Cinlar E., Vanderbei R. - Mathematical Methods of Engineering Analysis.pdf

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Mathematical Methods of Engineering Analysis
Erhan ¸ inlar
Robert J. Vanderbei
February 2, 2000
Contents
Sets and Functions
1
1
Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
Subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Set Operations . . . . . . . . . . . . . . . . . . . . . . . . .
2
Disjoint Sets . . . . . . . . . . . . . . . . . . . . . . . . . .
3
Products of Sets . . . . . . . . . . . . . . . . . . . . . . . .
3
2
Functions and Sequences . . . . . . . . . . . . . . . . . . . . . . . .
4
Injections, Surjections, Bijections . . . . . . . . . . . . . . .
4
Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
3
Countability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
4
On the Real Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Positive and Negative . . . . . . . . . . . . . . . . . . . . . . 9
Increasing, Decreasing . . . . . . . . . . . . . . . . . . . . . 9
Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Supremum and Infimum . . . . . . . . . . . . . . . . . . . . 9
Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Convergence of Sequences . . . . . . . . . . . . . . . . . . . 11
5
Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Ratio Test, Root Test . . . . . . . . . . . . . . . . . . . . . . 16
Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Absolute Convergence . . . . . . . . . . . . . . . . . . . . . 18
Rearrangements . . . . . . . . . . . . . . . . . . . . . . . . . 19
Metric Spaces
23
6
Euclidean Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Inner Product and Norm . . . . . . . . . . . . . . . . . . . . 23
Euclidean Distance . . . . . . . . . . . . . . . . . . . . . . . 24
7
Metric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Distances from Points to Sets and from Sets to Sets . . . . . . 26
Balls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
8
Open and Closed Sets . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Closed Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
Interior, Closure, and Boundary . . . . . . . . . . . . . . . . 30
i
Open Subsets of the Real Line . . . . . . . . . . . . . . . . .
31
9
Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
Subsequences . . . . . . . . . . . . . . . . . . . . . . . . . .
35
Convergence and Closed Sets . . . . . . . . . . . . . . . . .
36
10 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
Cauchy Sequences . . . . . . . . . . . . . . . . . . . . . . .
37
Complete Metric Spaces . . . . . . . . . . . . . . . . . . . .
38
11 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
Compact Subspaces . . . . . . . . . . . . . . . . . . . . . .
40
Cluster Points, Convergence, Completeness . . . . . . . . . .
41
Compactness in Euclidean Spaces . . . . . . . . . . . . . . .
42
Functions on Metric Spaces
45
12 Continuous Mappings . . . . . . . . . . . . . . . . . . . . . . . . . .
45
Continuity and Open Sets . . . . . . . . . . . . . . . . . . .
46
Continuity and Convergence . . . . . . . . . . . . . . . . . .
46
Compositions . . . . . . . . . . . . . . . . . . . . . . . . . .
47
Real-Valued Functions . . . . . . . . . . . . . . . . . . . . .
48
R n -Valued Functions . . . . . . . . . . . . . . . . . . . . . .
48
13 Compactness and Uniform Continuity . . . . . . . . . . . . . . . . .
50
Uniform Continuity . . . . . . . . . . . . . . . . . . . . . . .
51
14 Sequences of Functions . . . . . . . . . . . . . . . . . . . . . . . . .
53
Cauchy Criterion . . . . . . . . . . . . . . . . . . . . . . . .
54
Continuity of Limit Functions . . . . . . . . . . . . . . . . .
56
15 Spaces of Continuous Functions . . . . . . . . . . . . . . . . . . . .
57
Convergence in C . . . . . . . . . . . . . . . . . . . . . . . .
57
Lipschitz Continuous Functions . . . . . . . . . . . . . . . .
58
Completeness . . . . . . . . . . . . . . . . . . . . . . . . . .
60
Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
Differential and Integral Equations
63
16 Contraction Mappings . . . . . . . . . . . . . . . . . . . . . . . . .
63
Fixed Point Theorem . . . . . . . . . . . . . . . . . . . . . .
64
17 Systems of Linear Equations . . . . . . . . . . . . . . . . . . . . . .
69
Maximum Norm . . . . . . . . . . . . . . . . . . . . . . . .
69
Manhattan Metric . . . . . . . . . . . . . . . . . . . . . . . .
70
Euclidean Metric . . . . . . . . . . . . . . . . . . . . . . . .
70
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
18 Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
Fredholm Equation . . . . . . . . . . . . . . . . . . . . . . .
71
Volterra Equation . . . . . . . . . . . . . . . . . . . . . . . .
76
Generalization of the Fixed Point Theorem . . . . . . . . . .
77
19 Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . .
78
ii
Convex Analysis
83
20 Convex Sets and Convex Functions . . . . . . . . . . . . . . . . . . .
83
21 Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
86
22 Supporting Hyperplane Theorem . . . . . . . . . . . . . . . . . . . .
90
Measure and Integration 91
23 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
24 Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
Monotone Class Theorem . . . . . . . . . . . . . . . . . . . 94
25 Measurable Spaces and Functions . . . . . . . . . . . . . . . . . . . 96
Measurable Functions . . . . . . . . . . . . . . . . . . . . . 96
Borel Functions . . . . . . . . . . . . . . . . . . . . . . . . . 97
Compositions of Functions . . . . . . . . . . . . . . . . . . . 97
Numerical Functions . . . . . . . . . . . . . . . . . . . . . . 97
Positive and Negative Parts of a Function . . . . . . . . . . . 98
Indicators and Simple Functions . . . . . . . . . . . . . . . . 98
Approximations by Simple Functions . . . . . . . . . . . . . 99
Limits of Sequences of Functions . . . . . . . . . . . . . . . 100
Monotone Classes of Functions . . . . . . . . . . . . . . . . 100
Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
26 Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
Arithmetic of Measures . . . . . . . . . . . . . . . . . . . . . 104
Finite, -finite, -finite measures . . . . . . . . . . . . . . . 104
Specification of Measures . . . . . . . . . . . . . . . . . . . 105
Image of Measure . . . . . . . . . . . . . . . . . . . . . . . 106
Almost Everywhere . . . . . . . . . . . . . . . . . . . . . . 106
27 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
Definition of the Integral . . . . . . . . . . . . . . . . . . . . 109
Integral over a Set . . . . . . . . . . . . . . . . . . . . . . . 110
Integrability . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
Elementary Properties . . . . . . . . . . . . . . . . . . . . . 110
Monotone Convergence Theorem . . . . . . . . . . . . . . . 111
Linearity of Integration . . . . . . . . . . . . . . . . . . . . . 113
Fatou’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . 113
Dominated Convergence Theorem . . . . . . . . . . . . . . . 114
iii
Sets and Functions
This introductory chapter is devoted to general notions regarding sets, functions, se-
quences, and series. The aim is to introduce and review the basic notation, terminology,
conventions, and elementary facts.
1 Sets
A set is a collection of some objects. Given a set, the objects that form it are called its
elements . Given a set A , we write x2A to mean that x is an element of A . To say that
x2A , we also use phrases like x is in A , x is a member of A , x belongs to A , and A
includes x .
To specify a set, one can either write down all its elements inside curly brackets (if
this is feasible), or indicate the properties that distinguish its elements. For example,
A={a,b,c} is the set whose elements are a , b , and c , and B={x:x>2.7} is the
set of all numbers exceeding 2.7 . The following are some special sets:
; : The empty set . It has no elements.
N={1,2,3,...} : Set of natural numbers .
Z={0,1,−1,2,−2,...} : Set of integers .
Z + ={0,1,2,...} : Set of positive integers .
Q={ m n :m2Z,n2N} : Set of rationals .
R=(−1,1)={x:−1<x<+1} : Set of reals .
[a,b]={x2R:axb} : Closed intervals .
(a,b)={x2R:a<x<b} : Open intervals .
R + =[0,1)={x2R:x0} : Set of positive reals .
1

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