Partial Differential Equations: A First Course

Preface
0.1. Who Should Take a First Course in PDEs?
0.2. A Text for All Three Groups: Grounding in Core Concepts and Topics
0.3. Basic Structure of the Text
0.3.1. Presentation and Modularization of Material
0.3.2. Choice for the Orderings of Chapters
0.3.3. Codependence and Different Orderings of Chapters
0.4. Prerequisites
0.4.1. Advanced Calculus and the Appendix
0.4.2. Breadth and Nonrigidity
0.5. Acknowledgments
Chapter 1. Basic Definitions
1.1. ∙ Notation
1.2. ∙ What Are Partial Differential Equations and Why Are They Ubiquitous?
1.3. ∙ What Exactly Do We Mean by a Solution to a PDE?
1.4. ∙ Order, Linear vs. Nonlinear, Scalar vs. Systems
1.4.1. ∙ Order
1.4.2. ∙ Linear vs. Nonlinear
1.4.3. ∙ The Principle of Superposition for Linear PDEs
1.4.4. ∙ Scalar vs. Systems
1.5. ∙ General Solutions, Arbitrary Functions, Auxiliary Conditions, and the Notion of a Well-Posed Problem
1.5.1. ∙ General Solutions and Arbitrary Functions
1.5.2. ∙ Auxiliary Conditions: Boundary and Initial Conditions
1.5.3. ∙ General Auxiliary Conditions and the Cauchy Problem
1.5.4. ∙ A Well-Posed Problem
1.6. ∙ Common Approaches and Themes in Solving PDEs
1.6.1. ∙ Assuming We Have a Solution and Going Forward
1.6.2. ∙ Explicit vs. Nonexplicit Solutions, No Solutions, and Approximate Solutions
Exercises
Chapter 2. First-Order PDEs and the Method of Characteristics
2.1. ∙ Prelude: A Few Simple Examples Illustrating the Notion and Geometry of Characteristics
2.2. ∙ The Method of Characteristics, Part I: Linear Equations
2.2.1. ∙ A Few Examples
2.2.2. ∙ Temporal Equations: Using Time to Parametrize the Characteristics
2.2.3. ∙ More Than Two Independent Variables
2.2.4. ∙ Transport Equations in Three Space Dimensions with Constant Velocity
2.2.5. Transport Equations in Three Space Dimensions with Space Varying Velocity
2.2.6. The Continuity Equation in Three Space Dimensions: A Derivation
2.2.7. Semilinear Equations
2.2.8. Noncharacteristic Data and the Transversality Condition
2.3. ∙ An Important Quasilinear Example: The Inviscid Burgers Equation
2.4. ∙ The Method of Characteristics, Part II: Quasilinear Equations
2.4.1. ∙ A Few Examples
2.4.2. ∙ More Than Two Independent Variables
2.4.3. Pausing to Reflect on the Inherent Logic Behind the Method of Characteristics and Local Solutions of Quasilinear Equations
2.5. The Method of Characteristics, Part III: General First-Order Equations
2.5.1. The Notation
2.5.2. The Characteristic Equations
2.5.3. Linear and Quasilinear Equations in ? independent variables
2.5.4. Two Fully Nonlinear Examples
2.5.5. The Eikonal Equation
2.5.6. Hamilton-Jacobi Equations
2.5.7. The Level Set Equation and Interface Motion
2.6. ∙ Some General Questions
2.7. ∙ A Touch of Numerics, I: Computing the Solution of the Transport Equation
2.7.1. ∙ Three Consistent Schemes
2.7.2. ∙ von Neumann Stability Analysis
2.8. The Euler Equations: A Derivation
2.8.1. Conservation of Mass and the Continuity Equation
2.8.2. Conservation of Linear Momentum and Pressure
2.8.3. Gas Dynamics: The Compressible Euler Equations
2.8.4. An Ideal Liquid: The Incompressible Euler Equations
2.8.5. A Viscous Liquid: The Navier-Stokes Equations
2.8.6. Spatial vs. Material Coordinates and the Material Time Derivative
2.9. Chapter Summary
Exercises
Chapter 3. The Wave Equation in One Space Dimension
3.1. ∙ Derivation: A Vibrating String
3.2. ∙ The General Solution of the 1D Wave Equation
3.3. ∙ The Initial Value Problem and Its Explicit Solution: D’Alembert’s Formula
3.4. ∙ Consequences of D’Alembert’s Formula: Causality
3.4.1. ∙ The Domain of Dependence and Influence
3.4.2. ∙ Two Examples: A Plucked String and a Hammer Blow
3.5. ∙ Conservation of the Total Energy
3.6. ∙ Sources
3.6.1. ∙ Duhamel’s Principle
3.6.2. Derivation via Green’s Theorem
3.7. ∙ Well-Posedness of the Initial Value Problem and Time Reversibility
3.8. ∙ The Wave Equation on the Half-Line with a Fixed Boundary: Reflections
3.8.1. ∙ A Dirichlet (Fixed End) Boundary Condition and Odd Reflections
3.8.2. ∙ Causality with Respect to the Fixed Boundary
3.8.3. ∙ The Plucked String and Hammer Blow Examples with a Fixed Left End
3.9. ∙ Neumann and Robin Boundary Conditions
3.10. ∙ Finite String Boundary Value Problems
3.11. ∙ A Touch of Numerics, II: Numerical Solution to the Wave Equation
3.12. Some Final Remarks
3.12.1. Nonsmooth Solutions
3.12.2. Heterogeneous Media and Scattering
3.12.3. Finite Propagation Speed, Other “Wave” Equations, and Dispersion
3.12.4. Characterizing Dispersion in PDEs via the Ubiquitous Traveling Wave Solution
3.13. Chapter Summary
Exercises
Chapter 4. The Wave Equation in Three and Two Space Dimensions
4.1. ∙ Two Derivations of the 3D Wave Equation
4.1.1. ∙ Derivation 1: Electromagnetic Waves from Maxwell’s Equations
4.1.2. Derivation 2: Acoustics from the Euler Equations
4.2. ∙ Three Space Dimensions: The Initial Value Problem and Its Explicit Solution
4.2.1. ∙ Kirchhoff’s Formula
4.2.2. ∙ Consequences of Kirchhoff’s Formula: Causality and the Huygens Principle
4.2.3. ∙ Kirchhoff’s Formula via Spherical Means
4.2.4. Full Details: The Proof of Kirchhoff’s Formula
4.3. Two Space Dimensions: The 2D Wave Equation and Its Explicit Solution
4.3.1. The Solution via the Method of Descent
4.3.2. Causality in 2D
4.4. Some Final Remarks and Geometric Optics
4.4.1. The Wave Equation in Space Dimensions Larger Than Three
4.4.2. Regularity
4.4.3. Nonconstant Coefficients and Inverse Problems
4.4.4. Geometric Optics and the Eikonal Equation
4.5. Chapter Summary
Exercises
Chapter 5. The Delta “Function” and Distributions in One Space Dimension
5.1. ∙ Real-Valued Functions
5.1.1. ∙ What Is a Function?
5.1.2. ∙ Why Integrals (or Averages) of a Function Trump Pointwise Values
5.1.3. ∙ Singularities of Functions from the Point of View of Averages
5.2. ∙ The Delta “Function” and Why It Is Not a Function. Motivation for Generalizing the Notion of a Function
5.2.1. ∙ The Delta “Function” and the Derivative of the Heaviside Function
5.2.2. ∙ The Delta “Function” as a Limit of a Sequence of Functions Which Concentrate
5.3. ∙ Distributions (Generalized Functions)
5.3.1. ∙ The Class of Test Functions ?_{?}^{∞}(ℝ)
5.3.2. ∙ The Definition of a Distribution
5.3.3. ∙ Functions as Distributions
5.3.4. ∙ The Precise Definition of the Delta “Function” as a Distribution
5.4. ∙ Derivative of a Distribution
5.4.1. ∙ Motivation via Integration by Parts of Differentiable Functions
5.4.2. ∙ The Definition of the Derivative of a Distribution
5.4.3. ∙ Examples of Derivatives of Piecewise Smooth Functions in the Sense of Distributions
5.5. ∙ Convergence in the Sense of Distributions
5.5.1. ∙ The Definition
5.5.2. ∙ Comparisons of Distributional versus Pointwise Convergence of Functions
5.5.3. ∙ The Distributional Convergence of a Sequence of Functions to the Delta Function: Four Examples
5.5.4. ? vs. ? Proofs for the Sequences (5.23) and (5.24)
5.5.5. ∙ The Distributional Convergence of sin??
5.5.6. ∙ The Distributional Convergence of the Sinc Functions and the Dirichlet Kernel: Two Sequences Directly Related to Fourier Analysis
5.6. Dirac’s Intuition: Algebraic Manipulations with the Delta Function
5.6.1. Rescaling and Composition with Polynomials
5.6.2. Products of Delta Functions in Different Variables
5.6.3. Symmetry in the Argument
5.7. ∙ Distributions Defined on an Open Interval and Larger Classes of Test Functions
5.7.1. ∙ Distributions Defined over a Domain
5.7.2. ∙ Larger Classes of Test Functions
5.8. Nonlocally Integrable Functions as Distributions: The Distribution PV 1/?
5.8.1. Three Distributions Associated with the Function 1/?
5.8.2. A Second Way to Write ??1/?
5.8.3. A Third Way to Write ??1/?
5.8.4. The Distributional Derivative of ??1/?
5.9. Chapter Summary
Exercises
Chapter 6. The Fourier Transform
6.1. ∙ Complex Numbers
6.2. ∙ Definition of the Fourier Transform and Its Fundamental Properties
6.2.1. ∙ The Definition
6.2.2. ∙ Differentiation and the Fourier Transform
6.2.3. ∙ Fourier Inversion via the Delta Function
6.2.4. Finding the Fourier and Inverse Fourier Transforms of Particular Functions
6.2.5. The Fourier Transform of a Complex-Valued Function
6.3. ∙ Convolution of Functions and the Fourier Transform
6.3.1. ∙ The Definition of Convolution
6.3.2. ∙ Differentiation of Convolutions
6.3.3. ∙ Convolution and the Fourier Transform
6.3.4. Convolution as a Way of Smoothing Out Functions and Generating Test Functions in ?^{∞}_{?}
6.4. ∙ Other Important Properties of the Fourier Transform
6.5. Duality: Decay at Infinity vs. Smoothness
6.6. Plancherel’s Theorem and the Riemann-Lebesgue Lemma
6.6.1. The Spaces ?¹(ℝ) and ?²(ℝ)
6.6.2. Extending the Fourier Transform to Square-Integrable Functions and Plancherel’s Theorem
6.6.3. The Riemann-Lebesgue Lemma
6.7. The 2? Issue and Other Possible Definitions of the Fourier Transform
6.8. ∙ Using the Fourier Transform to Solve Linear PDEs, I: The Diffusion Equation
6.8.1. ∙ Prelude: Using the Fourier Transform to Solve a Linear ODE
6.8.2. ∙ Using the Fourier Transform to Solve the Diffusion Equation
6.9. The Fourier Transform of a Tempered Distribution
6.9.1. Can We Extend the Fourier Transform to Distributions?
6.9.2. The Schwartz Class of Test Functions and Tempered Distributions
6.9.3. The Fourier Transform of ?(?)≡1 and the Delta Function
6.9.4. The Fourier Transform of ?(?)=?^{???} and the Delta Function ?ₐ
6.9.5. The Fourier Transform of Sine, Cosine, and Sums of Delta Functions
6.9.6. The Fourier Transform of ?
6.9.7. The Fourier Transform of the Heaviside and Sgn Functions and PV 1/?
6.9.8. Convolution of a Tempered Distribution and a Function
6.10. Using the Fourier Transform to Solve PDEs, II: The Wave Equation
6.11. The Fourier Transform in Higher Space Dimensions
6.11.1. Definition and Analogous Properties
6.11.2. The Fourier Transform of a Radially Symmetric Function and Bessel Functions
6.12. Frequency, Harmonics, and the Physical Meaning of the Fourier Transform
6.12.1. Plane Waves in One Dimension
6.12.2. Interpreting the Fourier Inversion Formula
6.12.3. Revisiting the Wave Equation
6.12.4. Properties of the Fourier Transform, Revisited
6.12.5. The Uncertainty Principle
6.13. A Few Words on Other Transforms
6.14. Chapter Summary
6.15. Summary Tables
Exercises
Chapter 7. The Diffusion Equation
7.1. ∙ Derivation 1: Fourier’s/Fick’s Law
7.2. ∙ Solution in One Space Dimension and Properties
7.2.1. ∙ The Fundamental Solution/Heat Kernel and Its Properties
7.2.2. ∙ Properties of the Solution Formula
7.2.3. The Proof for the Solution to the Initial Value Problem: The Initial Conditions and the Delta Function
7.3. ∙ Derivation 2: Limit of Random Walks
7.3.1. ∙ Numerical Approximation of Second Derivatives
7.3.2. ∙ Random Walks
7.3.3. ∙ The Fundamental Limit and the Diffusion Equation
7.3.4. ∙ The Limiting Dynamics: Brownian Motion
7.4. Solution via the Central Limit Theorem
7.4.1. Random Variables, Probability Densities and Distributions, and the Normal Distribution
7.4.2. The Central Limit Theorem
7.4.3. Application to Our Limit of Random Walks and the Solution to the Diffusion Equation
7.5. ∙ Well-Posedness of the IVP and Ill-Posedness of the Backward Diffusion Equation
7.5.1. ∙ Nonuniqueness of the IVP of the Diffusion Equation
7.5.2. ∙ Ill-Posedness of the Backward Diffusion Equation
7.5.3. Deblurring in Image Processing
7.6. ∙ Some Boundary Value Problems in the Context of Heat Flow
7.6.1. ∙ Dirichlet and Neumann Boundary Conditions
7.6.2. ∙ The Robin Condition and Heat Transfer
7.7. ∙ The Maximum Principle on a Finite Interval
7.8. Source Terms and Duhamel’s Principle Revisited
7.8.1. An Intuitive and Physical Explanation of Duhamel’s Principle for Heat Flow with a Source
7.9. The Diffusion Equation in Higher Space Dimensions
7.10. ∙ A Touch of Numerics, III: Numerical Solution to the Diffusion Equation
7.11. Addendum: The Schrödinger Equation
7.12. Chapter Summary
Exercises
Chapter 8. The Laplacian, Laplace’s Equation, and Harmonic Functions
8.1. ∙ The Dirichlet and Neumann Boundary Value Problems for Laplace’s and Poisson’s Equations
8.2. ∙ Derivation and Physical Interpretations 1: Concentrations in Equilibrium
8.3. Derivation and Physical Interpretations 2: The Dirichlet Problem and Poisson’s Equation via 2D Random Walks/Brownian Motion
8.4. ∙ Basic Properties of Harmonic Functions
8.4.1. ∙ The Mean Value Property
8.4.2. ∙ The Maximum Principle
8.4.3. ∙ The Dirichlet Principle
8.4.4. Smoothness (Regularity)
8.5. ∙ Rotational Invariance and the Fundamental Solution
8.6. ∙ The Discrete Form of Laplace’s Equation
8.7. The Eigenfunctions and Eigenvalues of the Laplacian
8.7.1. Eigenvalues and Energy: The Rayleigh Quotient
8.8. The Laplacian and Curvature
8.8.1. Principal Curvatures of a Surface
8.8.2. Mean Curvature
8.8.3. Curvature and Invariance
8.9. Chapter Summary
Exercises
Chapter 9. Distributions in Higher Dimensions and Partial Differentiation in the Sense of Distributions
9.1. ∙ The Test Functions and the Definition of a Distribution
9.2. ∙ Convergence in the Sense of Distributions
9.3. ∙ Partial Differentiation in the Sense of Distributions
9.3.1. ∙ The Notation and Definition
9.3.2. ∙ A 2D Jump Discontinuity Example
9.4. ∙ The Divergence and Curl in the Sense of Distributions: Two Important Examples
9.4.1. ∙ The Divergence of the Gravitational Vector Field
9.4.2. The Curl of a Canonical Vector Field
9.5. ∙ The Laplacian in the Sense of Distributions and a Fundamental Example
9.6. Distributions Defined on a Domain (with and without Boundary)
9.7. Interpreting Many PDEs in the Sense of Distributions
9.7.1. Our First Example Revisited!
9.7.2. Burgers’s Equation and the Rankine-Hugoniot Jump Conditions
9.7.3. The Wave Equation with a Delta Function Source
9.7.4. Incorporating Initial Values into a Distributional Solution
9.7.5. Not All PDEs Can Be Interpreted in the Sense of Distributions
9.8. A View Towards Sobolev Spaces
9.9. Fourier Transform of an ?-dimensional Tempered Distribution
9.10. Using the Fourier Transform to Solve Linear PDEs, III: Helmholtz and Poisson Equations in Three Space
9.11. Chapter Summary
Exercises
Chapter 10. The Fundamental Solution and Green’s Functions for the Laplacian
10.1. ∙ The Proof for the Distributional Laplacian of 1over |?|
10.2. ∙ Unlocking the Power of the Fundamental Solution for the Laplacian
10.2.1. ∙ The Fundamental Solution Is Key to Solving Poisson’s Equation
10.2.2. ∙ The Fundamental Solution Gives a Representation Formula for Any Harmonic Function in Terms of Boundary Data
10.3. ∙ Green’s Functions for the Laplacian with Dirichlet Boundary Conditions
10.3.1. ∙ The Definition of the Green’s Function with Dirichlet Boundary Conditions
10.3.2. Using the Green’s Function to Solve the Dirichlet Problem for Laplace’s Equation
10.3.3. ∙ Uniqueness and Symmetry of the Green’s Function
10.3.4. ∙ The Fundamental Solution and Green’s Functions in One Space Dimension
10.4. ∙ Green’s Functions for the Half-Space and Ball in 3D
10.4.1. ∙ Green’s Function for the Half-Space
10.4.2. ∙ Green’s Function for the Ball
10.4.3. The Proof for Theorem 10.9
10.4.4. Green’s Functions for Other Domains and Differential Operators
10.5. Green’s Functions for the Laplacian with Neumann Boundary Conditions
10.5.1. Finding the Neumann Green’s Function for the Half-Space in ℝ³
10.5.2. Finding the Neumann Green’s Function for the Ball in ℝ³
10.6. A Physical Illustration in Electrostatics: Coulomb’s Law, Gauss’s Law, the Electric Field, and Electrostatic Potential
10.6.1. Coulomb’s Law and the Electrostatic Force
10.6.2. The Electrostatic Potential: The Fundamental Solution and Poisson’s Equation
10.6.3. Green’s Functions: Grounded Conducting Plates, Induced Charge Densities, and the Method of Images
10.6.4. Interpreting the Solution Formula for the Dirichlet Problem
10.7. Chapter Summary
Exercises
Chapter 11. Fourier Series
11.1. ∙ Prelude: The Classical Fourier Series —the Fourier Sine Series, the Fourier Cosine Series, and the Full Fourier Series
11.1.1. ∙ The Fourier Sine Series
11.1.2. ∙ The Fourier Cosine Series
11.1.3. ∙ The Full Fourier Series
11.1.4. ∙ Three Examples
11.1.5. ∙ Viewing the Three Fourier Series as Functions over ℝ
11.1.6. ∙ Convergence, Boundary Values, Piecewise Continuity, and Periodic Extensions
11.1.7. Complex Version of the Full Fourier Series
11.2. ∙ Why Cosines and Sines? Eigenfunctions, Eigenvalues, and Orthogonality
11.2.1. ∙ Finite Dimensions —the Linear Algebra of Vectors
11.2.2. ∙ Infinite Dimensions —the Linear Algebra of Functions
11.2.3. ∙ The Linear Operator ?=-?²over ??² and Symmetric Boundary Conditions
11.3. ∙ Fourier Series in Terms of Eigenfunctions of ? with a Symmetric Boundary Condition
11.3.1. ∙ The Eigenfunctions and Respective Fourier Series Associated with the Four Standard Symmetric Boundary Conditions
11.3.2. ∙ The Miracle: These Sets of Eigenfunctions Span the Space of All Reasonable Functions
11.4. ∙ Convergence, I: The ?² Theory, Bessel’s Inequality, and Parseval’s Equality
11.4.1. ∙ ?² Convergence of a Sequence of Functions
11.4.2. ∙ ?² Convergence of Fourier Series
11.4.3. ∙ Bessel’s Inequality and Reducing the ?² Convergence Theorem to Parseval’s Equality
11.4.4. The Riemann-Lebesgue Lemma and an Application of Parseval’s Equality
11.5. ∙ Convergence, II: The Dirichlet Kernel and Pointwise Convergence of the Full Fourier Series
11.5.1. ∙ Pointwise Convergence of a Sequence of Functions
11.5.2. ∙ Pointwise Convergence of the Full Fourier Series: The Dirichlet Kernel and the Delta Function
11.5.3. ∙ The Proof of Pointwise Convergence of the Full Fourier Series
11.6. Term-by-Term Differentiation and Integration of Fourier Series
11.6.1. Term-by-Term Differentiation
11.6.2. Term-by-Term Integration
11.7. Convergence, III: Uniform Convergence
11.7.1. Uniform Convergence of Functions
11.7.2. A Criterion for the Uniform Convergence of Fourier Series
11.7.3. The Proof of Theorem 11.9
11.7.4. The Gibbs Phenomenon
11.8. What Is the Relationship Between Fourier Series and the Fourier Transform?
11.8.1. Sending ?→∞ in the Full Fourier Series
11.8.2. Taking the Distributional Fourier Transform of a Periodic Function
11.9. Chapter Summary
Exercises
Chapter 12. The Separation of Variables Algorithm for Boundary Value Problems
12.1. ∙ The Basic Separation of Variables Algorithm
12.1.1. ∙ The Diffusion Equation with Homogeneous Dirichlet Boundary Conditions
12.1.2. ∙ The Diffusion Equation with Homogenous Neumann Boundary Conditions
12.2. ∙ The Wave Equation
12.2.1. ∙ The Wave Equation with Homogeneous Dirichlet Boundary Conditions
12.2.2. The Wave Equation with Homogeneous Neumann Boundary Conditions
12.3. ∙ Other Boundary Conditions
12.3.1. ∙ Inhomogeneous Dirichlet Boundary Conditions
12.3.2. ∙ Mixed Homogeneous Boundary Conditions
12.3.3. ∙ Mixed Inhomogeneous Boundary Conditions
12.3.4. ∙ Inhomogeneous Neumann Boundary Conditions
12.3.5. ∙ The Robin Boundary Condition for the Diffusion Equation
12.4. Source Terms and Duhamel’s Principle for the Diffusion and Wave Equations
12.5. ∙ Laplace’s Equations in a Rectangle and a Disk
12.5.1. ∙ Rectangle
12.5.2. ∙ The Disk
12.6. ∙ Extensions and Generalizations of the Separation of Variables Algorithm
12.7. ∙ Extensions, I: Multidimensional Classical Fourier Series: Solving the Diffusion Equation on a Rectangle
12.8. ∙ Extensions, II: Polar and Cylindrical Coordinates and Bessel Functions
12.8.1. ∙ Vibrations of a Drum and Bessel Functions
12.9. Extensions, III: Spherical Coordinates, Legendre Polynomials, Spherical Harmonics, and Spherical Bessel Functions
12.9.1. Separation of Variables for the 3D Laplace Equation in Spherical Coordinates
12.9.2. Legendre Polynomials and Associated Legendre Polynomials
12.9.3. Spherical Harmonics
12.9.4. Solving the 3D Diffusion Equation on the Ball
12.10. Extensions, IV: General Sturm-Liouville Problems
12.10.1. Regular Sturm-Liouville Problems
12.10.2. Singular Sturm-Liouville Problems
12.11. Separation of Variables for the Schrödinger Equation: Energy Levels of the Hydrogen Atom
12.12. Chapter Summary
Exercises
Chapter 13. Uniting the Big Three Second-Order Linear Equations, and What’s Next
13.1. Are There Other Important Linear Second-Order Partial Differential Equations? The Standard Classification
13.1.1. Classification of Linear Second-Order Partial Differential Equations
13.2. Reflection on Fundamental Solutions, Green’s Functions, Duhamel’s Principle, and the Role/Position of the Delta Function
13.2.1. Fundamental Solutions/Green’s Functions for the Laplacian
13.2.2. Fundamental Solutions/Green’s Functions of the Diffusion Equation
13.2.3. Fundamental Solutions/Green’s Functions of the 1D Wave Equation
13.2.4. Fundamental Solutions/Green’s Functions of the 3D Wave Equation
Exercises
13.3. What’s Next? Towards a Future Volume on This Subject
Appendix. Objects and Tools of Advanced Calculus
A.1. Sets, Domains, and Boundaries in ℝ^{ℕ}
A.2. Functions: Smoothness and Localization
A.2.1. Function Classes Sorted by Smoothness
A.2.2. Localization: Functions with Compact Support
A.2.3. Boundary Values for Functions Defined on a Domain
A.3. Gradient of a Function and Its Interpretations, Directional Derivatives, and the Normal Derivative
A.3.1. The Fundamental Relationship Between the Gradient and Directional Derivatives
A.3.2. Lagrange Multipliers: An Illuminating Illustration of the Meaning of the Gradient
A.3.3. An Important Directional Derivative: The Normal Derivative on an Orientable Surface
A.4. Integration
A.4.1. Bulk, Surface, and Line Integrals
A.4.2. Flux Integrals
A.4.3. Improper Integrals, Singularities, and Integrability
A.5. Evaluation and Manipulation of Integrals: Exploiting Radial Symmetry
A.5.1. Spherical (Polar) Coordinates in ℝ³
A.5.2. Integration of a Radially Symmetric Function
A.5.3. Integration of General Functions over a Ball via Spherical Shells
A.5.4. Rescalings and Translations
A.6. Fundamental Theorems of Calculus: The Divergence Theorem, Integration by Parts, and Green’s First and Second Identities
A.6.1. The Divergence Theorem
A.6.2. Two Consequences of the Divergence Theorem: Green’s Theorem and a Componentwise Divergence Theorem
A.6.3. A Match Made in Heaven: The Divergence + the Gradient = the Laplacian
A.6.4. Integration by Parts and Green’s First and Second Identities
A.7. Integral vs. Pointwise Results
A.7.1. IPW (Integral to Pointwise) Theorems
A.7.2. The Averaging Lemma
A.8. Convergence of Functions and Convergence of Integrals
A.9. Differentiation under the Integral Sign
A.9.1. General Conditions for Legality
A.9.2. Examples Where It Is Illegal
A.9.3. A Leibnitz Rule
A.10. Change in the Order of Integration
A.10.1. The Fubini-Tonelli Theorem
A.10.2. Examples Where It Is Illegal
A.11. Thinking Dimensionally: Physical Variables Have Dimensions with Physical Units
Exercises
Bibliography
Index