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Fundamentals of Advanced Mathematics(I)

样章
作者：
马知恩　王绵森　西安大学数学系
定价：
32.00 元
版面字数：
470千字
开本：
16开
装帧形式：
平装
版次：
1
最新版次
印刷时间：
2026年
ISBN：
978-7-04-015484-9
物料号：
15484-00
出版时间：
2005-01-14
读者对象：
高等教育
一级分类：
数学与统计学类
二级分类：
理工类专业数学基础课
三级分类：
高等数学
暂无
  • 目录
    • 前辅文
      • Introduction
        • Chapter 1 Theoretical Basis of Calculus
          • 1.1 Sets and Functions
            • 1.1.1 Sets and their operations
              • 1.1.2 Concepts of mappings and functions
                • 1.1.3 Composition of mappings and composition of functions
                  • 1.1.4 Inverse mappings and inverse functions
                    • 1.1.5 Elementary functions and hyperbolic functions
                      • 1.1.6 Some examples for modelling of functions in practical problems
                      • Exercises 1.1
                        • 1.2 Limit of Sequence
                          • 1.2.1 Concept of limit of a sequence
                            • 1.2.2 Conditions for convergence of a sequence
                              • 1.2.3 Rules of operations on convergent sequences
                              • Exercises 1.2
                                • 1.3 Limit of Function
                                  • 1.3.1 The concept of limit of a function
                                    • 1.3.2 The properties and operation rules of functional limits
                                      • 1.3.3 Two important limits
                                      • Exercises 1.3
                                        • 1.4 Infinitesimal and Infinite Quantities
                                          • 1.4.1 Infinitesimal quantities and their order
                                            • 1.4.2 Equivalence transformations of infinitesimals
                                              • 1.4.3 Infinite quantities
                                              • Exercises 1.4
                                                • 1.5 Continuous Functions
                                                  • 1.5.1 The concept of continuous function and classification of discontinuous points
                                                    • 1.5.2 Operations on continuous functions and the continuity of elementary functions
                                                      • 1.5.3 Properties of continuous functions on a closed interval
                                                      • Exercises 1.5
                                                      • Chapter 2 The Differential Calculus and Its Applications
                                                        • 2.1 Concept of Derivatives
                                                          • 2.1.1 Definition of derivatives
                                                            • 2.1.2 Relationship between derivability and continuity
                                                              • 2.1.3 Some examples of derivative problems in science and technology
                                                              • Exercises 2.1
                                                                • 2.2 Fundamental Derivation Rules
                                                                  • 2.2.1 Derivation rules for sum, difference, product and quotient of functions
                                                                    • 2.2.2 Derivation rule for composite functions
                                                                      • 2.2.3 The derivative of an inverse function
                                                                        • 2.2.4 Higherorder derivatives
                                                                        • Exercises 2.2
                                                                          • 2.3 Derivation of Implicit Functions and Functions Defined by Parametric Equations
                                                                            • 2.3.1 Method of derivation of implicit functions
                                                                              • 2.3.2 Method of derivation of a function defined by parametric equations
                                                                                • 2.3.3 Related rates of change
                                                                                • Exercises 2.3
                                                                                  • 2.4 The Differential
                                                                                    • 2.4.1 Concept of the differential
                                                                                      • 2.4.2 Geometric meaning of the differential
                                                                                        • 2.4.3 Rules of operations on differentials
                                                                                          • 2.4.4 Application of the differential in approximate computation
                                                                                          • Exercises 2.4
                                                                                            • 2.5 The Mean Value Theorem in Differential Calculus and L’Hospital‘s Rules
                                                                                              • 2.5.1 Mean value theorems in differential calculus
                                                                                                • 2.5.2 L'Hospital's rules
                                                                                                • Exercises 2.5
                                                                                                  • 2.6 Taylor's Theorem and Its Applications
                                                                                                    • 2.6.1 Taylor's theorem
                                                                                                      • 2.6.2 Maclaurin formulae for some elementary functions
                                                                                                        • 2.6.3 Some applications of Taylor‘s theorem
                                                                                                        • Exercises 2.6
                                                                                                          • 2.7 Study of Properties of Functions
                                                                                                            • 2.7.1 Monotonicity of functions
                                                                                                              • 2.7.2 Extreme values of functions
                                                                                                                • 2.7.3 Global maxima and minima
                                                                                                                  • 2.7.4 Convexity of functions
                                                                                                                  • Exercises 2.7
                                                                                                                    • Synthetic exercises
                                                                                                                    • Chapter 3 The Integral Calculus and Its Applications
                                                                                                                      • 3.1 Concept and Properties of Definite Integrals
                                                                                                                        • 3.1.1 Examples of definite integral problems
                                                                                                                          • 3.1.2 The definition of definite integral
                                                                                                                            • 3.1.3 Properties of definite integrals
                                                                                                                            • Exercises 3.1
                                                                                                                              • 3.2 The Newton Leibniz Formula and the Fundamental Theorems of Calculus
                                                                                                                                • 3.2.1 Newton Leibniz formula
                                                                                                                                  • 3.2.2 Fundamental theorems of Calculus
                                                                                                                                  • Exercises 3.2
                                                                                                                                    • 3.3 Indefinite Integrals and Integration
                                                                                                                                      • 3.3.1 Indefinite integrals
                                                                                                                                        • 3.3.2 Integration by substitutions
                                                                                                                                          • 3.3.3 Integration by parts
                                                                                                                                            • 3.3.4 Quadrature problems for elementary fundamental functions
                                                                                                                                            • Exercises 3.3
                                                                                                                                              • 3.4 Applications of Definite Integrals
                                                                                                                                                • 3.4.1 Method of elements for setting up integral representations
                                                                                                                                                  • 3.4.2 Some examples on the applications of the definite integral in geometry
                                                                                                                                                    • 3.4.3 Some examples of applications of the definite integral in physics
                                                                                                                                                    • Exercises 3.4
                                                                                                                                                      • 3.5 Some Types of Simple Differential Equations
                                                                                                                                                        • 3.5.1 Some fundamental concepts
                                                                                                                                                          • 3.5.2 First order differential equations with variables separable
                                                                                                                                                            • 3.5.3 Linear equations of first order
                                                                                                                                                              • 3.5.4 Equations of first order solvable by transformations of variables
                                                                                                                                                                • 3.5.5 Differential equations of second order solvable by reduced order methods
                                                                                                                                                                  • 3.5.6 Some examples of application of differential equations
                                                                                                                                                                  • Exercises 3.5
                                                                                                                                                                    • 3.6 Improper Integrals
                                                                                                                                                                      • 3.6.1 Integration on an infinite interval
                                                                                                                                                                        • 3.6.2 Integrals of unbounded functions
                                                                                                                                                                        • Exercises 3.6
                                                                                                                                                                        • Chapter 4 Infinite Series
                                                                                                                                                                          • 4.1 Series of Constant Terms
                                                                                                                                                                            • 4.1.1 Concepts and properties of series with constant terms
                                                                                                                                                                              • 4.1.2 Convergence tests for series of positive terms
                                                                                                                                                                                • 4.1.3 Series with variation of signs and tests for convergence
                                                                                                                                                                                • Exercises 4.1
                                                                                                                                                                                  • 4.2 Power Series
                                                                                                                                                                                    • 4.2.1 Concepts of series of functions
                                                                                                                                                                                      • 4.2.2 Convergence of power series and operations on power series
                                                                                                                                                                                        • 4.2.3 Expansion of functions in power series
                                                                                                                                                                                          • 4.2.4 Some examples of applications of power series
                                                                                                                                                                                            • *4.2.5 Uniform convergence of series of functions
                                                                                                                                                                                            • Exercises 4.2
                                                                                                                                                                                              • 4.3 Fourier Series
                                                                                                                                                                                                • 4.3.1 Periodic functions and trigonometric series
                                                                                                                                                                                                  • 4.3.2 Orthogonality of the system of trigonometric functions and Fourier series
                                                                                                                                                                                                    • 4.3.3 Fourier expansions of periodic functions
                                                                                                                                                                                                      • 4.3.4 Fourier expansion of functions defined on the interval
                                                                                                                                                                                                        • 4.3.5 Complex form of Fourier series
                                                                                                                                                                                                        • Exercises 4.3
                                                                                                                                                                                                          • Synthetic exercises
                                                                                                                                                                                                          • Appendix Answers and Hints for Exercises

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