University Mathematics Handbook

University Mathematics Handbook (2015)


I. Basic Concepts

II. Functions

Chapter 1. General Properties of Functions

Chapter 2. Classes of Elementary Functions

Chapter 3. Parametric Form of a Function

Chapter 4. Functions in Polar Coordinate System

III. Analytic Geometry and Vectors

Chapter 1. Coordinate Systems in the Plane

Chapter 2. Curves in the Plane

Chapter 3. Second-Order Curves in Plane - General Theory

Chapter 4. Coordinate Systems in Space, Space Vectors

Chapter 5. Plane in Space

Chapter 6. Straight Line in Space

Chapter 7. Straight Line and Plane

Chapter 8. Canonical Forms of Second-Order Surfaces in Space

Chapter 9. Cylindrical and Spherical Coordinates

IV. Single-Variable Differential Calculus

Chapter 1. Sequence, Limit of a Sequence

Chapter 2. Limits of Functions

Chapter 3. Continuity of Functions

Chapter 4. Derivative

V. Integral Calculus of Single-Variable Functions

Chapter 1. Indefinite Integral

Chapter 2. Definite Integral

Chapter 3. Improper Integral

VI. Series

Chapter 1. Basic Concepts

Chapter 2. Positive Series

Chapter 3. General Series

Chapter 4. Series of Functions

VII. Differential Calculus of Multivariable Functions

Chapter 1. Introduction

Chapter 2. Multivariable Function

Chapter 3. Limits and Continuity of Functions

Chapter 4. Partial Derivatives

Chapter 5. Differentiability. Taylor's Formula

Chapter 6. Chain Rule

Chapter 7. Directional Derivative

Chapter 8. Implicit Function

Chapter 9. Inverse Functions System

Chapter 10. Applications

Chapter 11. Extrema of Multivariable Functions

Chapter 12. Extrema with Constraints

VIII. Integral Calculus of Multivariable Functions

Chapter 1. Parameter-Dependent Integral

Chapter 2. Double Integral

Chapter 3. Double Integral Calculation

Chapter 4. Change of Variables in Double Integrals

Chapter 5. Geometric and Physical Applications

Chapter 6. Triple Integrals

IX. Vector Analysis

Chapter 1. Vector Function, Hodograph

Chapter 2. Limits and Continuity of Vector Functions

Chapter 3. Derivative of Vector Function

Chapter 4. Properties of the Derivatives

Chapter 5. Arc Length

Chapter 6. Vector Functions of Two Variables

Chapter 7. Scalar Field Gradient

Chapter 8. Vector Field

Chapter 9. Line Integrals

Chapter 10. Surface Integral

Chapter 11. Conservative Field in General

X. Algebra

Chapter 1. Complex Numbers

Chapter 2. Fields

Chapter 3. Polynomials

Chapter 4. Vector Spaces

Chapter 5. Matrices

Chapter 6. Determinants

Chapter 7. System of Linear Equations

Chapter 8. General Vector Spaces

Chapter 9. Linear Transformations

Chapter 10. Matrix Similarity

Chapter 11. Eigenvalues and Eigenvectors

Chapter 12. Quadratic Form

Chapter 13. Inner Product Spaces

XI. Ordinary Differential Equations, or ODE

Chapter 1. Classification of First-Order Ordinary Differential Equations

Chapter 2. Linear n-th Order Differential Equations

Chapter 3. Series Solutions of Second-Order ODE

Chapter 4. Systems of First-Order Linear Equations

Chapter 5. Sturm-Liouville Eigenvalue Problem

XII. Complex Functions

Chapter 1. Complex Numbers Sequence (see X.1)

Chapter 2. Complex Functions

Chapter 3. Elementary Functions

Chapter 4. Complex Function Derivative

Chapter 5. Integrals of Complex Functions

Chapter 6. Taylor and Laurent Series

Chapter 7. Isolated Singular Point

Chapter 8. Behavior of Analytical Functions at Infinity

Chapter 9. Residue and its Applications

Chapter 10. Poles and Zeroes of Meromorphic Functions

XIII. Fourier Series and Integral Transforms

Chapter 1. Trigonometric Fourier Series

Chapter 2. Fourier Integral and Fourier Transform

Chapter 3. Laplace Transformation Formulas

XIV. Partial Differential Equations (PDE)

Chapter 1. Introduction

Chapter 2. First-Order Linear PDE

Chapter 3. Quasi-Linear PDE

Chapter 4. Second-Order Linear PDE in Two Variables

Chapter 5. One-Dimensional Wave Equation

Chapter 6. Method of Separation of Variables

Chapter 7. Laplace Equation

XV. Combinatorics and Newton's Binomial

Chapter 1. Permutations

Chapter 2. Variations

Chapter 3. Combination

Chapter 4. Newton's Binomial

XVI. Table of Integrals

Constant c is added to each integral