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Double layer potential

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210: 361: 72: 253: 205:{\displaystyle u(\mathbf {x} )={\frac {-1}{4\pi }}\int _{S}\rho (\mathbf {y} ){\frac {\partial }{\partial \nu }}{\frac {1}{|\mathbf {x} -\mathbf {y} |}}\,d\sigma (\mathbf {y} )} 356:{\displaystyle u(\mathbf {x} )=\int _{S}\rho (\mathbf {y} ){\frac {\partial }{\partial \nu }}P(\mathbf {x} -\mathbf {y} )\,d\sigma (\mathbf {y} )} 446: 486: 467: 481: 462: 403: 503: 37: 457: 33: 476: 388: 29: 438: 442: 393: 372: 17: 416: 247: 398: 224:
denotes the directional derivative in the direction of the outward unit normal in the
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More generally, a double layer potential is associated to a
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in three-dimensions. Thus a double layer potential
355: 204: 8: 228:variable, and dĪƒ is the surface measure on 425:Methods of Mathematical Physics, Volume II 345: 335: 327: 319: 298: 290: 278: 263: 255: 194: 184: 176: 171: 163: 158: 152: 137: 129: 117: 93: 82: 74: 7: 304: 300: 143: 139: 14: 216:denotes the dipole distribution, 44:distribution on a closed surface 346: 328: 320: 291: 264: 195: 172: 164: 130: 83: 435:Foundations of potential theory 59:is a scalar-valued function of 350: 342: 332: 316: 295: 287: 268: 260: 199: 191: 177: 159: 134: 126: 87: 79: 1: 482:Encyclopedia of Mathematics 475:Solomentsev, E.D. (2001) , 463:Encyclopedia of Mathematics 520: 456:Shishmarev, I.A. (2001) , 404:Laplacian of the indicator 458:"Double-layer potential" 433:Kellogg, O. D. (1953), 477:"Multi-pole potential" 389:Single layer potential 357: 206: 26:double layer potential 358: 207: 32:corresponding to the 427:, Wiley-Interscience 254: 73: 439:Dover Publications 353: 202: 38:magnetic potential 30:Laplace's equation 448:978-0-486-60144-1 311: 182: 150: 111: 28:is a solution of 511: 504:Potential theory 489: 470: 451: 428: 417:Courant, Richard 394:Potential theory 373:Newtonian kernel 362: 360: 359: 354: 349: 331: 323: 312: 310: 299: 294: 283: 282: 267: 211: 209: 208: 203: 198: 183: 181: 180: 175: 167: 162: 153: 151: 149: 138: 133: 122: 121: 112: 110: 102: 94: 86: 68: 58: 40:associated to a 18:potential theory 519: 518: 514: 513: 512: 510: 509: 508: 494: 493: 474: 455: 449: 432: 415: 412: 385: 303: 274: 252: 251: 248:Euclidean space 157: 142: 113: 103: 95: 71: 70: 60: 49: 12: 11: 5: 517: 515: 507: 506: 496: 495: 492: 491: 472: 453: 447: 430: 421:Hilbert, David 411: 408: 407: 406: 401: 399:Electrostatics 396: 391: 384: 381: 352: 348: 344: 341: 338: 334: 330: 326: 322: 318: 315: 309: 306: 302: 297: 293: 289: 286: 281: 277: 273: 270: 266: 262: 259: 201: 197: 193: 190: 187: 179: 174: 170: 166: 161: 156: 148: 145: 141: 136: 132: 128: 125: 120: 116: 109: 106: 101: 98: 92: 89: 85: 81: 78: 13: 10: 9: 6: 4: 3: 2: 516: 505: 502: 501: 499: 488: 484: 483: 478: 473: 469: 465: 464: 459: 454: 450: 444: 440: 436: 431: 426: 422: 418: 414: 413: 409: 405: 402: 400: 397: 395: 392: 390: 387: 386: 382: 380: 378: 374: 370: 366: 339: 336: 324: 313: 307: 284: 279: 275: 271: 257: 249: 246:-dimensional 245: 241: 238: 233: 231: 227: 223: 219: 215: 188: 185: 168: 154: 146: 123: 118: 114: 107: 104: 99: 96: 90: 76: 67: 63: 56: 52: 47: 43: 39: 35: 34:electrostatic 31: 27: 23: 20:, an area of 19: 480: 461: 437:, New York: 434: 424: 379:dimensions. 376: 368: 364: 250:by means of 243: 239: 237:hypersurface 234: 229: 225: 221: 217: 213: 65: 61: 54: 50: 45: 25: 15: 22:mathematics 410:References 487:EMS Press 468:EMS Press 371:) is the 340:σ 325:− 308:ν 305:∂ 301:∂ 285:ρ 276:∫ 189:σ 169:− 147:ν 144:∂ 140:∂ 124:ρ 115:∫ 108:π 97:− 69:given by 498:Category 423:(1962), 383:See also 445:  363:where 212:where 42:dipole 443:ISBN 24:, a 375:in 242:in 36:or 16:In 500:: 485:, 479:, 466:, 460:, 441:, 419:; 232:. 222:∂ÎŊ 64:∈ 490:. 471:. 452:. 429:. 377:n 369:y 367:( 365:P 351:) 347:y 343:( 337:d 333:) 329:y 321:x 317:( 314:P 296:) 292:y 288:( 280:S 272:= 269:) 265:x 261:( 258:u 244:n 240:S 230:S 226:y 220:/ 218:∂ 214:Ī 200:) 196:y 192:( 186:d 178:| 173:y 165:x 160:| 155:1 135:) 131:y 127:( 119:S 105:4 100:1 91:= 88:) 84:x 80:( 77:u 66:R 62:x 57:) 55:x 53:( 51:u 46:S

Index

potential theory
mathematics
Laplace's equation
electrostatic
magnetic potential
dipole
hypersurface
Euclidean space
Newtonian kernel
Single layer potential
Potential theory
Electrostatics
Laplacian of the indicator
Courant, Richard
Hilbert, David
Dover Publications
ISBN
978-0-486-60144-1
"Double-layer potential"
Encyclopedia of Mathematics
EMS Press
"Multi-pole potential"
Encyclopedia of Mathematics
EMS Press
Category
Potential theory

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