数学分析解题法
Author: Zhiming Ou
Publisher: 3265 Public Way
ISBN: 978-1-0677109-8-9
Summary
Mathematical analysis studies the analytic properties of real-valued functions with one or more variables. These properties include convergence, continuity, differentiability, and Riemann integrability.
The format of a function is crucial. The so-called elementary functions involve only a finite number of operations (+, ‒, ×, ÷, and composition) on the 6 fundamental functions: (1) constant, (2) power, (3) exponential, (4) logarithmic, (5) trigonometric, (6) inverse of trigs. However, in real situations, the expression for a quantity usually do not have a closed form, especially the wavefunctions, the field equations.
If we add the factorial function x! into the fundamental list, many infinite series can be expressed in terms of the gamma function, or (x – 1)! by using integrals of the elementary fundamental functions, so that the conversion between discrete sums and continuous integrals can be carried out freely and easily.
Since one major purpose of mathematical analysis is to solve equations, the conversion between sums/integrals and products is still very hard; even functional analysis could not help; we need more mathematical tools. Unfortunately, we can only solve polynomial equations up to degree 4, a radical formula for degree 5 or more does not exist, any solution can only be given approximately.
In this book, I summarized all known types of problems in mathematical analysis and the methods to solve them. There are infinitely many problems. New types are emerging constantly. A teacher can change a number or the wording to make a new problem. Only mastering the methods and skills, one can solve all problems made all teachers.
Content
Chapter 1 Limits of sequences 4
List of formulas/rules
Typical types of Problems
/Methods to solve]
Chapter 2 Limits of Functions 11
List of formulas/rules
Typical types of Problems /Methods to solve]
Chapter 3 Derivatives for Single variable 15
List of formulas/rules
Typical types of Problems
--calculating derivatives
--limits
--proving inequalities
--extreme values 25
--Locating real zeros
--existent proofs
/Methods to solve]
Chapter 4 Vector Calculus 33
List of formulas/rules
Partial derivatives and exact differentials
Typical types of Problems
--tangent planes
--Lagrangian multipliers
/Methods to solve]
Chapter 5 Definite, Indefinite and Improper Integrals 37
List of formulas/rules
Typical types of Problems
--Integrate various types of integrands
--limit of integrals
-geometric applications
--physical applications
/Methods to solve]
Chapter 6 Line, Surface and Multiple Integrals 45
List of formulas/rules
Typical types of Problems /Methods to solve]
Chapter 7 Functional Equations 47
List of formulas/rules
Typical types of Problems
--explicit solutions, include functional equations
--word problems
/Methods to solve]
Chapter 8 Summation 64
List of formulas/rules
Typical types of Problems
--Convergence test
--Taylor and Fourier expansions
--Integral form, closed form
--Generating functions
/Methods to solve]
Chapter 9 Factoring 82
List of formulas/rules
Typical types of Problems /Methods to solve]
Chapter 10 Applications to Probability and Statistics 83
List of formulas/rules
Typical types of Problems /Methods to solve]
Chapter 11 Applications in Geometry and Physics
List of formulas/rules
Typical types of Problems /Methods to solve]