Advanced Mathematics

Author: Yan Zhanghang Xu Quejun Guo Jianping
Publisher:
Publishing Date: 2005-05-01
Features: The book is divided into four parts: Calculus of Univariate Functions, Fundamentals of Multivariate Function Calculus, Fundamentals of Ordinary Differential Equations, and Fundamentals of Infinite Series, totaling nine chapters. Its content covers the essential mathematical knowledge required by various professional majors in higher vocational and technical colleges, as well as methods for applying this knowledge to solve practical problems. Additionally, the book includes mathematical experiments that utilize mathematical software to solve practical calculations, making it available for optional study in institutions with the necessary resources. This textbook breaks from traditional teaching systems by selecting key content, emphasizing the main points, eliminating unnecessary details, focusing on practical use, and prioritizing effectiveness. It allows for flexible selection of content based on different professional majors and student categories, offering broad applicability. The selected examples and exercises are designed to help students understand concepts and master methods, while removing purely technical and highly challenging problems and adding thought-provoking, applied, and professionally relevant questions. This textbook is part of a comprehensive teaching package, accompanied by a supplementary textbook titled "Guidance for Exercises in Advanced Mathematics and Engineering Mathematics," which includes chapter summaries, categorized problem-solving methods, exercise answers, and solutions to typical problems, as well as self-assessment tests. Additionally, a digital teaching plan is available for free distribution to teachers, and a dedicated website is established to provide online services for faculty and students. This book can serve as the mathematics textbook for engineering and economics-related majors in three-year or two-year higher vocational and technical colleges, adult colleges, and secondary institutions within undergraduate universities, while also offering high reference value for professionals in various fields.
Table of Contents
Part I: Calculus of Univariate Functions
Chapter 1: Functions, Limits, and Continuity
Section 1: Functions
1. Concept of Functions
2. Properties of Functions
3. Composite Functions
4. Inverse Functions
5. Elementary Functions
6. Examples of Establishing Functional Relationships
7. Examples of Economic Functions
Section 2: Sequences and Their Limits
1. Limits of Sequences
2. Arithmetic Operations of Sequence Limits
3. Sum Formula for Infinite Geometric Series
4. Properties of Sequence Limits
Section 3: Limits of Functions
1. Limits of Functions as x → ∞
2. Limits of Functions as x → x?
3. Left and Right Limits
4. Properties of Function Limits
Section 4: Infinitesimals and Infinites
1. Definitions and Relationships of Infinitesimals and Infinites
2. Properties of Infinitesimals
Section 5: Limit Operations
Section 6: Two Important Limits
1. Limit lim(x→0) (sin x / x) = 1
2. Limit lim(x→∞) (1 / x^x) = e
Section 7: Comparison of Infinitesimals
Section 8: Continuity and Discontinuity of Functions
1. Concept of Continuity
2. Discontinuities of Functions
Section 9: Continuity of Elementary Functions
1. Continuity of Elementary Functions
2. Properties of Continuous Functions on Closed Intervals
Section 10: Mathematical Experiment 1: Introduction to Mathematica and Finding Limits of Univariate Functions
1. Introduction to Mathematica
2. Graphing Univariate Functions
3. Finding Limits of Univariate Functions
Chapter 2: Derivatives and Differentials
Section 1: Concept of Derivatives
1. Examples of Rate of Change Problems
2. Definition of Derivatives
3. Examples of Derivatives
4. Geometric Meaning of Derivatives
5. Relationship Between Differentiability and Continuity
Section 2: Derivatives of Sums, Differences, Products, and Quotients
Section 3: Derivatives of Composite Functions
Section 4: Derivatives of Elementary Functions
1. Derivatives of Inverse Functions
2. Derivatives of Elementary Functions
3. Derivatives of Piecewise Functions
Section 5: Derivatives of Implicit Functions and Parametric Equations
1. Derivatives of Implicit Functions
2. Derivatives of Power-Exponential Functions
3. Derivatives of Functions Defined by Parametric Equations
Section 6: Higher-Order Derivatives
Section 7: Differentials of Functions
1. Concept of Differentials
2. Operations of Differentials
3. Approximate Calculations
Section 8: Mathematical Experiment 2: Using Mathematica to Find Derivatives of Univariate Functions
1. Learning Mathematica Commands
2. Concept of Derivatives
3. Finding Derivatives of Univariate Functions
Chapter 3: Applications of Derivatives
Section 1: Lagrange Mean Value Theorem and Monotonicity Determination
1. Lagrange Mean Value Theorem
2. Monotonicity Determination of Functions
Section 2: Extrema of Functions and Their Determination
Section 3: Maximum and Minimum Values of Functions
Section 4: Convexity, Concavity, and Inflection Points of Curves
Section 5: Graphing Functions
Section 6: L'H?pital's Rule
Section 7: Applications of Derivatives in Economic Problems
1. Marginal Analysis
2. Elasticity Analysis
Chapter 4: Integral Calculus of Univariate Functions
Section 1: Concept and Properties of Indefinite Integrals
1. Antiderivatives
2. Indefinite Integrals
3. Geometric Meaning of Indefinite Integrals
4. Basic Integration Formulas
5. Basic Rules of Integration
Section 2: Methods of Indefinite Integration
1. First-Type Substitution Integration
2. Second-Type Substitution Integration
3. Integration by Parts
Section 3: Concept and Properties of Definite Integrals
1. Two Examples
2. Definition of Definite Integrals
3. Geometric Meaning of Definite Integrals
4. Properties of Definite Integrals
Section 4: Newton-Leibniz Formula
1. Integral Upper Limit Function
2. Newton-Leibniz Formula
Section 5: Substitution and Integration by Parts Methods for Definite Integrals
1. Substitution Method for Definite Integrals
2. Integration by Parts Method for Definite Integrals
Section 6: Improper Integrals
Section 7: Mathematical Experiment 3: Using Mathematica to Compute Integrals
1. Learning Mathematica Commands
2. Finding Indefinite Integrals
3. Finding Definite Integrals and Improper Integrals
Chapter 5: Applications of Definite Integrals
Section 1: Microelement Method of Definite Integrals
Section 2: Applications of Definite Integrals in Geometry
1. Area of Plane Figures
2. Volume of Revolution
3. Length of Plane Curves
Section 3: Applications of Definite Integrals in Physics
1. Work Done by Variable Force
2. Pressure of Liquids
3. Gravity
Section 4: Simple Applications of Definite Integrals in Economic Problems
1. Finding Total Functions from Marginal Functions
2. Present Value of Capital and Investment Problems
Part II: Fundamentals of Multivariate Function Calculus
Chapter 6: Fundamentals of Multivariate Function Differential Calculus
Section 1: Introduction to Spatial Analytic Geometry
1. Spatial Rectangular Coordinate System
2. Surfaces and Their Equations
3. Space Curves and Their Equations
Section 2: Concepts and Operations of Vectors
1. Concept of Vectors
2. Vector Addition and Subtraction
3. Multiplication of Scalars and Vectors
4. Coordinate Representation of Vectors
5. Scalar Product of Vectors
6. Vector Product of Vectors
Section 3: Planes, Lines, and Common Quadratic Surfaces in Space
1. Plane Equations and Angle Between Two Planes
2. Equations of Space Lines and Their Angles
3. Common Quadratic Surfaces and Their Equations
Section 4: Concept of Multivariate Functions
1. Definition of Bivariate Functions
2. Geometric Meaning of Bivariate Functions
3. Limits and Continuity of Bivariate Functions
Section 5: Partial Derivatives and Total Differentials
1. Definition and Calculation of Partial Derivatives
2. Higher-Order Partial Derivatives
3. Total Differentials
Section 6: Differentiation of Composite and Implicit Functions
1. Differentiation Rules for Composite Functions
2. Invariance of Total Differential Form
3. Differentiation Methods for Implicit Functions
Section 7: Extrema of Multivariate Functions
1. Extrema of Multivariate Functions
2. Conditional Extrema
Chapter 7: Fundamentals of Multivariate Function Integral Calculus
Section 1: Concept and Properties of Double Integrals
1. Two Examples
2. Definition of Double Integrals
3. Properties of Double Integrals
Section 2: Calculation of Double Integrals
1. Calculation of Double Integrals in Rectangular Coordinates
2. Calculation of Double Integrals in Polar Coordinates
Section 3: Applications of Double Integrals
1. Volume
2. Mass of Plane Slabs
3. Center of Mass of Plane Slabs
Section 4: Triple Integrals
1. Concept of Triple Integrals
2. Methods of Calculating Triple Integrals
Section 5: Line Integrals
1. Line Integrals of Arc Length
2. Line Integrals of Coordinates
3. Green's Theorem
4. Conditions for Independence of Line Integrals in the Plane
Section 6: Mathematical Experiment 4: Using Mathematica to Find Partial Derivatives and Compute Double Integrals
1. Learning Mathematica Commands
2. Calculation of Partial Derivatives
3. Calculation of Double Integrals
Part III: Fundamentals of Ordinary Differential Equations
Chapter 8: Ordinary Differential Equations
Section 1: Basic Concepts of Ordinary Differential Equations
Section 2: First-Order Differential Equations
1. Separable Differential Equations
2. Homogeneous Differential Equations
3. Differential Equations of the Form dy/dx = f(axby^c)
4. First-Order Linear Differential Equations
5. Bernoulli Equations
Section 3: Special Types of Higher-Order Differential Equations
1. Differential Equations of the Form d^n y/dx^n = f(x)
2. Differential Equations of the Form y'' = f(x, y')
3. Differential Equations of the Form y'' = f(y, y')
Section 4: Second-Order Linear Differential Equations
1. Structure of Solutions
2. Methods of Solving Second-Order Linear Differential Equations with Constant Coefficients
Part IV: Fundamentals of Infinite Series
Chapter 9: Infinite Series
Section 1: Concept and Basic Properties of Series
1. Concept of Series
2. Basic Properties of Series
Section 2: Convergence and Divergence of Series
1. Positive Series and Convergence Tests
2. Convergence and Divergence of Arbitrary Series
Section 3: Power Series
1. Concept of Series with Functions as Terms
2. Power Series and Their Convergence
3. Operations of Power Series
Section 4: Expansion of Functions into Power Series
1. Taylor Series
2. Expanding Functions into Power Series
3. Applications of Power Series Expansions
Section 5: Fourier Series
1. Orthogonality of Trigonometric Series and Trigonometric Function Systems
2. Fourier Series of Functions with Period 2π
3. Sine Series and Cosine Series
Section 6: Expansion of Functions with Period 2l into Fourier Series
1. Fourier Series of Functions with Period 2l
2. Complex Form of Fourier Series
References