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G12MAN Mathematical Analysis
Based on notes by Dr J. Zacharias
Modified by Dr J. F. Feinstein
1
About these notes
These notes include all of the definitions and theorems of
the module.
Most of them are examinable.
The notes omit a number of proofs and explanations of
examples given during the lectures.
There are gaps left where these should be filled in by you
during the lectures. Writing this material yourself is
an important part of studying this module.
Technology permitting, Dr Feinstein’s annotated slides
from the lectures will be made available from the module
web page.
The entire 2009-10 version of G12MAN (including
annotated slides and videos) can be downloaded from the
web. See the G12MAN Module Lecture Notes page
for details.
Downloading this material is not an adequate
substitute for attending classes.
2
Examinable and non-examinable material
Certain material in this module will be clearly marked as
Not examinable as bookwork, or NEB for short.
This does not mean that this material is irrelevant to
exam questions.
Some sections of some of the exam questions are likely to
be ‘unseen’.
It is possible that some of the NEB material could be
relevant to some unseen portions of exam questions.
For more information on the style of exam questions, see
the G12MAN Module Information page and the
G12MAN Module Feedback page.
3
Why do we do all these proofs?
Rigorous arguments and justification are essential
throughout mathematics, and not just in ‘Pure’
Mathematics.
Even in those modules where you do not justify all your
reasoning in full, you may well be told that certain facts
are true but that the proof is beyond the scope of that
module.
Here are some examples to illustrate the need for rigour,
and, in particular, some applications of Mathematical
Analysis.
4
• Dr Matthews tells me that in G12MDE in traffic
flow modelling and fluid dynamics, you need to
‘differentiate under the integral sign’ in a setting
where it would require mathematical analysis in order
to justify this manipulation.
• This interchange between differentiation and
integration also turns up in Professor Wood’s
module G13INF Statistical Inference.
Professor Wood says
‘This interchange is used to prove important
results in statistics such as the Cramer–Rao
lower bound.
In the module I do not provide rigorous
justification of this interchange though I say a
bit about when it can break down.’
Gap to fill in
5
Additionally:
• Dr Wilkinson tells me that advanced mathematical
analysis (measure theory) is needed
‘. . . in order to do real world things (such as date
the primate origins and the human-chimp split).’
• Dr Louko used several theorems from mathematical
analysis in his latest Mathematical Physics paper.
• Professor Fyodorov tells me that if it wasn’t for the
fact that we sometimes have
limX→0
limY→∞
f(X,Y ) 6= limY→∞
limX→0
f(X,Y )
then there would be no permanent magnets!
6
Here is an interesting example to think about that
illustrates some of the dangers.
For which constants α ≥ 0 is it true that
limn→∞
∫ 1
0
nαxn(1− x)dx
exists, and what value (possibly depending on α) does
the limit take in these cases?
Gap to fill in
7
1 Introduction to Rd
In this chapter we look at subsets of
Rd = {(x1, x2, . . . , xd) | x1, x2, . . . , xd ∈ R} and
establish some of the basic notation and terminology that
we will need in the rest of the module.
You have met R1 = R, R2, R3 and to some extent Rd
already, for instance when solving larger systems of linear
equations.
8
1.1 Set notation
We use the standard set theoretical notations from
G11ACF. In particular we denote the set of
• positive integers by N = {1, 2, 3, 4, . . . };
• integers by Z = {. . . ,−2,−1, 0, 1, 2, . . . };
• rational numbers (fractions) by
Q = {ab| a ∈ Z, b ∈ N};
• real numbers by R;
• non-negative real numbers by
R+ = {x ∈ R | x ≥ 0} = [0,∞).
9
Subsets are often specified by a certain condition shared
by all its elements.
For instance certain subsets of R can be written as
{x ∈ R | x2 < x}
or
{x ∈ R | x2 + 1 = 0}.
The former is another way to write
{x ∈ R | 0 < x < 1}.
The latter is another way to write the empty set ∅.
Many different looking conditions or properties define the
same set.
10
A set A is a subset of a set B, written A ⊆ B, if each
element in A lies also in B, i.e. x ∈ A implies x ∈ B (we
can write: x ∈ A⇒ x ∈ B).
Thus A = B if and only if A ⊆ B and B ⊆ A.
For sets C and D their intersection is
C ∩D = D ∩ C = {x | x ∈ C and x ∈ D}
= {x | x ∈ D, x ∈ C}
and their union is
C ∪D = D ∪ C = {x | x ∈ D or x ∈ C}.
11
Similarly if D1, D2, . . . , Dn are sets then
D1 ∩D2 ∩ · · · ∩Dn =n⋂i=1
Di
= {x | x ∈ D1, x ∈ D2, . . . , x ∈ Dn}
and
D1 ∪D2 ∪ · · · ∪Dn =n⋃i=1
Di
= {x | x ∈ D1 or x ∈ D2 or . . . or x ∈ Dn}
Intersections and unions can also be defined for infinitely
many Di’s, see later.
12
The difference between C and D is
C\D = {x ∈ C | x /∈ D}.
where x /∈ D means that x is not an element of D.
Note that this is often different from
D\C = {x ∈ D | x /∈ C} .
If A is a subset of a larger set B then its complement (in
B) is Ac = B\A.
Usually it is clear from the context what B is.
Examples: Complement of a subset of R.
Gap to fill in
13
1.2 Cartesian products
The Cartesian product of two sets A and B is the set
of all ordered pairs of elements in A and B, i.e.
A×B = {(a, b) | a ∈ A, b ∈ B}.
(a1, b1) = (a2, b2) if and only if a1 = a2 and b1 = b2.
Example 1.2.1 R× R is the set of ordered pairs of real
numbers.
Each (x, y) ∈ R× R can be visualised as a point in a
coordinate system.
Gap to fill in
14
The Cartesian product of two subsets A,B ⊆ R can be
visualised as well, e.g. if A = [1, 2] and B = [2, 4], then
A×B is represented by
Gap to fill in
15
We can produce a variety of subsets of R2 by taking
products of subsets of R, but there are many others such
as S = {(x, y) | 0 ≤ x ≤ 1, 0 ≤ y ≤ x} or
R = {(x, y) | x2 + y2 = 1} which can not be written as
Cartesian products.
(See question sheets for details.)
Gap to fill in
16
R× R is also denoted by R2.
Similarly A×B × C is the set of ordered triples
A×B × C = {(a, b, c) | a ∈ A, b ∈ B, c ∈ C}
etc.
We denote R× R× R by R3 and more generally
Rd = R× R× · · · × R︸ ︷︷ ︸d
= {(x1, x2, . . . , xd) | x1, x2, . . . , xd real numbers} ,
the set of all d-tuples of real numbers.
Here d is a positive integer. R1 is identified with R.
17
1.3 Revision of Rd
Elements of Rd may be regarded as points which are
positioned in a certain way or as vectors which we can
add or multiply.
The former is Rd as an analytic or geometrical object,
the latter is Rd as a vector space: an algebraic object.
We will not worry about the distinction in this module.
Thus elements in Rd are denoted by (for example)
x,y, z,p, q,
etc. where x = (x1, x2, . . . , xd), y = (y1, y2, . . . , yd) etc.
18
Sometimes we will write them as d-dimensional column
vectors.
x =
x1
x2...
xd
, y =
y1
y2...
yd
, z =
z1
z2...
zd
etc.
Column vectors may be added
x+ y =
x1 + y1
x2 + y2...
xd + yd
Column vectors may be multiplied by scalars λ ∈ R
λx = λ
x1
x2...
xd
=
λx1
λx2...
λxd
19
Since we identify (x1, x2, . . . , xd) and
x1...
xd
we may
well also write
(x1, x2, . . . , xd) + (y1, y2, . . . , yd)
= (x1 + y1, x2 + y2, . . . , xd + yd)
etc.
20
Recall the standard inner product on Rd:
〈x,y〉 =
⟨x1...
xd
,y1...
yd
⟩
=
d∑i=1
xiyi.
The norm corresponding to it is the Euclidean norm
defined by
‖x‖ =√〈x,x〉 =
(d∑i=1
|xi|2) 1
2
, for x =
x1...
xd
and may be regarded as the length of a vector.
Recall the Cauchy–Schwarz inequality
|〈x,y〉| ≤ ‖x‖ ‖y‖
which holds for all x,y ∈ Rd.
21
Proposition 1.3.1 The function || · || : Rd → R+ has the
following properties, for all x and y in Rd and all λ ∈ R:
(i) ‖x+ y‖ ≤ ‖x‖+ ‖y‖ (triangle inequality);
(ii) ‖λx‖ = |λ|‖x‖ (homogeneity);
(iii) ‖x‖ = 0⇐⇒ x = 0.
The statement and applications of this result are
examinable as bookwork.
The proof of this result is NEB (not examinable as
bookwork), but is available on request.
Gap to fill in
22
The three properties above are all you usually need to
know about the Euclidean norm ‖x‖ in this module.
Using our Euclidean norm ‖ · ‖, we define the Euclidean
distance between points x and y by
d(x,y) = ‖x− y‖ = ‖y − x‖ = d(y,x).
If z is another point in Rd we have
d(x,y) ≤ d(x, z) + d(z,y)
(triangle inequality for the Euclidean distance), since
Gap to fill in
23
Throughout this module, the Euclidean distance
d(x,y) = ‖x− y‖
=√
(x1 − y1)2 + · · ·+ (xd − yd)2
will play an important role.
In this module, unless otherwise specified, when we
refer to distance in Rd we will always mean the
Euclidean distance.
24