《The Theory of Potential and Spherical Harmonics》

INTRODUCTION1

Vectors and Scalars1

Ⅰ．THE NEWTONIAN LAW OF GRAVITY7

1．The Newtonian law7

2．The force field9

3．Examples and exercises11

4．Force fields．Lines of force．Vector fields．Velocity fields15

Ⅱ．CONCEPT OF POTENTIAL19

1．Work．potential．Gradient of a scalar19

2．Newtonian potential of a body24

3．Logarithmic potential26

4．Newtonian potential of surfacedistributions and of double layers28

5．The Laplace equation31

6．Behaviour of the potential at infinity33

7．Harmonic functions．Regularity in the finite regions and at infinity39

8．Equipotential surfaces and lines of force42

9．Examples and problems43

Ⅲ．THE INTEGRAL THEOREMS OF POTENTIAL THEORY48

1．Gauss＇s theorem or divergence theorem48

2．Divergence．Solenoidal fields52

3．The flux of force through a closed surface56

4．Stokes＇ theorem58

5．Curl60

6．Green＇s formulas63

7．Representation of a harmonic function as a potential67

8．Gauss＇s mean value theorem74

Ⅳ．ANALYTIC CHARACTER OF THE POTENTIAL．SPHERICAL HARMONICS77

1．Analytic character of the potential77

2．Expansion of 1/r in spherical harmonics．Legendre polynomials83

3．Expansion of the Newtonian potential92

4．Derivation of the spherical harmonics in rectangular coordinates from Laplace＇s equation95

5．Surface spherical harmonics．Their differential equation．Associated functions97

6．Orthogonality102

7．The addition theorem for spherical harmonics109

8．Expansion of the potentials at infinity115

9．Exercises concerning Legendre functions123

Ⅴ．BEHAVIOUR OF THE POTENTIAL AT POINTS OF THE MASS126

1．Auxiliary considerations126

2．Continuity of the potential of a body and of its first derivatives130

3．Poisson＇s equation132

4．Continuity of potential of surface distribution135

5．Discontinuity of potential of a double layer136

6．Discontinuity of normal derivative at a surface distribution140

7．Normal derivative of potential of a double layer145

8．Analogous theorems for logarithmic potential153

9．Dirichlet＇s characteristic properties of potential156

10．Applications159

Ⅵ．RELATION OF POTENTIAL TO THEORY OF FUNCTIONS167

1．The conjugate potential167

2．Expansion in Cartesian and polar coordinates173

3．Converse of the ideas in Art．1175

4．Invariance of potential under conformal mapping177

Ⅶ．THE BOUNDARY VALUE PROBLEMS OF POTENTIAL THEORY180

1．Statement of the problems180

2．Uniqueness theorems183

3．The exterior problems185

4．The Dirichlet principle．Direct methods of calculus of variations187

Ⅷ．THE POISSON INTEGRAL IN THE PLANE193

1．Solution of the Dirichlet problem for the circle193

2．Expansion in the circle196

3．Expansion on the circumference of the circle199

4．Expansion of arbitrary functions in Fourier series．Bessel＇s and Schwarz＇s inequalities201

5．Expansion in a circular ring209

6．The equipotentials are analytic curves214

7．Harnack＇s theorems216

8．Harmonic continuation220

9．Green＇s function in the plane224

10．Green＇s function for the circle231

11．Green＇s function of the second kind．Characteristic function233

12．Conformal mapping and the Green＇s function235

Ⅸ．THE POISSON INTEGRAL IN SPACE245

1．Solution of the first boundary value problem for the sphere．The Harnack theorems for space245

2．Green＇s function in space247

3．Expansion of a harmonic function in spherical harmonics250

4．Expansion of an arbitrary function in surface spherical harmonics254

5．Expansion in Legendre polynomials257

Ⅹ．THE FREDHOLM THEORY OF INTEGRAL EQUATIONS259

1．The problem of integral equations259

2．The first theorem of Fredholm261

3．The minor determinants of D(λ)267

4．The second theorem of Fredholm270

5．The third theorem of Fredholm276

6．The iterated kernels280

7．Proof of Hadamard＇s theorem281

8．Kernels which are not bounded283

Ⅺ．GENERAL SOLUTION OF THE BOUNDARY VALUE PROBLEMS287

1．Reduction to integral equations287

2．The existence theorems291

3．The first boundary value problem for the exterior295

4．Boundedness of the third iterated kernel305

INDEX311