本书是“索茱菲理论物理教程”的第六卷，主题是物理学中的偏微分方程。“索末菲理论物理教程”包括力学、变形介质力学、电动力学、光学、热力学与统计物理、物理学中的偏微分方程六卷，是作者给Muenchen大学和理工学院物理专业大三、大四学生讲课的手稿整理而成的。索末菲老师教书是物理数学融合在一起的，关键是他还能实验物理和理论物理一起教。索末菲的教学应该是深刻地影响到了不少人。他的教学非同于一般的教书匠，其教科书里融入了自己的理解还有自己对学术的贡献。索未菲对物理学的贡献是多方面的，即便面对爱因斯坦这样的。

阿诺德·索末菲（Arnold Sommerfeld，1868-1951），Sommerfeld是德国伟大的理论物理学家、应用数学家、流体力学家、教育家、原子物理与量子物理的创始人之一。他对理论物理多个领域，包括力学、光学、热力学、统计物理、原子物理、固体物理（包括金属物理）等有重大贡献，在偏微分方程、数学物理等应用数学领域也有重要贡献。他引进了第二量子数（角量子数）、第四量子数（自旋量子数）和精细结构常数，等等。20世纪最伟大的物理学家之一Planck在获得1918年度诺贝尔物理学奖的颁奖典礼的仪式上的演讲中指出：“Sommerfeld…便可以得到一个重要公式，这个公式能够解开氢与氢光谱的精细结构之谜，而且现在最精确的测量……一般地也能通过这个公式来解释……这个成就完全可以和海王星的著名发现相媲美。早在人类看到这颗行星之前Leverrier就计算出它的存在和轨道。”
Sommerfeld思想深刻，研究成果影响深远。例如，他去世后发展起来的数值广义相对论和新近崛起的引力波理论研究中，还引用“Sommerfeld条件”，该条件在求解中发挥了重要作用。这再次彰显了他的科学工作的巨大价值。

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. Anhar-monic Fourier Analysis. An Example of Non-Final Determination of Coefficients
A. Oscillating and Osculating Approximation
B. Anharmonic Fourier Analysis
C. An Example of a 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
A. Hyperbolic Differential Equations
B. Elliptic Differential Equations
C. Parabolic Differential Equations
10. Green's Theorem and Green's Function for Linear, and, in Particular, for Elliptic Differential Equations
A. Definition of the Adjoint Differential Expression
B. Green's Theorem for an Elliptic Differential Equation in its NormalForm
C. Definition of a Unit Source and of the Principal Solution
D. The Analytic Character of the Solution of an Elliptic Differential Equation
E. The Principal Solution for an Arbitrary Number of Dimensions
F. Definition of Green's Function for Self-Adjoint Differential Equations
11. Riemann's Integration of the Hyperbolic Differential Equation
12. Green's Theorem in Heat Conduction. The Principal Solution of HeatConduction
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
A. The Bessel Function and its Integral Representation
B. The Hankel Function and its Integral Representation
C. Series Expansion at the Origin
D. Recursion Formulas
E. Asymptotic Representation of the Hankel Functions
20. Heat Equalization in a Cylinder
A. One-Dimensional Case f = f(r)
B.Two-Dimensional Case f =f(r,)
C. Thrce-Dimensional Case f = f(r,P,z)
21. More About Bessel Functions
A. Generating Function and Addition Theorems
B. Integral Representations in Terms of Bessel Functions
C. The Indices n + ? and n ±
D. Generalization of the Saddle-Point Method According to Debye
22. Spherical Harmonics and Potential Theory
A. The Generating Function
B. Differential and Difference Equation
C. Associated Spherical Harmonics
D. On Associated Harmonics with Negative Index m
E. Surface Spherical Harmonics and the Representation of Arbitrary Functions
F. Integral Representation of Spherical Harmonics
G. A Recursion Formula for the Associated Harmonics
H. On the Normalization of Associated Harmonics
J. The Addition Theorem of Spherical Harmonics
23. Green's Function of Potential Theory for the Sphere. Sphere and Circle Problems for Other Differential Equations
A. Geometry of Reciprocai Radii
B. The Boundary Value Problem of Potential Theory for the Sphere, ThePoisson Integral
C. General Remarks about Transformations by Reciprocal Radii
D. Spherical Inversion in Potential Theory
E. The Breakdown of Spherical Inversion for the Wave Equation
24. More About Spherical Harmonics
A. The Plane Wave and the Spherical Wave in Space
B. Asymptotic Behavior
C. The Spherical Harmonic as an Electric Multipole
D. Some Remarks about the Hypergeometric Function
E. Spherical Harmoni