1 defines multiple integrals, first over rectangular parallelepipeds in Rn and
then over more gen