Elements of Partial Differential Equations

Chapter 1  ORDINARY DIFFERENTIAL EQUATIONS IN MORE THAN TWO VARIABLES
     1. Surfaces and Curves in Three Dimensions
          PROBLEMS
     2. Simultaneous Differential Equations of the First Order and the First Degree in Three Variables
     3. Methods of Solution of dx/P = dy/Q = dz /R
          PROBLEMS
     4. Orthogonal Trajectories of a System of Curves on a Surface
          PROBLEMS
     5. Pfaffian Differential Forms and Equations
          PROBLEMS
     6. Solution of Pfaffian Differential Equations in Three Variables
          (a) By Inspection
          (b) Variables Separable.
          (c) One Variable Separable
          (d) Homogeneous Equations
          (e). Natani\'s Method
          (f) Reduction to an Ordinary Differential Equation.
          PROBLEMS
     7. Carathéodory\'s Theorem
     8. Application. to Thermodynamics
     MISCELLANEOUS PROBLEMS

Chapter 2  PARTIAL DIFFERENTIAL EQUATIONS OF THE FIRST ORDER
     1. Partial Differential Equations
     2. Origins of First -order Partial Differential Equations
          PROBLEMS
     3. Cauchy\'s Problem for First -order Equations
     4. Linear Equations of the First Order
          PROBLEMS
     5. Integral Surfaces Passing through a Given Curve
          PROBLEMS
     6. Surfaces Orthogonal to a Given System of Surfaces
          PROBLEMS
     7. Nonlinear Partial Differential Equations of the First Order
          PROBLEMS
     8. Cauchy\'s Method of Characteristics
          PROBLEMS
     9. Compatible Systems of First -order Equations
          PROBLEMS
     10. Charpit\'s Method
          PROBLEMS
     11. Special Types of First -order Equations
          PROBLEMS
     12. Solutions Satisfying Given Conditions
          PROBLEMS
     13. Jacobi\'s Method
          PROBLEMS
     14. Applications of First -order Equations
          PROBLEMS
     MISCELLANEOUS PROBLEMS

Chapter 3  PARTIAL DIFFERENTIAL EQUATIONS OF THE SECOND ORDER
     1. The Origin of Second -order Equations
          PROBLEMS
     2. Second -order Equations in Physics
          PROBLEMS
     3. Higher -order Equations in Physics
          PROBLEMS
     4. Linear Partial Differential Equations with Constant Coefficients
          PROBLEMS
     5. Equations with Variable Coefficients
          PROBLEMS
     6. Characteristic Curves of Second -order Equations
          PROBLEMS
     7. Characteristics of Equations in Three Variables
          PROBLEMS
     8. The Solution of Linear Hyperbolic Equations
          PROBLEMS
     9. Separation of Variables
          PROBLEMS
     10. The Method of Integral Transforms
          PROBLEMS
     11. Nonlinear Equations of the Second Order
          PROBLEMS
     MISCELLANEOUS PROBLEMS

Chapter 4  LAPLACE\'S EQUATION
     1. The Occurrence of Laplace\'s Equation in Physics
          PROBLEMS
     2. Elementary Solutions of Laplace\'s Equation
          PROBLEMS
     3. Families of Equipotential Surfaces
          PROBLEMS
     4. Boundary Value Problems
          PROBLEMS
     5. Separation of Variables
          PROBLEMS
     6. Problems with Axial Symmetry
          PROBLEMS
     7. Kelvin\'s Inversion Theorem
          PROBLEMS
     8. The Theory of Green\'s Function for Laplace\'s Equation
          PROBLEMS
     9. The Relation of Dirichlet\'s Problem to the Calculus of Variations
     10. \"Mixed\" Boundary Value Problems
          PROBLEMS
     11. The Two -dimensional Laplace Equation
          PROBLEMS
     12. Relation of the Logarithmic Potential to the Theory of Functions
          PROBLEMS
     13. Green\'s Function for the Two -dimensional Equation
          PROBLEMS
     MISCELLANEOUS PROBLEMS

Chapter 5  THE WAVE EQUATION
     1. The Occurrence of the Wave Equation in Physics
          PROBLEMS
     2. Elementary Solutions of the One -dimensional Wave Equation
          PROBLEMS
     3. The Riemann- Volterra Solution of the One -dimensional Wave Equation
          PROBLEMS
     4. Vibrating Membranes: Application of the Calculus of Variations
          PROBLEMS
     5. Three -dimensional Problems
          PROBLEMS
     6. General Solutions of the Wave Equation
          PROBLEMS
     7. Green\'s Function for the Wave Equation
          PROBLEMS
     8. The Nonhomogeneous Wave Equation
          PROBLEMS
     9. Riesz\'s Integrals
          PROBLEMS
     10. The Propagation of Sound Waves of Finite Amplitude
          PROBLEMS
     MISCELLANEOUS PROBLEMS

Chapter 6  THE DIFFUSION EQUATION
     1. The Occurrence of the Diffusion Equation in Physics
          PROBLEMS
     2. The Resolution of Boundary Value Problems for the Diffusion Equation
          PROBLEMS\'
     3. Elementary Solutions of the Diffusion Equation
          PROBLEMS
     4. Separation of Variables
          PROBLEMS
     5. The Use of Integral Transforms
          PROBLEMS
     6. The Use of Green\'s Functions
          PROBLEMS
     7. The Diffusion Equation with Sources
          PROBLEMS
     MISCELLANEOUS PROBLEMS

APPENDIX  SYSTEMS OF SURFACES
     1. One- parameter Systems
     2. Two -parameter Systems
     3. The Edge of Regression
     4. Ruled Surfaces

SOLUTIONS TO THE ODD-NUMBERED PROBLEMS
     
INDEX