University Mathematics — Contents Advanced Resources AD 02 · 49 chapters + 2 appendices

University Mathematics— From Calculus to Partial Differential Equations: University Mathematics for Precocious High-School Students —

Based on the nine volumes of Yasuhide Mochizuki's handwritten notes, “Mathematics Notes” (Calculus, Linear Algebra, Fundamentals of Probability, Vector Analysis, Complex Analysis, Fourier Series and Transforms, Laplace Transforms, Ordinary Differential Equations, and Partial Differential Equations), this book is an educational resource that brings together, in a single volume, the whole of the mathematics studied at universities of science and engineering. The definitions, theorems, proofs, worked computations, and figures of the notes are included without omitting a single one; in addition, so that even precocious high-school students who want to understand university mathematics can read it, we have supplemented the text with explanations of how to read new symbols, the steps between the lines of formula manipulations, and concrete examples and figures.

Each section contains worked examples that omit no intermediate steps, and each chapter ends with exercises. The only prerequisite is high school mathematics, and where it is needed we have added links to the corresponding chapters of the companion volume “High School Mathematics” (AD 01). In the part on partial differential equations in the second half of the book, we go as far as applications to quantum mechanics (the Schrödinger equation and the hydrogen atom) and statistical mechanics (Planck's radiation law).

How to read this book
  • Part I (Calculus) and Part II (Linear Algebra) are the foundation for everything else. Read them first, in order.
  • From Part IV (Vector Analysis) onward, taking Parts I and II as prerequisites, you may start with whichever part interests you. Ordinary differential equations (Part VIII) are a prerequisite for partial differential equations (Part IX).
  • Unfamiliar symbols and Greek letters can be looked up in Appendix B (Symbols, Greek Letters, and Terms). Solutions to the exercises are in Appendix A.
Part I Calculus
  1. Chapter 1The Rigorous Definition of Limits and an Introduction to Differentiation Writing the Convergence of a Sequence as a Logical Formula: The $\varepsilon$-$N$ Argument; Writing the Limit of a Function as a Logical Formula: The $\varepsilon$-$\delta$ Argument; Mastering the $\varepsilon$-$N$ Argument; Introducing Differentiation: From the Average Rate of Change to the Instantaneous Rate of Change; The Definition of Differentiation and the Introduction of Napier's Constant $e$
  2. Chapter 2Continuity of Functions, the Mean Value Theorem, and Hyperbolic Functions Fundamental Limits of Trigonometric Functions and Representations in the Style of Euler's Formula; Continuity of Functions; Rolle's Theorem and the Mean Value Theorem; L'Hôpital's Rule; Addition Formulas and Properties of Hyperbolic Functions; Inverse Hyperbolic Functions
  3. Chapter 3Derivatives of Elementary Functions Basic Differentiation Formulas for Exponential Functions; Differentiation Formulas for Products and Quotients; Summary of Differentiation Formulas for Logarithmic Functions; Differentiation Formulas for Trigonometric Functions; Differentiation Formulas for Hyperbolic Functions and Their Inverses; Differentiation Formulas for Inverse Trigonometric Functions
  4. Chapter 4Taylor's Theorem and Expansions of Functions Taylor's Theorem — From First-Order to General-Order Approximation; Taylor and Maclaurin Expansions and Concrete Examples; An Application of Taylor Expansion: Computing $\displaystyle\lim_{x\to+0}x^x$; Higher-Order Derivatives and Leibniz's Rule; Criteria for Extrema (Using Higher-Order Derivatives)
  5. Chapter 5Basic Integration Formulas and Properties of Definite Integrals Basic Indefinite Integral Formulas — Reversing Differentiation; Properties of Definite Integrals; Defining the Definite Integral by Riemann Sums; Improper Integrals and Integrals over Infinite Intervals; Geometric Applications of Integration
  6. Chapter 6Partial Derivatives (Differentiation of Multivariable Functions) Functions of Two Variables and Their Limits; Definition and Basic Properties of Partial Derivatives; Schwarz's Theorem and Total Differentiability; Partial Derivative Formulas for Composite Functions; Change to Polar Coordinates and the Laplacian; Taylor's Theorem and Extrema for Functions of Two Variables; The Method of Lagrange Multipliers
  7. Chapter 7Multiple Integrals (Integration of Multivariable Functions) Definition of Multiple Integrals and Iterated Integrals; Examples of Improper Multiple Integrals; The Jacobian (Change of Variables and the Area Element); Examples of Computing Multiple Integrals: Applications of Change of Variables; Proof of the Gaussian Integral; Further Applications of Change of Variables and the Volume of a Tetrahedron; Volume and Surface Area of a Sphere
Part II Linear Algebra
  1. Chapter 8Matrices and Determinants Basic Operations on Matrices and Patterns in Matrix Powers; Permutations and Their Signs; Definition of the Determinant; Cofactors and Expansion of Determinants; Properties of Determinants; Geometric Meaning of the Determinant
  2. Chapter 9Systems of Linear Equations The Adjugate Matrix and the Formula for the Inverse Matrix; Rank and Row Echelon Matrices; Degrees of Freedom and Homogeneous Systems of Linear Equations; Nonhomogeneous Systems of Linear Equations and Degrees of Freedom; Solving Systems of Linear Equations by Gaussian Elimination (the Sweep-Out Method); Finding Inverse Matrices, Conditions for Invertibility, and the Normal Form
  3. Chapter 10Vector Spaces (Linear Spaces) Definition and Axioms of Vector Spaces; Linear Independence and Linear Dependence; Basis and Dimension; Subspaces; Spanned Subspaces, Solution Spaces, Intersections, and Sums of Subspaces; Theorems on Subspaces and the Dimension Theorem
  4. Chapter 11Linear Maps Basic Vocabulary of Maps: Surjective, Injective, and Bijective Maps; Definition and Basic Properties of Linear Maps; The Image ($\mathrm{Im}\,f$) and the Kernel ($\mathrm{Ker}\,f$); Representation Matrices; Composite Maps and Their Representation Matrices
  5. Chapter 12Eigenvalues, Eigenvectors, and Diagonalization of Matrices Definition of Eigenvalues and Eigenvectors; Properties of Eigenvalues and Eigenspaces; Diagonalization of Matrices; An Application of Diagonalization — Powers of Matrices
Part III Fundamentals of Probability
  1. Chapter 13Sets and the Probability of Events Basic Operations on Sets and Venn Diagrams; De Morgan's Laws; The Language of Trials and Events; Definition of Probability and Its Properties; Mutually Exclusive Events, Complementary Events, Independence, and Repeated Trials
  2. Chapter 14Expected Value — From Discrete to Continuous Discrete Random Variables and Expected Value; Probability Density Functions — Thinking in Intervals Rather Than at Points; The Riemann-Sum Method — Reading the Limit of a Sum as an Integral; Expected Value of Continuous Random Variables; Advanced Topic: Expectation Values in Materials Science and Physics
Part IV Vector Analysis
  1. Chapter 15Inner Products and Cross Products of Vectors What Is a Vector?—Magnitude and Direction, the Zero Vector, and Unit Vectors; The Inner Product (Scalar Product) and Its Properties; Definition of the Cross Product (Vector Product) and the Right-Hand Screw Rule; Component Expression of the Cross Product—Determinants and the Area of a Parallelogram; How the Meanings of the Inner Product and the Cross Product Differ
  2. Chapter 16Coordinate Systems and Basis Vectors Rectangular (Cartesian) Coordinate Systems and Basis Vectors; Cylindrical Coordinate Systems — Basis Vectors and Conversion to Rectangular Coordinates; Spherical Coordinate Systems (Polar Coordinates)—Basis Vectors and Conversion; Rotation Matrices of Coordinate Axes and Euler Angles; Orthogonality and Partial Derivatives of the Spherical Basis Vectors, and the Nabla Operator; Velocity and Acceleration in Polar Coordinates (Circular Motion)
  3. Chapter 17Calculus of Vector-Valued Functions Differentiation of Vector-Valued Functions and Tangent Vectors; Length of a Curve; Integration of Vector-Valued Functions; First-Order Linear Differential Equations for Vector-Valued Functions; Vector-Valued Functions of Two Variables—Partial Derivatives and Total Differentials; Area of a Surface
  4. Chapter 18Scalar Fields, Vector Fields, and the Gradient Scalar Fields and Vector Fields; Level Curves and Level Surfaces; Streamlines; Definition of the Gradient Vector; Formulas for the Gradient Vector; Geometric Meaning of the Gradient Vector; Equation of the Tangent Plane via the Gradient Vector
  5. Chapter 19Divergence and Curl Definition and Basic Formulas of Divergence; Physical Meaning of Divergence — Thinking in Terms of the Flow of Water; The Laplace Operator (the Laplacian); Definition and Basic Formulas of the Curl (Rotation); Identities Involving the Curl and Their Physical Meaning
  6. Chapter 20Line Integrals, Surface Integrals, and the Integral Theorems Line Integrals of Scalar Fields; Line Integrals of Vector Fields (Tangential Line Integrals) and Applications; Tangential Line Integrals of Gradient Vectors and Path Independence; Surface Integrals of Scalar Fields; The Vector Surface Element and the Gauss and Stokes Theorems
Part V Complex Analysis
  1. Chapter 21Complex Numbers and the Complex Plane Absolute Value and Complex Conjugates of Complex Numbers; The Polar Form of a Complex Number — Expressing It by Length and Angle; Products and Quotients in Polar Form — Adding and Subtracting Angles; De Moivre's Theorem and Euler's Formula; The nth Roots of Unity and $i$ as a Rotation Operator; Connecting Trigonometric and Hyperbolic Functions; Equations of Lines and Circles in the Complex Plane
  2. Chapter 22Complex Functions and Mappings What Is a Complex Function? — Understanding It as a Mapping; One-to-One and Many-to-One Mappings; A Two-to-One Mapping: $W=z^2$; $W=1/z$ and the Riemann Sphere; Complex Polynomial Functions of Degree $n$ and Linear Fractional Functions
  3. Chapter 23Elementary Complex Functions — Exponential, Logarithm, Power, and Trigonometric Functions The Complex Exponential Function — Definition and Properties; Mappings Produced by the Complex Exponential Function; The Complex Logarithm; Complex Powers; Mappings Produced by the Complex Logarithm; Complex Trigonometric Functions — Definition by Power Series and Euler's Formula; Why Consider Complex Functions? — The Hidden Imaginary Axis
  4. Chapter 24Holomorphic Functions and the Cauchy–Riemann Equations Neighborhoods and Domains — The Vocabulary of the Complex Plane; Continuity and Holomorphy of Complex Functions — The Cauchy–Riemann Equations; Holomorphy Is Equivalent to Differentiability; Derivatives and Differentiation Formulas for Various Complex Functions
  5. Chapter 25Complex Integration and Cauchy's Integral Theorem Definition and Properties of Complex Integrals; Contour Integrals over Circles — $z^n$ and $\log z$; Green's Theorem and Cauchy's Integral Theorem; The Principle of Path Deformation and Multiply Connected Domains; The Contour Integral Formula
  6. Chapter 26Cauchy's Integral Formula A Concrete Example of Green's Theorem — Verification on the Upper Half-Disk; Path Integrals of $\log z$ in a Multiply Connected Domain and Branch Points; Cauchy's Integral Formula — Statement and Proof; Putting Cauchy's Integral Formula to Work — Examples and Corollaries
Part VI Fourier Series and Transforms
  1. Chapter 27Orthogonality of Trigonometric Functions and Inner Products of Functions Orthogonality of Trigonometric Functions — Basic Formulas; Inner Products and Norms of Functions; The Analogy Between Trigonometric Functions and Vectors; Piecewise Continuous Functions and Piecewise Smooth Functions
  2. Chapter 28Definition of Fourier Series and Worked Examples Fourier Series of Period 2π; Fourier Series of General Period 2L; Fourier Cosine and Sine Series and the Complex Fourier Series; Examples of Computing Fourier Series
  3. Chapter 29Convergence Theory of Fourier Series The Riemann–Lebesgue Lemma; Fourier's Theorem—Proof of Pointwise Convergence; Pointwise Convergence and Uniform Convergence
  4. Chapter 30The Fourier Transform Definition of the Fourier Transform — From Fourier Series to Integrals; The Fourier Transform of a Rectangular Pulse and the Gateway to the Delta Function; The Fourier Transform of a Complex Exponential and the Definition of the Delta Function; The Fourier Transform of the Gaussian Function; The Fourier Transform of the Two-Sided Exponential and the Lorentzian; Advanced Topic: The Fourier Transform of $1/x$ and Parity
Part VII The Laplace Transform
  1. Chapter 31Definition of the Laplace Transform and Basic Formulas Definition of the Laplace Transform; Laplace Transforms of Power Functions; Laplace Transforms of Trigonometric, Hyperbolic, and Exponential Functions; Existence Conditions for the Laplace Transform; The Delta Function, the Unit Step Function, and the Shift Theorem; Laplace Transforms of Periodic Functions and of Derivatives
  2. Chapter 32The Gamma Function, the Beta Function, and the Laplace Transform The Error Function and the Complementary Error Function; Definition and Basic Properties of the Gamma Function; Stirling's Formula; The Beta Function
Part VIII Ordinary Differential Equations
  1. Chapter 33Basic Solution Methods for First-Order Ordinary Differential Equations What Is a Differential Equation? (Classification and Terminology); Separable Equations and Homogeneous-Type Equations; First-Order Linear Differential Equations (Variation of Parameters); First-Order Linear Differential Equations (Integrating Factor Method) and the Validity of Variation of Parameters; Bernoulli's Differential Equation and Riccati's Differential Equation
  2. Chapter 34Exact Differential Equations, Integrating Factors, and Singular Solutions Families of Orthogonal Curves — Deriving the Equation of a Circle with the Power of Differential Equations; Exact Differential Equations and the Test for Exactness; Why the Test and the Solution Formula Hold; Integrating Factors; Clairaut's Differential Equation and Singular Solutions; Lagrange's Differential Equation; The Logistic Equation — A Differential Equation for S-Shaped Curves
  3. Chapter 35Fundamentals of Second-Order Linear Homogeneous Differential Equations Structure of the General Solutions of Nonhomogeneous and Homogeneous Equations; Linear Independence and the Wronskian; Structure of the Solution Space — A Picture in Terms of Vector Spaces; Second-Order Homogeneous Equations with Constant Coefficients and the Characteristic Equation; The Three Cases of the Characteristic Equation
  4. Chapter 36Second-Order Linear Nonhomogeneous Differential Equations and Euler's Equation Structure of the General Solution of a Nonhomogeneous Equation and Variation of Parameters; Solving the Equation When Only One Fundamental Solution Is Known (Reduction of Order); Euler's Differential Equation — Discovering the Trial Solution $x^\lambda$; Solving Euler's Differential Equation — Real Roots, Repeated Roots, and Complex Roots; General Second-Order Linear Differential Equations — Conditions for Finding Fundamental Solutions; Reduction to Normal Form; Phase Space and Differential Equations of Simple-Harmonic Type
  5. Chapter 37Higher-Order Linear Differential Equations Higher-Order Exact Differential Equations; Structure of Solutions of Higher-Order Linear Differential Equations; Solving nth-Order Homogeneous Differential Equations with Constant Coefficients; Fundamental Solutions When the Characteristic Equation Has a Repeated Root; Higher-Order Euler Equations; Reflections on Variation of Parameters and the Completion of the Repeated-Root Theorem (Advanced)
  6. Chapter 38The Operator Method Why Study the Operator Method?—Definition of the Differential Operator $D$; Properties of Differential Operators; The Inverse Operator $\dfrac{1}{D-\alpha}$—Reducing Particular Solutions to First-Order Linear Equations; Substitution Formulas for Exponential Functions; The Shift Theorem; Exponential Functions at Resonance; Substitution Formulas for Trigonometric Functions; Trigonometric Functions at Resonance
  7. Chapter 39The Power Series Method and Frobenius Theory Why Do We Need Series Solutions? — The Limits of the Maclaurin Expansion; The Procedure for Series Solutions and Fundamentals of Taylor Expansion; Frobenius Theory — Ordinary Points and Regular Singular Points; Roots of the Indicial Equation and Fundamental Solutions; Examples of Series Solutions
  8. Chapter 40Legendre's Differential Equation and Legendre Polynomials Legendre's Differential Equation and Its Frobenius Series Solutions; Deriving the Recurrence Relation and Computing the Coefficients; The General Solution and $U_n(x)$, $V_n(x)$; Legendre Polynomials; Rodrigues's Formula and Orthogonality
  9. Chapter 41Special Functions — Bessel Functions and Laguerre Polynomials Bessel's Differential Equation and Its Frobenius Series Solutions; The Gamma Function and Bessel Functions of the First Kind; The General Solution of Bessel's Differential Equation; Laguerre's Differential Equation and Laguerre Polynomials; The Associated Laguerre Differential Equation (Advanced)
Part IX Partial Differential Equations
  1. Chapter 42Introduction to Partial Differential Equations and Lagrange's First-Order Partial Differential Equation What Is a Partial Differential Equation? — The Appearance of Arbitrary Functions; Lagrange's First-Order Partial Differential Equation and the Characteristic Equations; The Generalized Lagrange Partial Differential Equation; Total Differential Equations and Exact Differential Equations; Partial Differential Equations for Traveling Waves
  2. Chapter 43The Wave Equation and the Heat Equation — Derivation and Separation of Variables Derivation of the Wave Equation (from Maxwell's Equations); Derivation of the One-Dimensional Wave Equation (Vibrating String); Derivation of the Heat Equation; Solving the One-Dimensional Wave Equation by Separation of Variables; Solving the Two-Dimensional Wave Equation by Separation of Variables; Solving the One-Dimensional Heat Equation by Separation of Variables, with Simulation; Solving the Two-Dimensional Heat Equation by Separation of Variables; Examples for Laplace's Equation
  3. Chapter 44Coordinate Transformations and the Laplacian Why Change Coordinates? — Symmetry and the Laplacian; Deriving the Laplacian in Polar Coordinates; The Laplacian in Cylindrical Coordinates; The Laplacian in Spherical Coordinates
  4. Chapter 45The Schrödinger Equation (Cartesian Coordinates) and the Infinite Square Well Potential Operators and the Hamiltonian in Quantum Mechanics; Derivation of the Schrödinger Equation; The Infinite Square Well Potential (One Dimension, Stationary State); The Infinite Square Well Potential (One Dimension, Time-Dependent); The Infinite Square Well Potential (Two Dimensions, Stationary State); The Infinite Square Well Potential (Two Dimensions, Time-Dependent)
  5. Chapter 46The Harmonic Oscillator and the Morse Potential The Harmonic Oscillator Potential and Nondimensionalization; The Power Series Method and Energy Levels (Hermite Polynomials); The Harmonic Oscillator on a Half-Infinite Domain; The Three-Dimensional Harmonic Oscillator; The Morse Potential
  6. Chapter 47The Schrödinger Equation (Polar and Cylindrical Coordinates) and Bessel Functions The Infinite Cylindrical Well Potential: The Radial Solution; The Radial Solution and Bessel Functions (Normalization and Energy Levels); Graphs and Concrete Examples of Bessel Functions; The Infinite Cylinder Potential (Three Dimensions)
  7. Chapter 48The Schrödinger Equation for the Hydrogen Atom (Spherical Coordinates) The Schrödinger Equation in Spherical Coordinates and Separation of Variables—The Example of the Infinite Spherical Well Potential; The Hydrogen Atom Potential and Separation of the Radial Equation; Solving the Radial Equation—Associated Laguerre Polynomials and Energy Levels; The Angular Solutions—Associated Legendre Functions and the Complete Wave Function
  8. Chapter 49Statistical Mechanics and Black-Body Radiation — The Boltzmann Distribution and Planck's Radiation Law The Boltzmann Distribution — The Altitude Distribution of Gas Molecules; The Rayleigh–Jeans Law and Wien's Radiation Law; Planck's Radiation Law and Its Two Limits; Deriving the Mean Energy from the Quantum Hypothesis; Counting the Standing Waves Confined in a Cavity — The Density of States and the Completion of Planck's Formula
Appendices
  1. Appendix ASolutions to the Exercises Complete solutions to the exercises at the end of each chapter
  2. Appendix BSymbols, Greek Letters, and Terms How to read the symbols used in this book, their meanings, the names of the Greek letters, and an index to the chapters where each first appears

Source

Mochizuki Research Group, Department of Materials Science and Technology, Faculty of Advanced Engineering, Tokyo University of Science — Advanced Resources AD 02
← Back to Educational Resources