Partial Differential Equations in Physics - 1st Edition - ISBN: 9780126546569, 9780080873091

Partial Differential Equations in Physics, Volume 1

1st Edition

Series Volume Editors: Arnold Sommerfeld
eBook ISBN: 9780080873091
Imprint: Academic Press
Published Date: 1st January 1949
Page Count: 334
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Table of Contents

Pure and Applied Mathematics


Editors’ Foreword


Chapter I: Fourier Series and Integrals

§1 Fourier Series

§2 Example of a Discontinuous Function. Gibbs’ Phenomenon and Non-Uniform Convergence

§3 On the Convergence of Fourier Series

§4 Passage to the Fourier Integral

§5 Development by Spherical Harmonics

§6 Generalizations: Oscillating and Osculating Approximations. Anharmonic Fourier Analysis. An Example of Non-Final Determination of Coefficients

Chapter II: Introduction to Partial Differential Equations

§ 7 How The Simplest Partial Differential Equations Arise

§ 8 Elliptic, Hyperbolic and Parabolic Type. Theory of Characteristics

§ 9 Differences Among Hyperbolic, Elliptic, and Parabolic Differential Equations. The Analytic Character of Their Solutions

§10 Green’s Theorem and Green’s Function for Linear, and, in Particular, for Elliptic Differential Equations

§11 Riemann’s Integration of the Hyperbolic Differential Equation

§12 Green’s Theorem in Heat Conduction. The Principal Solution of Heat Conduction

Chapter III: Boundary Value Problems in Heat Conduction

§13 Heat Conductors Bounded on One Side

§14 The Problem of the Earth’s Temperature

§15 The Problem of a Ring-Shaped Heat Conductor

§16 Linear Heat Conductors Bounded on Both Ends

§ 17 Reflection in the Plane and in Space

§ 18 Uniqueness of Solution for Arbitrarily Shaped Heat Conductors

Chapter IV: Cylinder and Sphere Problems

§ 19 Bessel and Hankel Functions

§ 20 Heat Equalization in a Cylinder

§ 21 More about Bessel Functions

§ 22 Spherical Harmonics and Potential Theory

§ 23 Green’s Function of Potential Theory for the Sphere. Sphere and Circle Problems for Other Differential Equations

§ 24 More about Spherical Harmonics

Appendix I Reflection on a Circular-Cylindrical or Spherical Mirror

Appendix II Additions to the Riemann Problem of Sound Waves in §11

Chapter V: Eigenfunctions and Eigen Values

§ 25 Eigen Values and Eigenfunctions of the Vibrating Membrane

§ 26 General Remarks Concerning the Boundary Value Problems of Acoustics and of Heat Conduction

§ 27 Free and Forced Oscillations. Green’s Function for the Wave Equation

§ 28 Infinite Domains and Continuous Spectra of Eigen Values. The Condition of Radiation

§ 29 The Eigen Value Spectrum of Wave Mechanics. Balmer’s Term

§ 30. Green’s Function for the Wave Mechanical Scattering Problem. The Rutherford Formula of Nuclear Physics

Appendix I Normalization of yhe Eigenfunctions in the Infinite Domain

Appendix II A New Method for the Solution of the Exterior Boundary Value Problem of the Wave Equation Presented for the Special Case of the Sphere

Appendix III The Wave Mechanical Eigenfunctions of the Scattering Problem in Parabolic Coordinates

Appendix IV Plane And Spherical Waves in Unlimited Space of an Arbitrary Number of Dimensions

Chapter VI: Problems of Radio

§31 The Hertz Dipole in a Homogeneous Medium Over a Completely Conductive Earth

§32 The Vertical Antenna Over an Arbitrary Earth

§33 The Horizontal Antenna Over an Arbitrary Earth

§ 34 Errors in Range Finding for an Electric Horizontal Antenna

§ 35 § 35. The Magnetic or Frame Antenna

§ 36 Radiation Energy and Earth Absorption

Appendix Radio Waves On The Spherical Earth

Exercises For Chapter I

Exercises For Chapter II

Exercises For Chapter III

Exercises For Chapter IV

Exercises For Chapter V

Exercises For Chapter VI

Hints for Solving the Exercises



The topic with which I regularly conclude my six-term series of lectures in Munich is the partial differential equations of physics. We do not really deal with mathematical physics, but with physical mathematics; not with the mathematical formulation of physical facts, but with the physical motivation of mathematical methods. The oftmentioned “prestabilized harmony” between what is mathematically interesting and what is physically important is met at each step and lends an esthetic - I should like to say metaphysical -- attraction to our subject.

The problems to be treated belong mainly to the classical matherhatical literature, as shown by their connection with the names of Laplace, Fourier, Green, Gauss, Riemann, and William Thomson. In order to show that these methods are adequate to deal with actual problems, we treat the propagation of radio waves in some detail in Chapter VI.


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About the Series Volume Editors

Arnold Sommerfeld Series Volume Editor

Affiliations and Expertise

University of Munich