The Theory of Potential and Spherical HarmonicsPDF电子书下载

INTRODUCTION 1

Vectors and Scalars 1

Ⅰ．THE NEWTONIAN LAW OF GRAVITY 7

1．The Newtonian law 7

2．The force field 9

3．Examples and exercises 11

4．Force fields．Lines of force．Vector fields．Velocity fields 15

Ⅱ．CONCEPT OF POTENTIAL 19

1．Work．potential．Gradient of a scalar 19

2．Newtonian potential of a body 24

3．Logarithmic potential 26

4．Newtonian potential of surface distributions and of double layers 28

5．The Laplace equation 31

6．Behaviour of the potential at infinity 33

7．Harmonic functions．Regularity in the finite regions and at infinity 39

8．Equipotential surfaces and lines of force 42

9．Examples and problems 43

Ⅲ．THE INTEGRAL THEOREMS OF POTENTIAL THEORY 48

1．Gauss＇s theorem or divergence theorem 48

2．Divergence．Solenoidal fields 52

3．The flux of force through a closed surface 56

4．Stokes＇ theorem 58

5．Curl 60

6．Green＇s formulas 63

7．Representation of a harmonic function as a potential 67

8．Gauss＇s mean value theorem 74

Ⅳ．ANALYTIC CHARACTER OF THE POTENTIAL．SPHERICAL HARMONICS 77

1．Analytic character of the potential 77

2．Expansion of 1/r in spherical harmonics．Legendre polynomials 83

3．Expansion of the Newtonian potential 92

4．Derivation of the spherical harmonics in rectangular coordinates from Laplace＇s equation 95

5．Surface spherical harmonics．Their differential equation．Associated functions 97

6．Orthogonality 102

7．The addition theorem for spherical harmonics 109

8．Expansion of the potentials at infinity 115

9．Exercises concerning Legendre functions 123

Ⅴ．BEHAVIOUR OF THE POTENTIAL AT POINTS OF THE MASS 126

1．Auxiliary considerations 126

2．Continuity of the potential of a body and of its first derivatives 130

3．Poisson＇s equation 132

4．Continuity of potential of surface distribution 135

5．Discontinuity of potential of a double layer 136

6．Discontinuity of normal derivative at a surface distribution 140

7．Normal derivative of potential of a double layer 145

8．Analogous theorems for logarithmic potential 153

9．Dirichlet＇s characteristic properties of potential 156

10．Applications 159

Ⅵ．RELATION OF POTENTIAL TO THEORY OF FUNCTIONS 167

1．The conjugate potential 167

2．Expansion in Cartesian and polar coordinates 173

3．Converse of the ideas in Art．1 175

4．Invariance of potential under conformal mapping 177

Ⅶ．THE BOUNDARY VALUE PROBLEMS OF POTENTIAL THEORY 180

1．Statement of the problems 180

2．Uniqueness theorems 183

3．The exterior problems 185

4．The Dirichlet principle．Direct methods of calculus of variations 187

Ⅷ．THE POISSON INTEGRAL IN THE PLANE 193

1．Solution of the Dirichlet problem for the circle 193

2．Expansion in the circle 196

3．Expansion on the circumference of the circle 199

4．Expansion of arbitrary functions in Fourier series．Bessel＇s and Schwarz＇s inequalities 201

5．Expansion in a circular ring 209

6．The equipotentials are analytic curves 214

7．Harnack＇s theorems 216

8．Harmonic continuation 220

9．Green＇s function in the plane 224

10．Green＇s function for the circle 231

11．Green＇s function of the second kind．Characteristic function 233

12．Conformal mapping and the Green＇s function 235

Ⅸ．THE POISSON INTEGRAL IN SPACE 245

1．Solution of the first boundary value problem for the sphere．The Harnack theorems for space 245

2．Green＇s function in space 247

3．Expansion of a harmonic function in spherical harmonics 250

4．Expansion of an arbitrary function in surface spherical harmonics 254

5．Expansion in Legendre polynomials 257

Ⅹ．THE FREDHOLM THEORY OF INTEGRAL EQUATIONS 259

1．The problem of integral equations 259

2．The first theorem of Fredholm 261

3．The minor determinants of D(λ) 267

4．The second theorem of Fredholm 270

5．The third theorem of Fredholm 276

6．The iterated kernels 280

7．Proof of Hadamard＇s theorem 281

8．Kernels which are not bounded 283

Ⅺ．GENERAL SOLUTION OF THE BOUNDARY VALUE PROBLEMS 287

1．Reduction to integral equations 287

2．The existence theorems 291

3．The first boundary value problem for the exterior 295

4．Boundedness of the third iterated kernel 305

INDEX 311