Working Analysis (Hardcover)

Description:

The text is for a two semester course in advanced calculus. It develops the basic ideas of calculus rigorously but with an eye to showing how mathematics connects with other areas of science and engineering. In particular, effective numerical computation is developed as an important aspect of mathematical analysis.

 

Table of Contents:

Preface

1. Foundations
1.1 Ordered Fields
1.2 Completeness
1.3 Using Inequalities
1.4 Induction
1.5 Sets and Functions

2. Sequences of Real Numbers
2.1 Limits of Sequences
2.2 Criteria for Convergence
2.3 Cauchy Sequences

3. Continuity
3.1 Limits of Functions
3.2 Continuous Functions
3.3 Further Properties of Continuous Functions
3.4 Golden-Section Search
3.5 The Intermediate Value Theorem

4. The Derivative
4.1 The Derivative and Approximation
4.2 The Mean Value Theorem
4.3 The Cauchy Mean Value Theorem and l’Hopital’s Rule
4.4 The Second Derivative Test

5. Higher Derivatives and Polynomial Approximation
5.1 Taylor Polynomials
5.2 Numerical Differentiation
5.3 Polynomial Inerpolation
5.4 Convex Funtions

6. Solving Equations in One Dimension
6.1 Fixed Point Problems
6.2 Computation with Functional Iteration
6.3 Newton’s Method

7. Integration
7.1 The Definition of the Integral
7.2 Properties of the Integral
7.3 The Fundamental Theorem of Calculus and Further Properties of the Integral
7.4 Numerical Methods of Integration
7.5 Improper Integrals

8. Series
8.1 Infinite Series
8.2 Sequences and Series of Functions
8.3 Power Series and Analytic Functions

Appendix I
I.1 The Logarithm Functions and Exponential Functions
I.2 The Trigonometric Funtions

Part II
9. Convergence and Continuity in Rn
9.1 Norms
9.2 A Little Topology
9.3 Continuous Functions of Several Variables

10. The Derivative in Rn
10.1 The Derivative and Approximation in Rn
10.2 Linear Transformations and Matrix Norms
10.3 Vector-Values Mappings

11. Solving Systems of Equations
11.1 Linear Systems
11.2 The Contraction Mapping Theorem
11.3 Newton’s Method
11.4 The Inverse Function Theorem
11.5 The Implicit Function Theorem
11.6 An Application in Mechanics

12. Quadratic Approximation and Optimization
12.1 Higher Derivatives and Quadratic Approximation
12.2 Convex Functions
12.3 Potentials and Dynamical Systems
12.4 The Method of Steepest Descent
12.5 Conjugate Gradient Methods
12.6 Some Optimization Problems

13. Constrained Optimization
13.1 Lagrange Multipliers
13.2 Dependence on Parameters and Second-order Conditions
13.3 Constrained Optimization with Inequalities
13.4 Applications in Economics

14. Integration in Rn
14.1 Integration Over Generalized Rectangles
14.2 Integration Over Jordan Domains
14.3 Numerical Methods
14.4 Change of Variable in Multiple Integrals
14.5 Applications of the Change of Variable Theorem
14.6 Improper Integrals in Several Variables
14.7 Applications in Probability

15. Applications of Integration to Differential Equations
15.1 Interchanging Limits and Integrals
15.2 Approximation by Smooth Functions
15.3 Diffusion
15.4 Fluid Flow

Appendix II
A Matrix Factorization

Solutions to Selected Exercises

References

Index

描述：
這本書是一門為期兩個學期的高級微積分課程教材。它以嚴謹的方式發展微積分的基本概念，同時關注數學如何與其他科學和工程領域相關聯。特別是，它將有效的數值計算作為數學分析的重要方面進行了深入探討。

目錄：
前言
1. 基礎
1.1 有序域
1.2 完備性
1.3 不等式的應用
1.4 數學歸納法
1.5 集合和函數

2. 實數數列
2.1 數列的極限
2.2 收斂的條件
2.3 柯西數列

3. 連續性
3.1 函數的極限
3.2 連續函數
3.3 連續函數的進一步性質
3.4 黃金分割搜索
3.5 中值定理

4. 導數
4.1 導數和近似
4.2 平均值定理
4.3 柯西中值定理和洛必達法則
4.4 二階導數測試

5. 高階導數和多項式逼近
5.1 泰勒多項式
5.2 數值微分
5.3 多項式插值
5.4 凸函數

6. 一維方程的求解
6.1 不動點問題
6.2 函數迭代計算
6.3 牛頓法

7. 積分
7.1 積分的定義
7.2 積分的性質
7.3 微積分基本定理和積分的進一步性質
7.4 數值積分方法
7.5 不定積分

8. 級數
8.1 無窮級數
8.2 函數序列和級數
8.3 冪級數和解析函數

附錄 I
I.1 對數函數和指數函數
I.2 三角函數

第二部分
9. Rn 中的收斂和連續性
9.1 范數
9.2 一點點拓撲學
9.3 多變量連續函數

10. Rn 中的導數
10.1 Rn 中的導數和近似
10.2 線性變換和矩陣范數
10.3 向量值映射

11. 方程組的求解
11.1 線性方程組
11.2 收縮映射定理
11.3 牛頓法
11.4 逆函數定理
11.5 隱函數定理
11.6 在力學中的應用

12. 二次逼近和最佳化
12.1 高階導數和二次逼近
12.2 凸函數
12.3 势能和動力系統
12.4 最速下降法
12.5 共軛梯度法
12.6 一些最佳化問題

13. 有約束的最佳化
13.1 拉格朗日乘數法
13.2 參數的依賴性和二階條件
13.3 帶有不等式約束的最佳化
13.4 在經濟學中的應用

14. Rn 中的積分
14.1 對廣義矩形的積分
14.2 對喬登區域的積分
14.3 數值方法
14.4 多重積分的變量變換
14.5 變量變換定理的應用
14.6 多變量的不定積分
14.7 在概率中的應用

15. 積分在微分方程中的應用
15.1 交換極限和積分
15.2 平滑函數的逼近
15.3 扩散
15.4 流體流動

附錄 II
A 矩陣分解

解答