The theory of the potential:
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| Format: | Book |
| Language: | English |
| Published: |
New York, NY
Dover Publ.
1958
|
| Edition: | Unabridged and unaltered republ. of the 1. ed. |
| Series: | Dover books on physics, engineering
Theoretical mechanics |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIII, 469 S. graph. Darst. |
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MARC
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| 100 | 1 | |a MacMillan, William D. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a The theory of the potential |c by William Duncan MacMillan |
| 250 | |a Unabridged and unaltered republ. of the 1. ed. | ||
| 264 | 1 | |a New York, NY |b Dover Publ. |c 1958 | |
| 300 | |a XIII, 469 S. |b graph. Darst. | ||
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| 490 | 0 | |a Dover books on physics, engineering | |
| 490 | 0 | |a Theoretical mechanics | |
| 650 | 4 | |a Potential theory (Mathematics) | |
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Record in the Search Index
| _version_ | 1804116679177797632 |
|---|---|
| adam_text | CONTENTS
Paqb
Preface v
CHAPTER I
The Attraction of Finite Bodies
Section
1. The Law of Gravitation 1
2. The Attraction of Systems of Particles 1
3. The Components of Attraction 1
4. The Elements of Mass 2
5. Attraction on a Point 3
6. The Attraction of a Circular Arc on Its Center 3
7. The Attraction of a Straight Line on a Point 4
8. The Attraction of a Thin Sheet on Its Axis of Symmetry 5
9. The Attraction of the Frustum of a Cone on Its Apex 7
10. Perspectivity 8
11. The Attraction of an Ellipsoidal Homoeoid upon an Interior Point 10
12. The Attraction of a Spherical Shell upon an Exterior Particle 11
13. The Attraction of a Solid Sphere upon an Exterior Point .... 14
14. The Mutual Attraction of Two Straight Collinear Rods 14
15. The Attraction of a Circular Disk on Its Axis 15
16. The Attraction of a Body of Revolution on a Point in Its Axis 16
17. Example—The Oblate Spheroid 17
18. The Attraction of a Uniform Rectangular Plate on a Point in Its
Own Plane 19
19. The Attraction betweerf Two Rigid Bodies 21
Problems 21
CHAPTER II
The Newtonian Potential Function
20. The Potential Function Denned 24
21. The Significance of the Potential Function 25
22. The Potential Function Exists 26
23. The Existence of Derivatives of the Potential 27
24. Existence of Derivatives at Exterior Points 27
25. Existence of Derivatives at Interior Points 29
26. The Equation of Laplace 32
27. Equipotential Surfaces, or Level Surfaces 34
28. The Logarithmic Potential 35
29. The Potential of a Spherical Shell 36
vii
viii CONTENTS
Section Page
30. Potential of a Uniform Circular Disk along Its Axis 41
31. Potential of a Homogeneous Straight Rod 42
32. The Potential of a Homogeneous Solid Ellipsoid for Interior Points 45
33. The Equipotential Surfaces 50
34. The Components of Attraction at an Interior Point 51
35. The Attraction of a Solid Homogeneous Ellipsoid upon an Exterior
Point—Ivory s Method 52
36. The Potential of a Homogeneous Solid Ellipsoid at Exte¬
rior Points 56
37. Evaluation of the Elliptic Integrals 58
38. MacLaurin s Theorem 60
39. The Potential of Spheroids at Exterior Points 62
40. The Attraction of a Spheroid at the Surface 63
41. The Attraction Is a Maximum 66
42. The Potential of Homogeneous Elliptic Cylinders 69
43. The Potential of a Homogeneous, Rectangular Parallelopiped 72
44. The Components of Force for the Right Parallelopiped 79
45. A Generalization Regarding Derivatives of a Potential 80
46. The Potential of a Body at a Distant Point 81
47. The Terms of Higher Degrees 85
48. The Expansion for the Homogeneous Ellipsoid 87
49. The Right Parallelopiped 88
50. The Inertial Integrals 89
51. The Inertial Integrals Cannot All Vanish 91
52. A Body Is Uniquely Denned by Its Inertial Integrals 94
Problems 94
CHAPTER III
Vector Fields. Theorems of Green and Gauss
53. Definitions 96
54. The Normal Derivative 97
55. Relations between Certain Volume and Surface Integrals. ... 99
56. A Vector Interpretation • 101
57. Generalized Orthogonal Coordinates 102
58. Green s Theorem 104
59. The Potential of Homogeneous Bodies 106
60. Example—A Non Homogeneous Spherical Shell 107
61. Existence of Higher Derivatives of Potential Functions 109
62. Harmonic Functions Ill
63. An Extension of Green s Theorem for Harmonic Functions . . . Ill
64. Reduction to Two Dimensions 115
65. Analogy with Cauchy s Theory of Residues 117
66. The Surface Integral of the Normal Derivatives of 1/p 120
67. The Contour Integral of the Normal Derivative of log p 122 ;
68. A Theorem of Gauss 123
69. Poisson s Equation 124
70. Poisson s Equation in Two Dimensions 126
CONTENTS ix
Section Page
71. An Extension of Gauss Theorem 126
72. Green s Theorem Applied to Two Potential Functions 128
73. Characteristic Properties of a Potential Function 130
74. The Average Value of a Potential Function over a Sphere . . . 132
75. Maxima and Minima of Harmonic Functions 133
76. The Potential Energy of a Finite Mass 136
77. The Potential Energy of a Homogeneous Sphere 138
78. The Heat of the Sun 139
79. Relation between Certain Surface and Line Integrals 140
80. Stokes Theorem 143
81. Examples of Vector Curls 144
82. The Vector and Its Curl Are Orthogonal 145
83. Condition That a Line Integral Shall Be Independent of the Path
of Integration 148
84. Condition That a Surface Integral Shall Depend upon the Contour
Only 149
Problems 152
CHAPTER IV
The Attuactions op Surfaces and Lines
85. The Occasion for Their Study 153
Attractions of Surfaces
86. A Uniform Disk 153
87. An Infinite Homogeneous Universe 155
88. Proper and Improper Integrals 157
89. Semi convergent Integrals 161
90. The Potential at a Point of the Surface 165
91. The Potential Is Continuous across the Surface 166
92. The Normal Component of the Attraction Is Discontinuous across
the Surface 168
93. The Tangential Components of the Attraction Are Continuous 171
94. Discontinuities in the Derivatives of Surface Potentials 174
95. Example—A Non homogeneous Disk 176
96. Discontinuities in the Second Derivatives of Surface Potentials . 178
97. Singular Points of the Surface 189
Attractions of Lines
98. A Straight Rod 190
99. The Components of Attraction 191
100. Attraction in the Line Is Not Well Denned 193
101. Asymptotic Expression for the Potential 194
102. The Potential of a Uniform Hoop 195
103. Evaluation of the Potential According to Gauss 197
104. Asymptotic Expression for the Potential 200
CHAPTER V
Surface Distributions of Matter
105. Transformation by Reciprocal Radii 204
106. Application of the Transformation to Potentials 207
X CONTENTS
Suction Page
107. The Potential of a Uniform Distribution of Matter on a Sphere 209
108. A Non uniform Spherical Distribution 209
109. Inversion of a Homogeneous Ellipsoidal Shell 211
110. Centrobaric Bodies 212
111. The Center of Gravity of Centrobaric Bodies 213
112. The Central Ellipsoid of Inertia 214
113. A System of Detached Masses Cannot Be Centrobaric 215
114. Theorems Relating to Electric Images 216
115. Level Layers 219
116. Families of Level Layers 221
117. Level Layer on an Arbitrarily Given Surface 222
118. Robin s Integral Equation 227
119. Picard s Solution of Robin s Equation 228
120. Example of a Level Layer 231
121. Level Layers on Prolate Spheroids 233
122. Level Layers on Ellipsoids 235
123. The Potential of Ellipsoidal Level Layers 239
124. Layers of Finite Thickness 240
125. A Finite Shell Bounded by Confocal Spheroids 241
126. The Surface Density Necessary to Produce Given Potentials 244
127. Green s Problem 246
128. Certain Physical Considerations 247
129. The Existence of Green s Function 248
130. Miscellaneous Properties of Green s Function 249
131. The Green Function Is Symmetric 252
132. The Normal Derivative of Green s Function Is Harmonic. . . . 253
133. The Green Function for the Sphere 254
134. The Normal Derivative on the Sphere 257
135. Green s Equation for the Sphere 260
136. A Generalization for the Sphere 261
137. Green s Equation for any Closed Surface 265
138. A General Theorem of Green s 269
Green s Problem for the Logarithmic Potential
139. Statement of the Problem 269
140. Electric Images for the Logarithmic Potential 270
141. Electric Images of Centrobaric Bodies 271
142. The Existence of Green s Function 274
143. Green s Function for the Circle 276
144. The Principle of Dirichlet and Lord Kelvin 277
145. The Equivalent Problem of Poincar6 280
Problems 282
CHAPTER VI
Two layer Surfaces
The Methods of Neumann and Poincar6
146. Various Types of Mass 283
147. The Magnetic Doublet 283
CONTENTS xi
Suction Pagb
148. The Bar Magnet 285
149. Magnetic Sheets—Two layer Surfaces 285
150. Closed, Uniform, Two layer Surfaces 286
151. Uniform Surfaces Not Closed 288
152. Surfaces with Variable Moments 290
153. Discontinuities in the First Derivatives 292
154. The Configuration Constant of a Closed Surface 293
155. The Spread of Values of the Potential on a Closed, Convex Surface 295
156. Neumann s Proof of Dirichlet s Principle 296
157. The Limiting Values of the Potentials Wn on S 299
158. Harnack s Theorem for Harmonic Functions 300
159. Case I—The Constant C Is Zero 305
160. The Interior and Exterior Functions as Potentials of the Same
Simple Layer 307
161. The Constant C Is Not Zero 309
162. The Construction of a Simple, Level Layer on S 311
Poincar^ s Me thode du Balayage
163. The Balayage of a Sphere 313
164. Existence of a Level Layer on a Given Surface 314
165. Application of Harnack s Theorem 318
166. Construction of an Infinite System of Spheres within S 319
167. The Existence of the Required Harmonic Function 320
Problems 323
CHAPTER VII
Spherical Harmonics
168. Definitions 325
169. Examples of Spherical Harmonics 326
170. Homogeneous, Harmonic Polynomials 326
171. Relation between Certain Harmonics 328
172. The Expansion of a Potential 328
173. Rotation about an Imaginary Axis 330
174. Harmonics Which Depend upon r and z Alone 331
175. The Equation of Laplace for Surface Harmonics 332
176. Zonal Harmonics 333
177. The Polynomials of Legendre 335
178. The Expansion in Taylor s Series 337
179. The Expansion in Lagrange s Series 338
180. Zonal Harmonics Given Explicitly 339
181. The Zeros of the Zonal Harmonics Are All Real 340
182. Certain Useful Relations 342
183. The Zonal Harmonics Are Orthogonal Functions 343
184. A Generalization of the Preceding Formulas 345
185. A Recursion Formula for Zonal Harmonics 346
186. The General Formula for Zonal Harmonics 347
187. The General Expression for #» » 349
188. Zonal Harmonics Expressed by Cosines of Multiples of the
Argument 350
xii CONTENTS
Section Page
189. Powers of m Expressed in Terms of Zonal Harmonics 352
190. A Definite Integral Representation of Zonal Harmonics 355 ,
191. An Important Property of Zonal Harmonics 356
192. Expansion of Sin trup in a Series of Zonal Harmonics 357
193. The Potential of a Solid of Revolution 360
194. The Oblate Spheroid 363
195. The Apparent Size of a Plane Circular Disk 365
196. The Potential of a Zonal Distribution of Matter on a Spherical
Surface 366
197. Tesseral Harmonics 368
198. Examples of Solid Tesseral Harmonics 371
199. The Zeros of the Tesseral Harmonics 372
200. The Surface Integral of the Product of Two Spherical Harmonics
of Different Degrees 373
201. The Surface Integral of the Product of Two Spherical Harmonics
of the Same Degree 374
202. The Expansion of 1/p in a Series of Tesseral Harmonics 376
203. The Expansion of the Potential of a Finite Body in a Series of
Tesseral Harmonics 380
204. The Expansion of the Potential of a Finite Body in a Series of
Inertial Integrals 382
205. Laplace s Integral Equation 384
206. The Expansion of an Arbitrary Function in a Series of Spherical
Harmonics 387
207. The Representation of a Rational, Integral Function 390
208. Green s Problem for the Sphere 392
209. The Potential of a Surface Distribution of Matter on a Sphere . . 393
210. Differentiation with Respect to Poles 395
211. Derivation of the Tesseral Harmonics by Polar Differentiation 398
Problems 404
CHAPTER VIII
Ellipsoidal Harmonics
212. Introduction • 407
213. Definition of the Elliptic Coordinates 407
214. Differential Relations 409
215. The Equation of Laplace 410
216. The Elliptic Functions of Weierstrass 412
217. Spherical Harmonics in Elliptic Coordinates 415
218. The Inverse Problem 416
219. The Functions of Lame1 418
220. Determination of the Constant N 419
221. Existence of Solutions for Class 1 420
222. Existence of Solutions for Class II 422
223. Existence of Solutions for Class III 423
224. Existence of .Solutions for Class IV 425
225. The Products of Lame 426
226. Liouville s Proof That All of the Roots Are Real 427
CONTENTS xiii
Section Paoe
227. Particular Examples of Lame s Functions 429
228. The Pattern as a Function of q, 435
229. Parametric Representations of a Sphere 436
230. The Ellipsoidal Harmonics as a Function of X 438
231. The Surface Harmonic V^V, 439
232. The Spheroidal Surface Harmonic VtVs 441
233. The Roots of the Characteristic Equation Considered as Functions
of t 443
234. The Characteristic Equation Has No Multiple Roots 445
235. The Functions of Lame Are Linearly Independent 446
236. The Expansion of An Arbitrary Function in Terms of the Ellip¬
soidal Harmonics 447
237. Surface Integrals 449
238. The Coefficients of an Expansion in Terms of Ellipsoidal
Harmonics 451
239. The Roots of Lamp s Polynomials Are Real, Distinct, and Lie
between a! and c* 452
240. Ellipsoidal Harmonics of the Second Kind 454
241. The Potential of an Ellipsoidal Harmonic Surface Distribution
of Matter 456
242. The Potential of an Ellipsoidal Homoeoid 457
243. Green s Problem on an Ellipsoid 459
Extension of the General Theory
244. Fundamental Functions 460
Bibliography 463
Index 467
|
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| id | DE-604.BV002224080 |
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| indexdate | 2024-07-09T15:42:20Z |
| institution | BVB |
| language | English |
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| physical | XIII, 469 S. graph. Darst. |
| publishDate | 1958 |
| publishDateSearch | 1958 |
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| publisher | Dover Publ. |
| record_format | marc |
| series2 | Dover books on physics, engineering Theoretical mechanics |
| spelling | MacMillan, William D. Verfasser aut The theory of the potential by William Duncan MacMillan Unabridged and unaltered republ. of the 1. ed. New York, NY Dover Publ. 1958 XIII, 469 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Dover books on physics, engineering Theoretical mechanics Potential theory (Mathematics) Potenzialtheorie (DE-588)4046939-6 gnd rswk-swf Potenzialtheorie (DE-588)4046939-6 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001461422&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | MacMillan, William D. The theory of the potential Potential theory (Mathematics) Potenzialtheorie (DE-588)4046939-6 gnd |
| subject_GND | (DE-588)4046939-6 |
| title | The theory of the potential |
| title_auth | The theory of the potential |
| title_exact_search | The theory of the potential |
| title_full | The theory of the potential by William Duncan MacMillan |
| title_fullStr | The theory of the potential by William Duncan MacMillan |
| title_full_unstemmed | The theory of the potential by William Duncan MacMillan |
| title_short | The theory of the potential |
| title_sort | the theory of the potential |
| topic | Potential theory (Mathematics) Potenzialtheorie (DE-588)4046939-6 gnd |
| topic_facet | Potential theory (Mathematics) Potenzialtheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001461422&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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