Mathematical Analysis
Author: Zhiming Ou
Publisher: 3265 Public Way
ISBN: 978-0-9866371-0-0
Summary
Much of our understanding of the world depends on describing how things change. Algebra and geometry are useful tools for describing relationships among static quantities, but they do not include concepts appropriate for describing change. Calculus is the study of change in a function, when the independent variable changes, how does the dependent variable change. The basic concept is rate of change, and the basic tool is limit.
Mathematical analysis provides the deepest understanding for the concept of limits. It deals with the existence and uniqueness of limits; the necessary and condition for differentiability and Riemann integrability; the conditions for interchanging the order of different operations.
Remind that, there are many other types of integrals, and there are types of limits. In topology, we have the net limit, using the concept of neighborhood. Once we unite all concepts of limits, it is the grand unification of mathematics.
Content
Stage 1 Limits of Sequences
1 What is the limit of a Sequence? 5
2 Existence of limits: the 8 axioms 10
3 Cluster Points 13
4 Stolz Theorem 15
5 Convergence of Infinite Series 16
Stage 2 Limits of Functions
6 Different Situations for the limit of a function 23
7 Limit Laws and Theorems 28
8 Two Fundamental Limits 30
9 Continuity of a Function 33
10 Skills for Calculating Limits of Functions 39
Stage 3 Derivatives
11 What is a Derivative? 41
12 How to Calculate Derivatives? 48
13 Differentiating 3 Specific Types of Functions 52
14 Higher Order Derivatives 58
15 The Differential Notation 61
Stage 4 Mean-Value Theorems
16 Fermat’s Theorem 63
17 Rolle’s Theorem 64
18 Lagrange’s Mean Value Theorem 65
19 Cauchy’s Mean Value Theorem 68
20 Taylor’s Formula 70
Stage 5 Applying Derivatives
21 Related Rates 72
22 Optimization 74
23 Concavity and Curvature 78
24 Roots of Functions 86
Stage 6 Anti-derivatives
25 Notations 89
26 Techniques for Calculating Indefinite 96
Stage 7 Definite Integrals
27 The Idea of Partition 112
28 The Fundamental Theorem of Calculus 116
29 Formulas and Properties of Riemann Integrals 118
30 Mean Value Theorems for Definite Integrals 123
Stage 8 Improper Integrals
31 Unbounded Integrands 125
32 Infinite Limits 127
Stage 9 Applications of Riemann Integrals
33 Geometric Measurements 128
34 Applications in Physics 147
Stage 10 Representations for Non-elementary Functions
35 Functional Infinite Series 153
36 Infinite Products 176