{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 8 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "" 0 1 0 0 176 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "" 0 1 0 0 96 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "" 0 1 139 137 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 271 "" 0 1 80 76 84 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 272 "" 0 1 65 82 65 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 273 "" 0 1 0 0 32 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 274 "" 0 1 147 137 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 276 "" 0 1 0 0 72 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE " " -1 277 "" 0 1 146 137 2 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 278 "" 0 1 101 32 97 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 279 "" 0 1 0 0 22 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 280 "" 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 281 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE " " -1 282 "" 0 1 31 137 15 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 283 "" 0 1 104 101 116 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 284 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 285 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 286 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE " " -1 287 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 288 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 289 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 290 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 291 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 292 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 293 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 294 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 295 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 296 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 297 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 298 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 299 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 300 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 301 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 302 "" 0 1 101 91 116 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 303 "" 0 1 97 100 105 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 304 "" 0 1 105 116 105 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 305 "" 0 1 0 0 64 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 306 "" 0 1 123 137 117 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 307 "" 0 1 0 0 240 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 308 "" 0 1 0 0 22 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 309 "" 0 1 0 0 64 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 310 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 311 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 312 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 313 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 314 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 315 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 316 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 317 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 318 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 319 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 320 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 321 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 322 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 323 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 324 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 325 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 326 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 327 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 328 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 329 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 330 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 331 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 332 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 333 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 334 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 335 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 336 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 337 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 338 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 339 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 340 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 341 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 342 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 343 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 344 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 345 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 346 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 347 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 348 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 349 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 350 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 351 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 352 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 353 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 354 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 355 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 356 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 357 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 358 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 359 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 360 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 361 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 362 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 363 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 364 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 365 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 366 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 367 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 368 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 369 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 370 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 109 137 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 4 4 0 0 0 0 0 0 -1 0 } {PSTYLE "R3 Font 0" -1 256 1 {CSTYLE "" -1 -1 "Times" 0 14 0 0 0 1 2 1 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 2 " -1 257 1 {CSTYLE "" -1 -1 "Courier" 0 11 0 0 0 1 2 1 2 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 262 1 {CSTYLE " " -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 263 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 264 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 265 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 266 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 267 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 268 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 269 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE " " 0 270 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 271 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 272 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 273 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 274 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 275 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 276 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 277 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 278 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 279 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 280 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 281 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 282 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 283 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 284 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 285 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 286 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 287 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 288 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 289 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 290 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 291 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 292 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 293 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 294 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 295 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 296 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 297 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 298 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 299 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 300 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 301 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "Unit 4: Vector Calculus" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "Chapter \+ 22: Non-Cartesian Coordinates" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 52 "Section 22.2: vector operators in polar \+ coordinates" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 " " {TEXT -1 9 "Copyright" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "Copyright * 2001 by Addison Wesley Longman, Inc." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 302 "All righ ts reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, elec tronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Printed in the United Stat es of America." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 15 "Initializations" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 61 "with(linalg):\nwith(plots):\nwith(plottools) :\nread(`pvac.txt`):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 29 "Gradient in Polar Coordinates" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 " In cartesian coordinates, the gradient of the scalar " }{XPPEDIT 18 0 "u(x,y)" "6#-%\"uG6$%\"xG%\"yG" }{TEXT -1 14 " is the vector" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "g rad(u(x,y),[x,y]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "which is interpreted as the vector " }}{PARA 273 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u[x]" "6#&%\"uG6# %\"xG" }{TEXT -1 1 " " }{TEXT 256 1 "i" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "u[y]" "6#&%\"uG6#%\"yG" }{TEXT -1 1 " " }{TEXT 257 2 "j " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "where " } {TEXT 258 1 "i" }{TEXT -1 5 " and " }{TEXT 259 1 "j" }{TEXT -1 107 " a re unit basis vectors for the cartesian system. What does it mean to \+ compute the gradient of the scalar " }{XPPEDIT 18 0 "u(r,theta)" "6#-% \"uG6$%\"rG%&thetaG" }{TEXT -1 45 " in polar coordinates? What does M aple give?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 40 "grad(u(r,theta),[r,theta],coords=polar);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "Next, we explore how this vector was obtained, and what i t actually means." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 42 "Gra dient in Polar Coordinates - Derivation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "Recall the equations defining polar coordinates" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "X := r*cos(theta);\nY := r*sin(theta);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "and the equations defining the inverse transformation" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "R := sqrt(x^2+y^2);\nT := arctan(y/x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "Notice t he care with which name clashes were avoided." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "Write the radius vector \+ " }{TEXT 270 1 "R" }{TEXT -1 3 " = " }{TEXT 263 1 "x" }{TEXT -1 1 " " }{TEXT 271 1 "i" }{TEXT -1 3 " + " }{TEXT 298 1 "y" }{TEXT -1 1 " " } {TEXT 272 1 "j" }{TEXT -1 13 " in Maple as " }{TEXT 324 2 "Rc" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "x=x(r,theta)" "6#/%\"xG-F$6$%\"rG%&the taG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y=y(r,theta)" "6#/%\"yG-F$6$% \"rG%&thetaG" }{TEXT -1 13 ", obtaining " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "Rc := vector([X,Y]); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 22 " is held constant and " }{TEXT 288 1 "r" }{TEXT -1 18 " allowed to vary, " }{TEXT 273 1 "R" }{TEXT -1 3 " = " }{TEXT 274 1 "R" }{TEXT -1 1 "(" }{TEXT 323 1 "r" }{TEXT -1 9 "), so an " }{TEXT 284 1 "r" } {TEXT -1 35 "-coordinate curve is traced in the " }{TEXT 285 2 "xy" } {TEXT -1 26 "-plane. For example, set " }{XPPEDIT 18 0 "theta" "6#%&t hetaG" }{TEXT -1 6 " = 1, " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "R1 := subs(theta=1,op(Rc));" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "and plot the resulting " }{TEXT 301 1 "r" }{TEXT -1 80 "- coordinate curve which turns out to be a radial line emanating from th e origin." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 58 "plot([R1[1], R1[2], r = 0..1], color=black, labels= [x,y]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Alternatively, set " }{TEXT 286 1 "r" } {TEXT -1 13 " = 1 and let " }{XPPEDIT 18 0 "theta" "6#%&thetaG" } {TEXT -1 40 " vary, so the radius vector defines the " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 17 "-coordinate curve" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "R2 := sub s(r = 1, op(Rc));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "which, when plotted, is, a circle \+ centered at the origin." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "plot([R2[1], R2[2], theta = 0..2*Pi ], color=black, labels=[x,y], scaling=constrained);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 205 " A fundamental idea of vector calculus, now long familiar, is that diff erentiation of a curve with respect to its parameter yields a vector t angent to that curve. Apply that principle to the radius vector " } {TEXT 275 2 "R(" }{TEXT 322 1 "r" }{TEXT -1 1 "," }{XPPEDIT 18 0 "thet a" "6#%&thetaG" }{TEXT -1 41 "), first differentiating with respect to " }{TEXT 289 1 "r" }{TEXT -1 27 ", and then with respect to " } {XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 52 ". In the first case, obtain a vector tangent to an " }{TEXT 290 1 "r" }{TEXT -1 75 "-coord inate curve (a radial line) and in the second, a vector tangent to a \+ " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 45 "-coordinate curve, a circle about the origin." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "Figure 22.7, created below, shows a sketch of t he vectors " }{XPPEDIT 18 0 "diff(``,r)" "6#-%%diffG6$%!G%\"rG" } {TEXT 325 1 "R" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "diff(``,theta)" "6 #-%%diffG6$%!G%&thetaG" }{TEXT 326 1 "R" }{TEXT -1 10 " drawn at " } {XPPEDIT 18 0 "``(x,y)=``(sqrt(2),sqrt(2))" "6#/-%!G6$%\"xG%\"yG-F%6$- %%sqrtG6#\"\"#-F,6#F." }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 641 "p1 := circle([0,0],2): \np2 := plot([[0,0],[4*cos(Pi/4),4*sin(Pi/4)]],color=black):\np3 := ar row([sqrt(2),sqrt(2)],vector([1,1]),.1,.3,.2,color=black):\np4 := arro w([sqrt(2),sqrt(2)],vector([-2,2]),.1,.3,.2,color=black):\np5 := textp lot([.5,.2,`q`],font=[SYMBOL,10]):\np6 := textplot(\{[2.25,1.78,`R `], [-1,2.9,`R `]\},font=[TIMES,BOLD,12]):\np7 := textplot([2.45,1.72,`r ` ],font=[TIMES,ITALIC,10]):\np8 := textplot([-.8,2.8,`q `],font=[SYMBOL ,10]):\np9 := textplot([.2,3.6,`y`], font=[TIMES,ITALIC,12]):\ndisplay ([p||(1..9)],scaling=constrained, xtickmarks=5, ytickmarks=[-2,-1,0,1, 2,4], view=[-2..3,-2..4], labels=[x,``], labelfont=[TIMES,ITALIC,12]); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "The vector tangent to the " }{TEXT 287 1 "r" }{TEXT -1 77 "-coordinate curve is automatically a unit vector. The vector t angent to the " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 156 "-co ordinate curve needs to be normalized since it does not automatically \+ have unit length. Typical notation for these unit vectors in polar co ordinates is " }}{PARA 274 "" 0 "" {TEXT 329 1 "e" }{XPPEDIT 18 0 "``[ r]" "6#&%!G6#%\"rG" }{TEXT -1 3 " = " }{TEXT 334 1 "R" }{XPPEDIT 18 0 "``[r]" "6#&%!G6#%\"rG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "cos(theta) \+ " "6#-%$cosG6#%&thetaG" }{TEXT -1 1 " " }{TEXT 327 1 "i" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "sin(theta)" "6#-%$sinG6#%&thetaG" }{TEXT -1 1 " \+ " }{TEXT 328 1 "j" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 3 "and" }}{PARA 275 "" 0 "" {TEXT -1 1 " " } {TEXT 330 1 "e" }{XPPEDIT 18 0 "``[theta]=1/r" "6#/&%!G6#%&thetaG*&\" \"\"F)%\"rG!\"\"" }{TEXT -1 1 " " }{TEXT 331 1 "R" }{XPPEDIT 18 0 "``[ theta]=-sin(theta)" "6#/&%!G6#%&thetaG,$-%$sinG6#F'!\"\"" }{TEXT -1 1 " " }{TEXT 332 1 "i" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "cos(theta) " "6 #-%$cosG6#%&thetaG" }{TEXT -1 1 " " }{TEXT 333 1 "j" }{TEXT -1 1 " " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 145 "However , to avoid violating Lopez's Large Law (Never use as a name on the lef t, a variable in use on the right), name these unit tangent vectors " }{XPPEDIT 18 0 "E[r]" "6#&%\"EG6#%\"rG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "E[theta]" "6#&%\"EG6#%&thetaG" }{TEXT -1 10 " in Maple." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "First, di fferentiate with respect to " }{TEXT 291 1 "r" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "E[r] := map(diff,Rc,r);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 " Second, differentiate w ith respect to " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "Q := map( diff,Rc,theta);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "and normalize, obtaining" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "E [theta] := map(simplify,normalize(Q),symbolic);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "Solvin g for " }{TEXT 335 1 "i" }{TEXT -1 5 " and " }{TEXT 336 1 "j" }{TEXT -1 13 " in terms of " }{TEXT 337 1 "e" }{XPPEDIT 18 0 "``[r]" "6#&%!G6 #%\"rG" }{TEXT -1 5 " and " }{TEXT 338 1 "e" }{XPPEDIT 18 0 "``[theta] " "6#&%!G6#%&thetaG" }{TEXT -1 9 ", we have" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 276 "" 0 "" {TEXT 339 1 "i" }{TEXT -1 3 " = " } {XPPEDIT 18 0 "cos(theta)" "6#-%$cosG6#%&thetaG" }{TEXT -1 1 " " } {TEXT 340 1 "e" }{XPPEDIT 18 0 "``[r]" "6#&%!G6#%\"rG" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "-sin(theta)" "6#,$-%$sinG6#%&thetaG!\"\"" }{TEXT -1 1 " " }{TEXT 341 1 "e" }{XPPEDIT 18 0 "``[theta]" "6#&%!G6#%&thetaG" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "and" }}{PARA 277 "" 0 "" {TEXT 342 1 "j" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "sin(theta) " "6#-%$s inG6#%&thetaG" }{TEXT -1 1 " " }{TEXT 343 1 "e" }{XPPEDIT 18 0 "``[r]+ cos(theta) " "6#,&&%!G6#%\"rG\"\"\"-%$cosG6#%&thetaGF(" }{TEXT -1 1 " \+ " }{TEXT 344 1 "e" }{XPPEDIT 18 0 "``[theta]" "6#&%!G6#%&thetaG" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "which we obtain in Maple as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "Introduce the " }{TEXT 264 1 "i" }{TEXT -1 5 " and " }{TEXT 265 1 "j" }{TEXT -1 62 " basis ve ctors of the cartesian frame by the following device." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "q1 := e[ r] = dotprod(E[r],[i,j], orthogonal);\nq2 := e[theta] = dotprod(E[thet a],[i,j], orthogonal);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "These are now two equations in \+ the two unknowns " }{TEXT 266 1 "i" }{TEXT -1 5 " and " }{TEXT 267 1 " j" }{TEXT -1 25 " which can be solved for " }{TEXT 276 1 "i" }{TEXT -1 5 " and " }{TEXT 277 1 "j" }{TEXT -1 48 " in terms of the polar coo rdinate basis vectors " }{XPPEDIT 18 0 "e[r]" "6#&%\"eG6#%\"rG" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "e[theta]" "6#&%\"eG6#%&thetaG" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "q3 := solve(\{q1,q2\}, \{i,j\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "To emphasize this result, induce Maple to write " }{TEXT 278 1 "i" }{TEXT -1 5 " and " }{TEXT 279 1 "j" }{TEXT -1 13 " in terms of " }{XPPEDIT 18 0 "e[r]" "6#&%\"eG6#%\"rG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "e[theta]" "6#&%\"eG6#%&thetaG" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "i = subs( q3, i);\nj = subs(q3, j);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 346 "The point here is to discov er the proper expressions for the gradient and laplacian of a scalar-v alued function, and the divergence of a vector-valued function, when e ach is given in polar coordinates. Since the expression for the gradi ent in cartesian coordinates is already known, use that knowledge to d educe the result for polar coordinates." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 "Start with the expression for the gr adient in cartesian coordinates." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "q4 := grad(u(x,y),[x,y]);" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 85 "This gradient vector, given explicitly in terms of the \+ fixed cartesian basis vectors " }{TEXT 268 1 "i" }{TEXT -1 5 " and " } {TEXT 269 1 "j" }{TEXT -1 4 ", is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "q5 := q4[1]*i + q4[2]*j;" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "The conversion process begins by replacing the vectors " }{TEXT 280 1 "i" }{TEXT -1 5 " and " }{TEXT 281 1 "j" }{TEXT -1 36 " w ith their equivalents in terms of " }{XPPEDIT 18 0 "e[r]" "6#&%\"eG6#% \"rG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "e[theta]" "6#&%\"eG6#%&theta G" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 19 "q6 := subs(q3, q5);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "This loo ks a bit better if the coefficients of the basis vectors " }{XPPEDIT 18 0 "e[r]" "6#&%\"eG6#%\"rG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "e[th eta]" "6#&%\"eG6#%&thetaG" }{TEXT -1 18 " are collected via" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "q 7 := collect(q6, [e[r], e[theta]]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "Then, we have to transform the partial derivatives by the use of the chain rule." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "The seco nd step expresses " }{XPPEDIT 18 0 "u(x,y)" "6#-%\"uG6$%\"xG%\"yG" } {TEXT -1 4 " as " }{XPPEDIT 18 0 "U(r(x,y),theta(x,y))" "6#-%\"UG6$-% \"rG6$%\"xG%\"yG-%&thetaG6$F)F*" }{TEXT -1 90 " prior to using the cha in rule for determining the equivalents of the partial derivatives " } {XPPEDIT 18 0 "u[x]" "6#&%\"uG6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "u[y]" "6#&%\"uG6#%\"yG" }{TEXT -1 59 " in polar coordinates. Fo r insight, consider the function " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "h := (x,y) -> x*y^2;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "and change to polar coordinates, obtaining " }{XPPEDIT 18 0 "h(x(r,theta),y(r,theta))=H(r,theta)" "6#/-%\"hG6$-%\"xG6$%\"rG%& thetaG-%\"yG6$F*F+-%\"HG6$F*F+" }{TEXT -1 15 ", which is then" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "H := unapply(h(X,Y),r,theta);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "The introduct ion of " }{XPPEDIT 18 0 "H(r,theta)" "6#-%\"HG6$%\"rG%&thetaG" }{TEXT -1 51 " is essential, since it is a different function of " }{XPPEDIT 18 0 "r" "6#%\"rG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "theta" "6#%&the taG" }{TEXT -1 6 " than " }{XPPEDIT 18 0 "h(x,y)" "6#-%\"hG6$%\"xG%\"y G" }{TEXT -1 19 " was a function of " }{XPPEDIT 18 0 "x" "6#%\"xG" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "y" "6#%\"yG" }{TEXT -1 3 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "In " } {XPPEDIT 18 0 "h(x,y)" "6#-%\"hG6$%\"xG%\"yG" }{TEXT -1 21 " the first variable, " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 47 ", simply multi plies the square of the second. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 3 "In " }{XPPEDIT 18 0 "H(r,theta)" "6#-%\"HG 6$%\"rG%&thetaG" }{TEXT -1 22 ", the first variable, " }{XPPEDIT 18 0 "r" "6#%\"rG" }{TEXT -1 13 ", is cubed. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "Thus, two different letters, " } {XPPEDIT 18 0 "f" "6#%\"fG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "F" "6# %\"FG" }{TEXT -1 47 " are more than appropriate, they are necessary." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "Now, go back the other way by computing " }{XPPEDIT 18 0 "H(r(x,y),theta(x,y) )=h(x,y)" "6#/-%\"HG6$-%\"rG6$%\"xG%\"yG-%&thetaG6$F*F+-%\"hG6$F*F+" } {TEXT -1 23 ", obtained in Maple via" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "simplify(H(R,T),symbolic);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "This second identity, namely," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 278 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "h(x,y)=H(x( r,theta),y(r,theta))" "6#/-%\"hG6$%\"xG%\"yG-%\"HG6$-F'6$%\"rG%&thetaG -F(6$F.F/" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 " is the starting point for the partial differentiations about to t ake place." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "Thus, start with " }}{PARA 266 "" 0 "" {XPPEDIT 18 0 "u(x,y)=U( r(x,y),theta(x,y))" "6#/-%\"uG6$%\"xG%\"yG-%\"UG6$-%\"rG6$F'F(-%&theta G6$F'F(" }}{PARA 0 "" 0 "" {TEXT -1 21 "entered into Maple as" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "u := U(R, T);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "Differentiate with respect to " } {TEXT 292 1 "x" }{TEXT -1 5 " and " }{TEXT 293 1 "y" }{TEXT -1 24 " to obtain the partials " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 279 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u[x]=U[r]*r[x]+U[theta]*theta[x]" "6#/&%\"uG6#%\"xG,&*&&%\"UG6#%\"rG\"\"\"&F-6#F'F.F.*&&F+6#%&thetaGF.&F 46#F'F.F." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "and" }}{PARA 280 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u[y]=U[r]*r[y]+U[theta]*the ta[y]" "6#/&%\"uG6#%\"yG,&*&&%\"UG6#%\"rG\"\"\"&F-6#F'F.F.*&&F+6#%&the taGF.&F46#F'F.F." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "Since " }}{PARA 281 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "r[x]=cos(theta)" "6#/&%\"rG6#%\"xG-%$cosG6#%&thetaG" }{TEXT -1 4 " " }}{PARA 282 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "theta[x]=-sin(theta)/r" "6#/&%&the taG6#%\"xG,$*&-%$sinG6#F%\"\"\"%\"rG!\"\"F/" }{TEXT -1 1 " " }}{PARA 283 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "r[y]=sin(theta)" "6#/&%\"rG 6#%\"yG-%$sinG6#%&thetaG" }{TEXT -1 4 " " }}{PARA 284 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "theta[y]=cos(theta)/r" "6#/&%&thetaG6#% \"yG*&-%$cosG6#F%\"\"\"%\"rG!\"\"" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 45 "the gradient in cartesian coordinates becomes" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 285 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "[U[r]*cos(theta)-U[theta]*``(sin(theta)/r)]" "6#7#,&*&&%\"UG6#% \"rG\"\"\"-%$cosG6#%&thetaGF*F**&&F'6#F.F*-%!G6#*&-%$sinG6#F.F*F)!\"\" F*F9" }{TEXT -1 2 " [" }{XPPEDIT 18 0 "cos(theta)" "6#-%$cosG6#%&theta G" }{TEXT -1 1 " " }{TEXT 345 1 "e" }{XPPEDIT 18 0 "``[r]-sin(theta)" "6#,&&%!G6#%\"rG\"\"\"-%$sinG6#%&thetaG!\"\"" }{TEXT -1 1 " " }{TEXT 346 1 "e" }{XPPEDIT 18 0 "``[theta]" "6#&%!G6#%&thetaG" }{TEXT -1 4 "] + " }{XPPEDIT 18 0 "[U[r]*sin(theta)+U[theta]*``(cos(theta)/r)]" "6#7 #,&*&&%\"UG6#%\"rG\"\"\"-%$sinG6#%&thetaGF*F**&&F'6#F.F*-%!G6#*&-%$cos G6#F.F*F)!\"\"F*F*" }{TEXT -1 2 " [" }{XPPEDIT 18 0 "sin(theta)" "6#-% $sinG6#%&thetaG" }{TEXT -1 1 " " }{TEXT 347 1 "e" }{XPPEDIT 18 0 "``[r ]+cos(theta)" "6#,&&%!G6#%\"rG\"\"\"-%$cosG6#%&thetaGF(" }{TEXT -1 1 " " }{TEXT 348 1 "e" }{XPPEDIT 18 0 "``[theta]" "6#&%!G6#%&thetaG" } {TEXT -1 2 "] " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 286 "" 0 "" {TEXT -1 3 " = " }{XPPEDIT 18 0 "U[r]" "6#&%\"UG6#%\"rG" }{TEXT -1 1 " " }{TEXT 349 1 "e" }{XPPEDIT 18 0 "``[r]+1/r" "6#,&&%!G6#%\"rG\"\"\"* &F(F(F'!\"\"F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "U[theta]" "6#&%\"UG6#% &thetaG" }{TEXT -1 1 " " }{TEXT 350 1 "e" }{XPPEDIT 18 0 "``[theta]" " 6#&%!G6#%&thetaG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 69 "To implement these calculations in Maple, begin with the derivatives " }{XPPEDIT 18 0 "u[x]" "6#&%\"uG6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "u[y]" "6#&%\"uG6#%\"yG" }{TEXT -1 12 " in the form" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "ux := diff(u,x);\nuy := diff(u,y);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 99 "These results are nothing more than the chain rule, which in subscript notation would be written as" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 268 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u[x]=U[r]*r [x]+U[theta]*theta[x]" "6#/&%\"uG6#%\"xG,&*&&%\"UG6#%\"rG\"\"\"&F-6#F' F.F.*&&F+6#%&thetaGF.&F46#F'F.F." }{TEXT -1 1 "\n" }{XPPEDIT 18 0 "u[y ]=U[r]*r[y]+U[theta]*theta[y]" "6#/&%\"uG6#%\"yG,&*&&%\"UG6#%\"rG\"\" \"&F-6#F'F.F.*&&F+6#%&thetaGF.&F46#F'F.F." }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "Use these express ions for " }{XPPEDIT 18 0 "u[x]" "6#&%\"uG6#%\"xG" }{TEXT -1 5 " and \+ " }{XPPEDIT 18 0 "u[y]" "6#&%\"uG6#%\"yG" }{TEXT -1 4 " in " }{TEXT 299 2 "q7" }{TEXT -1 12 ", namely, in" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 267 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(diff(u(x,y),x)*c os(theta)+diff(u(x,y),y)*sin(theta))*e[r]+(-diff(u(x,y),x)*sin(theta)+ diff(u(x,y),y)*cos(theta))*e[theta]" "6#,&*&,&*&-%%diffG6$-%\"uG6$%\"x G%\"yGF-\"\"\"-%$cosG6#%&thetaGF/F/*&-F(6$-F+6$F-F.F.F/-%$sinG6#F3F/F/ F/&%\"eG6#%\"rGF/F/*&,&*&-F(6$-F+6$F-F.F-F/-F:6#F3F/!\"\"*&-F(6$-F+6$F -F.F.F/-F16#F3F/F/F/&F=6#F3F/F/" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 9 "to obtain" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "q8 := subs(diff(u(x,y), x)=ux, diff(u(x,y),y)=uy, q7);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "This represents " } {XPPEDIT 18 0 "grad(u)" "6#-%%gradG6#%\"uG" }{TEXT -1 51 " as a vector in the polar coordinate basis vectors " }{XPPEDIT 18 0 "e[r]" "6#&%\" eG6#%\"rG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "e[theta]" "6#&%\"eG6#%& thetaG" }{TEXT -1 40 ". However, all traces of the variables " } {XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y" "6# %\"yG" }{TEXT -1 43 " are to be removed. So, to begin, replace " } {XPPEDIT 18 0 "sqrt(x^2+y^2)" "6#-%%sqrtG6#,&*$%\"xG\"\"#\"\"\"*$%\"yG F)F*" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "arctan(y/x)" "6#-%'arctanG6# *&%\"yG\"\"\"%\"xG!\"\"" }{TEXT -1 21 " in the arguments of " }{TEXT 295 1 "U" }{TEXT -1 6 " with " }{TEXT 296 1 "r" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 15 ", respectively." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "q9 := subs(R=r,T=theta,q8);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "There are sti ll other " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 7 "'s and " } {XPPEDIT 18 0 "y" "6#%\"yG" }{TEXT -1 28 "'s left. Replace them with \+ " }{XPPEDIT 18 0 "r*cos(theta)" "6#*&%\"rG\"\"\"-%$cosG6#%&thetaGF%" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "r*sin(theta)" "6#*&%\"rG\"\"\"-%$si nG6#%&thetaGF%" }{TEXT -1 23 " respectively,obtaining" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "q10 := s implify(subs(x=X,y=Y,q9), assume=positive);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 " A convers ion from D-notation to the " }{TEXT 300 8 "partials" }{TEXT -1 17 " no tation yields " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 25 "q11 := convert(q10,diff);" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 162 "Sin ce this is actually a vector, split the fraction into two parts to iso late each component of the gradient vector now expressed completely in polar coordinates." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "q12 := expand(q11);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "Expre ssion " }{TEXT 297 3 "q12" }{TEXT -1 81 " represents the polar coodina te equivalent to the cartesian gradient vector grad(" }{TEXT 294 1 "u " }{TEXT -1 4 ") = " }{XPPEDIT 18 0 "u[x]" "6#&%\"uG6#%\"xG" }{TEXT 283 1 "i" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "u[y]" "6#&%\"uG6#%\"yG" } {TEXT 282 1 "j" }{TEXT -1 42 ". Writing the result as the column vect or" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "vector([coeff(q12,e[r]),coeff(q12,e[theta])]);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "allows a direct comparison with Maple's" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "grad(U(r, theta),[r,theta], coords=polar);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 31 "Divergence in Polar Coordinates" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "In cartesian coordinates, the vector field" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "F := vector([f(x,y),g(x,y)]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "has as its di vergence the scalar field" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "diverge(F,[x,y]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "T he divergence of the vector " }}{PARA 287 "" 0 "" {TEXT 351 1 "B" } {TEXT -1 3 " = " }{XPPEDIT 18 0 "U(r,theta)" "6#-%\"UG6$%\"rG%&thetaG " }{TEXT -1 1 " " }{TEXT 352 1 "e" }{XPPEDIT 18 0 "``[r]" "6#&%!G6#%\" rG" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "V(r,theta)" "6#-%\"VG6$%\"rG%&th etaG" }{TEXT -1 1 " " }{TEXT 353 1 "e" }{XPPEDIT 18 0 "``[theta]" "6#& %!G6#%&thetaG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 22 "that is, of the vector" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 45 "U:='U':\nB := vector([U(r,theta),V(r,theta)]); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "is the scalar" }}{PARA 288 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "U[r]+1/r" "6#,&&%\"UG6#%\"rG\"\"\"*&F(F(F'!\"\"F(" } {TEXT -1 1 " " }{XPPEDIT 18 0 "U+1/r" "6#,&%\"UG\"\"\"*&F%F%%\"rG!\"\" F%" }{TEXT -1 1 " " }{XPPEDIT 18 0 "V[theta]" "6#&%\"VG6#%&thetaG" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 8 "that is," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "expand(d iverge(B,[r,theta],coords=polar));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "Notice the te rm without any derivative! There is something to discover here!" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 44 "Divergence in Polar Coo rdinates - Derivation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "To derive the expression for the divergence of \+ the polar vector" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 289 "" 0 "" {TEXT 354 1 "B" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "U(r,theta)" "6#-%\"U G6$%\"rG%&thetaG" }{TEXT -1 1 " " }{TEXT 355 1 "e" }{XPPEDIT 18 0 "``[ r]" "6#&%!G6#%\"rG" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "V(r,theta)" "6#- %\"VG6$%\"rG%&thetaG" }{TEXT -1 1 " " }{TEXT 356 1 "e" }{XPPEDIT 18 0 "``[theta]" "6#&%!G6#%&thetaG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 10 "transform " }{TEXT 357 1 "B" }{TEXT -1 165 " completely t o cartesian coordinates, compute the divergence in cartesian coordinat es, then transform that result back to polar coordinates. Therefore, \+ in terms of " }{TEXT 358 1 "i" }{TEXT -1 5 " and " }{TEXT 359 1 "j" } {TEXT -1 9 ", we have" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 290 "" 0 "" {TEXT 360 1 "B" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "F(r,theta)" "6# -%\"FG6$%\"rG%&thetaG" }{TEXT -1 1 " " }{TEXT 361 1 "i" }{TEXT -1 3 " \+ + " }{XPPEDIT 18 0 "G(r,theta)" "6#-%\"GG6$%\"rG%&thetaG" }{TEXT -1 1 " " }{TEXT 362 1 "j" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "whe re" }}{PARA 291 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "F=U*cos(theta)- V*sin(theta)" "6#/%\"FG,&*&%\"UG\"\"\"-%$cosG6#%&thetaGF(F(*&%\"VGF(-% $sinG6#F,F(!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "and" } }{PARA 292 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "G=U*sin(theta)+V*cos (theta)" "6#/%\"GG,&*&%\"UG\"\"\"-%$sinG6#%&thetaGF(F(*&%\"VGF(-%$cosG 6#F,F(F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "F" "6#%\"FG" }{TEXT -1 5 " \+ and " }{XPPEDIT 18 0 "G" "6#%\"GG" }{TEXT -1 18 " are functions of " } {XPPEDIT 18 0 "r" "6#%\"rG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "theta " "6#%&thetaG" }{TEXT -1 8 ", define" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 293 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x,y)=F(r(x,y),the ta(x,y))" "6#/-%\"fG6$%\"xG%\"yG-%\"FG6$-%\"rG6$F'F(-%&thetaG6$F'F(" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "and" }}{PARA 294 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "g(x,y)=G(r(x,y),theta(x,y))" "6#/-%\"gG 6$%\"xG%\"yG-%\"GG6$-%\"rG6$F'F(-%&thetaG6$F'F(" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 33 "so that the cartesian version of " } {TEXT 363 1 "B" }{TEXT -1 10 " is really" }}{PARA 295 "" 0 "" {TEXT 364 1 "B" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "f(x,y) " "6#-%\"fG6$%\"xG% \"yG" }{TEXT -1 1 " " }{TEXT 365 1 "i" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "g(x,y) " "6#-%\"gG6$%\"xG%\"yG" }{TEXT -1 1 " " }{TEXT 366 1 "j" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "Then, the partial derivatives " }{XPPEDIT 18 0 "f[x]" "6# &%\"fG6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "g[y]" "6#&%\"gG6#% \"yG" }{TEXT -1 94 ", computed by the chain rule, are precisely what a re needed for determining the divergence of " }{TEXT 367 1 "B" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "By the chain rule," }}{PARA 296 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f[x]=F[r]*r[x]+F[theta]*theta[x]" "6#/&%\"fG6#%\"xG,&*&&%\"FG6#% \"rG\"\"\"&F-6#F'F.F.*&&F+6#%&thetaGF.&F46#F'F.F." }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 3 "and" }}{PARA 297 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "g[y]=G[r]*r[y]+G[theta]*theta[y]" "6#/&%\"gG6#%\"yG,&*& &%\"GG6#%\"rG\"\"\"&F-6#F'F.F.*&&F+6#%&thetaGF.&F46#F'F.F." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 2 "so" }}{PARA 298 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f[x]=[U[r]*cos(theta)-V[r]*sin(theta)]*cos(theta )+[U[theta]*cos(theta)-U*sin(theta)-V[theta]*sin(theta)-V*cos(theta)]* ``(-sin(theta)/r)" "6#/&%\"fG6#%\"xG,&*&7#,&*&&%\"UG6#%\"rG\"\"\"-%$co sG6#%&thetaGF1F1*&&%\"VG6#F0F1-%$sinG6#F5F1!\"\"F1-F36#F5F1F1*&7#,**&& F.6#F5F1-F36#F5F1F1*&F.F1-F;6#F5F1F=*&&F86#F5F1-F;6#F5F1F=*&F8F1-F36#F 5F1F=F1-%!G6#,$*&-F;6#F5F1F0F=F=F1F1" }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 3 "and" }}{PARA 299 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " g[y]=[U[r]*sin(theta)+V[r]*cos(theta)]*sin(theta)+[U[theta]*sin(theta) +U*cos(theta)+V[theta]*cos(theta)-V*sin(theta)]*``(cos(theta)/r)" "6#/ &%\"gG6#%\"yG,&*&7#,&*&&%\"UG6#%\"rG\"\"\"-%$sinG6#%&thetaGF1F1*&&%\"V G6#F0F1-%$cosG6#F5F1F1F1-F36#F5F1F1*&7#,**&&F.6#F5F1-F36#F5F1F1*&F.F1- F;6#F5F1F1*&&F86#F5F1-F;6#F5F1F1*&F8F1-F36#F5F1!\"\"F1-%!G6#*&-F;6#F5F 1F0FRF1F1" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 8 "The sum " }}{PARA 300 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "f[x]+g[y]=U[r]+1/r" "6#/,&&%\"fG6#%\"xG\"\"\"&%\"gG6#% \"yGF),&&%\"UG6#%\"rGF)*&F)F)F2!\"\"F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "U+1/r" "6#,&%\"UG\"\"\"*&F%F%%\"rG!\"\"F%" }{TEXT -1 1 " " } {XPPEDIT 18 0 "V[theta]" "6#&%\"VG6#%&thetaG" }{TEXT -1 7 " = div(" } {TEXT 368 1 "B" }{TEXT -1 2 ") " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 39 "is the divergence in polar coordinates." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 133 "To imp lement these calculations in Maple, we begin by writing an arbitrary v ector in polar coordinates, using the unit basis vectors " }{TEXT 369 1 "e" }{XPPEDIT 18 0 "``[r]" "6#&%!G6#%\"rG" }{TEXT -1 5 " and " } {TEXT 370 1 "e" }{XPPEDIT 18 0 "``[theta]" "6#&%!G6#%&thetaG" }{TEXT -1 22 " in polar coordinates." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "B := U*e[r] + V*e[theta];" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 18 "The basis vectors " }{XPPEDIT 18 0 "e[r]" "6#&%\"eG6#% \"rG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "e[theta]" "6#&%\"eG6#%&theta G" }{TEXT -1 66 " have already been given in terms of the cartesian ba sis vectors, " }{TEXT 302 1 "i" }{TEXT -1 5 " and " }{TEXT 303 1 "j" } {TEXT -1 3 " by" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 7 "q1;\nq2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 210 "Transform the given p olar vector completely to cartesian coordinates, compute the divergenc e in cartesian coordinates, then transform that result back to polar c oordinates. That will clarify what replaces div(" }{TEXT 310 1 "B" } {TEXT -1 23 ") in polar coordinates." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "This is the vector " }{TEXT 311 1 "B" } {TEXT -1 13 " in terms of " }{TEXT 304 1 "i" }{TEXT -1 5 " and " } {TEXT 305 1 "j" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "Bij := subs(q1, q2, B);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "Collecting the " }{TEXT 306 1 "i" }{TEXT -1 5 " and " } {TEXT 307 1 "j" }{TEXT -1 23 " components is helpful." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "Bij_coll ected := collect(Bij, [i,j]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "Next, extract the " } {TEXT 308 1 "i" }{TEXT -1 5 " and " }{TEXT 309 1 "j" }{TEXT -1 58 " co mponents of this vector which is now in cartesian form." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "F := c oeff(Bij_collected,i);\nG := coeff(Bij_collected,j);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "T he polar vector " }{TEXT 314 1 "B" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "U " "6#%\"UG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "e[r]" "6#&%\"eG6#%\"rG" } {TEXT -1 3 " + " }{XPPEDIT 18 0 "V" "6#%\"VG" }{TEXT -1 1 " " } {XPPEDIT 18 0 "e[theta]" "6#&%\"eG6#%&thetaG" }{TEXT -1 33 " has becom e the cartesian vector " }{XPPEDIT 18 0 "F" "6#%\"FG" }{TEXT -1 1 " " }{TEXT 315 1 "i" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "G" "6#%\"GG" } {TEXT -1 1 " " }{TEXT 316 1 "j" }{TEXT -1 18 ". Unfortunately, " } {XPPEDIT 18 0 "F" "6#%\"FG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "G" "6# %\"GG" }{TEXT -1 18 " are functions of " }{XPPEDIT 18 0 "r" "6#%\"rG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 11 " , so define" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 271 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x,y)=F(r(x,y),theta(x,y))" "6#/-%\"fG 6$%\"xG%\"yG-%\"FG6$-%\"rG6$F'F(-%&thetaG6$F'F(" }}{PARA 272 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "g(x,y)=G(r(x,y),theta(x,y))" "6#/-%\"gG 6$%\"xG%\"yG-%\"GG6$-%\"rG6$F'F(-%&thetaG6$F'F(" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "so that the cartesian ver sion of " }{TEXT 317 1 "B" }{TEXT -1 11 " is really " }{XPPEDIT 18 0 " f(x,y)" "6#-%\"fG6$%\"xG%\"yG" }{TEXT -1 1 " " }{TEXT 319 1 "i" } {TEXT -1 3 " + " }{XPPEDIT 18 0 "g(x,y)" "6#-%\"gG6$%\"xG%\"yG" } {TEXT -1 1 " " }{TEXT 320 1 "j" }{TEXT -1 33 ". Then, the partial der ivatives " }{XPPEDIT 18 0 "f[x]" "6#&%\"fG6#%\"xG" }{TEXT -1 5 " and \+ " }{XPPEDIT 18 0 "g[y]" "6#&%\"gG6#%\"yG" }{TEXT -1 65 " are precisely what are needed for determining the divergence of " }{TEXT 318 1 "B" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "To express " }{XPPEDIT 18 0 "F" "6#%\"FG" }{TEXT -1 5 " a nd " }{XPPEDIT 18 0 "G" "6#%\"GG" }{TEXT -1 17 ", as well as sin(" } {XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 10 ") and cos(" } {XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 15 "), in terms of " } {XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y" "6# %\"yG" }{TEXT -1 8 ", define" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 269 "" 0 "" {XPPEDIT 18 0 "U(sqrt(x^2+y^2), arctan(y/x))=u(x,y)" "6#/- %\"UG6$-%%sqrtG6#,&*$%\"xG\"\"#\"\"\"*$%\"yGF-F.-%'arctanG6#*&F0F.F,! \"\"-%\"uG6$F,F0" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }}{PARA 270 "" 0 "" {XPPEDIT 18 0 "V(sqrt(x^2+y^2), arctan(y/x))=v(x,y )" "6#/-%\"VG6$-%%sqrtG6#,&*$%\"xG\"\"#\"\"\"*$%\"yGF-F.-%'arctanG6#*& F0F.F,!\"\"-%\"vG6$F,F0" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "Efficiency is achieved by setting up these substitut ions in a template." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "q13 := \{U=U(R,T),V=V(R,T),theta=T\};" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "Making these substitutions in the polar components " } {XPPEDIT 18 0 "F" "6#%\"FG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "G" "6# %\"GG" }{TEXT -1 33 " yields the cartesian components " }{XPPEDIT 18 0 "f(x,y)" "6#-%\"fG6$%\"xG%\"yG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 " g(x,y)" "6#-%\"gG6$%\"xG%\"yG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "f := simpli fy(subs(q13, F), symbolic);\ng := simplify(subs(q13, G), symbolic);" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 24 "The partial derivatives " }{XPPEDIT 18 0 "f[x]" "6#&%\" fG6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "g[y]" "6#&%\"gG6#%\"yG " }{TEXT -1 27 " can now be taken, yielding" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "fx := diff(f,x); \ngy := diff(g,y);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "The sum " }{XPPEDIT 18 0 "f[x]+g[y] " "6#,&&%\"fG6#%\"xG\"\"\"&%\"gG6#%\"yGF(" }{TEXT -1 8 " is div(" } {TEXT 312 1 "B" }{TEXT -1 162 ") in cartesian coordinates, the system \+ in which the divergence operator was originally defined. In that sum, begin the process of returning to polar coordinates." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "q14 := s ubs(R=r,T=theta, fx + gy);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "There are yet more " } {XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 7 "'s and " }{XPPEDIT 18 0 "y" " 6#%\"yG" }{TEXT -1 35 "'s to convert to polar coordinates." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "q15 := simplify(subs(x=X,y=Y,q14),symbolic);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "Next, chang e notation for the derivatives, from the D-notation to the " }{TEXT 321 8 "partials" }{TEXT -1 20 " notation, obtaining" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "q16 := co nvert(q15,diff);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "Making separate fractions yields t he final result." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "q17 := expand(q16);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "Once aga in turning to Maple for a comparison, write the general vector as" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "B2 := vector([U(r,theta), V(r,theta)]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "and appl y the built-in " }{TEXT 313 7 "diverge" }{TEXT -1 19 " command, obtain ing" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "`div(B2)` := diverge(B2,[r,theta],coords=polar);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 "This is the result just derived from first principles. A gain creating separate fractions, " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "expand(`div(B2)`);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "the results are seen to agree." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "q17;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 30 "Laplacian in Polar Coordinates" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "In rectangular coordinat es, the laplacian of " }{XPPEDIT 18 0 "f(x,y)" "6#-%\"fG6$%\"xG%\"yG" }{TEXT -1 3 " is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "unassign('f','F');\nlaplacian(f(x,y),[x,y]); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "In polar coordinates, the laplacian of " }{XPPEDIT 18 0 "F(r,theta)" "6#-%\"FG6$%\"rG%&thetaG" }{TEXT -1 3 " is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 301 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "F[rr]+1/r" "6#,&&%\"FG6#%#rrG\"\"\"*&F(F(%\"rG!\"\"F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "F[r]+1/r^2" "6#,&&%\"FG6#%\"rG\"\"\"*&F(F(*$F '\"\"#!\"\"F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "F[theta theta]" "6#&%\" FG6#*&%&thetaG\"\"\"F'F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 8 "that is," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "expand(laplacian(F(r,theta),[r,theta],coords=polar ));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 12 "Example 22.2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "For example, the laplacian of the \+ function " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 13 "f := x + y^3;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "qc \+ := laplacian(f, [x,y]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Converting the function " } {XPPEDIT 18 0 "f(x,y)=x+y^3" "6#/-%\"fG6$%\"xG%\"yG,&F'\"\"\"*$F(\"\"$ F*" }{TEXT -1 28 " to polar coordinates gives " }{XPPEDIT 18 0 "F(r,th eta) = f(x(r,theta),y(r,theta))" "6#/-%\"FG6$%\"rG%&thetaG-%\"fG6$-%\" xG6$F'F(-%\"yG6$F'F(" }{TEXT -1 10 ", that is," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "F := subs(x = r*cos(theta), y = r*sin(theta), f);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "In polar coor dinates, the laplacian of " }{XPPEDIT 18 0 "F(r,theta)" "6#-%\"FG6$%\" rG%&thetaG" }{TEXT -1 3 " is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "qp := laplacian(F, [r,theta] , coords = polar);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "an expression which actually simpl ifies to" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "combine(qp,trig);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "If " } {XPPEDIT 18 0 "6*y" "6#*&\"\"'\"\"\"%\"yGF%" }{TEXT -1 81 ", the lapla cian in rectangular coordinates, were converted to polar coordinates, \+ " }{XPPEDIT 18 0 "6*r*sin(theta)" "6#*(\"\"'\"\"\"%\"rGF%-%$sinG6#%&th etaGF%" }{TEXT -1 141 " would result, in perfect agreement with the la placian computed directly in polar coordinates. In fact, this compari son is made in Maple via" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "simplify(6*r*sin(theta) - qp);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 16 "Maple Derivation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "Instead of the very general " } {XPPEDIT 18 0 "F(r,theta) = f(x(r,theta),y(r,theta))" "6#/-%\"FG6$%\"r G%&thetaG-%\"fG6$-%\"xG6$F'F(-%\"yG6$F'F(" }{TEXT -1 6 ", let " } {TEXT 260 1 "x" }{TEXT -1 5 " and " }{TEXT 261 1 "y" }{TEXT -1 52 " be specifically given in polar coordinates. Then, " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x,y ) = F(r(x,y),theta(x,y))" "6#/-%\"fG6$%\"xG%\"yG-%\"FG6$-%\"rG6$F'F(-% &thetaG6$F'F(" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "F(sqrt(x^2+y^2),arcta n(y/x))" "6#-%\"FG6$-%%sqrtG6#,&*$%\"xG\"\"#\"\"\"*$%\"yGF,F--%'arctan G6#*&F/F-F+!\"\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "unassign('F');\nf := F(sqrt(x^2+y^2), arctan( y/x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "The cartesian derivatives can computed as" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "fx := diff(f,x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 " fxx := diff(f,x,x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "fy \+ := diff(f,y);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "fyy := dif f(f,y,y);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 81 "Hence, the laplacian in cartesian coordin ates has the following polar equivalent." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "q := fxx + fyy;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "This unwieldy expression can be simplified to the much sm aller" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "q1 := simplify(q);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "Further simpl ification comes notationally, by expressing the arguments of F as " } {XPPEDIT 18 0 "r" "6#%\"rG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "theta " "6#%&thetaG" }{TEXT -1 17 ", rather than as " }{XPPEDIT 18 0 "sqrt(x ^2+y^2)" "6#-%%sqrtG6#,&*$%\"xG\"\"#\"\"\"*$%\"yGF)F*" }{TEXT -1 5 " a nd " }{XPPEDIT 18 0 "arctan(y/x)" "6#-%'arctanG6#*&%\"yG\"\"\"%\"xG!\" \"" }{TEXT -1 72 ", respectively. This step is done by making the fol lowing replacements." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "q2 := subs(R=r,T=theta,q1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "Still more simplification is achieved by changing the remaining apearances of " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "y" "6#%\"yG" }{TEXT -1 54 " to their counterparts in po lar coordinates, obtaining" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "q3 := simplify(subs(x=X,y=Y,q2),sym bolic);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "Finally, convert from the D-notation for \+ derivatives to the " }{TEXT 262 8 "partials" }{TEXT -1 18 " notation, \+ getting" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "expand(convert(q3,diff));" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "Comparing t his result to Maple's built-in version" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "expand(laplacian(F(r,th eta),[r,theta],coords=polar));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "shows the results to b e identical." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 33 "Tradit ional Chain Rule Derivation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 240 "The Maple derivation above might seem in timidating and complex, but it is actually easier than the traditional derivation based on a direct use of the chain rule. The following is a Maple-implemented version of the traditional computation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "The starting po int for the differentiations is the statement" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x,y )=F(r(x,y),theta(x,y))" "6#/-%\"fG6$%\"xG%\"yG-%\"FG6$-%\"rG6$F'F(-%&t hetaG6$F'F(" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "An application of the chain rule gives" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f[x]= F[r]*r[x]+F[theta]*theta[x]" "6#/&%\"fG6#%\"xG,&*&&%\"FG6#%\"rG\"\"\"& F-6#F'F.F.*&&F+6#%&thetaGF.&F46#F'F.F." }}{PARA 0 "" 0 "" {TEXT -1 3 " and" }}{PARA 261 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f[y]=F[r]*r[y] +F[theta]*theta[y]" "6#/&%\"fG6#%\"yG,&*&&%\"FG6#%\"rG\"\"\"&F-6#F'F.F .*&&F+6#%&thetaGF.&F46#F'F.F." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 85 "Second derivatives pose a greater challen ge unless the dependence of the derivatives " }{XPPEDIT 18 0 "F[r]" "6 #&%\"FG6#%\"rG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "F[theta]" "6#&%\"F G6#%&thetaG" }{TEXT -1 4 " on " }{XPPEDIT 18 0 "r(x,y)" "6#-%\"rG6$%\" xG%\"yG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "theta(x,y)" "6#-%&thetaG6 $%\"xG%\"yG" }{TEXT -1 84 " is properly appreciated. Again using the \+ chain rule, the second derivatives become" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 262 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f[xx]=diff(F[r ],x)*r[x]+F[r]*r[xx]+diff(F[theta],x)*theta[x]+F[theta]*theta[xx]" "6# /&%\"fG6#%#xxG,**&-%%diffG6$&%\"FG6#%\"rG%\"xG\"\"\"&F06#F1F2F2*&&F.6# F0F2&F06#F'F2F2*&-F+6$&F.6#%&thetaGF1F2&F?6#F1F2F2*&&F.6#F?F2&F?6#F'F2 F2" }{TEXT -1 42 " " }}{PARA 263 "" 0 "" {TEXT -1 2 "= " }{XPPEDIT 18 0 "[F[rr]*r[x]+F[r theta]*the ta[x]]*r[x]+F[r]*r[xx]+[F[theta r]*r[x]+F[theta theta]*theta[x]]*theta [x]+F[theta]*theta[xx]" "6#,**&7#,&*&&%\"FG6#%#rrG\"\"\"&%\"rG6#%\"xGF ,F,*&&F)6#*&F.F,%&thetaGF,F,&F56#F0F,F,F,&F.6#F0F,F,*&&F)6#F.F,&F.6#%# xxGF,F,*&7#,&*&&F)6#*&F5F,F.F,F,&F.6#F0F,F,*&&F)6#*&F5F,F5F,F,&F56#F0F ,F,F,&F56#F0F,F,*&&F)6#F5F,&F56#F?F,F," }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "and" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 264 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f[yy]=diff(F[r],y)*r [y]+F[r]*r[yy]+diff(F[theta],y)*theta[y]+F[theta]*theta[yy]" "6#/&%\"f G6#%#yyG,**&-%%diffG6$&%\"FG6#%\"rG%\"yG\"\"\"&F06#F1F2F2*&&F.6#F0F2&F 06#F'F2F2*&-F+6$&F.6#%&thetaGF1F2&F?6#F1F2F2*&&F.6#F?F2&F?6#F'F2F2" } {TEXT -1 41 " " }}{PARA 265 " " 0 "" {TEXT -1 2 "= " }{XPPEDIT 18 0 "[F[rr]*r[y]+F[r theta]*theta[y] ]*r[y]+F[r]*r[yy]+[F[theta r]*r[y]+F[theta theta]*theta[y]]*theta[y]+F [theta]*theta[yy]" "6#,**&7#,&*&&%\"FG6#%#rrG\"\"\"&%\"rG6#%\"yGF,F,*& &F)6#*&F.F,%&thetaGF,F,&F56#F0F,F,F,&F.6#F0F,F,*&&F)6#F.F,&F.6#%#yyGF, F,*&7#,&*&&F)6#*&F5F,F.F,F,&F.6#F0F,F,*&&F)6#*&F5F,F5F,F,&F56#F0F,F,F, &F56#F0F,F,*&&F)6#F5F,&F56#F?F,F," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 46 "We can represent these derivatives in Map le as" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 269 "fxx := (F[rr]*r[x]+F[r*theta]*theta[x])*r[x] + F[r]* r[xx] + (F[r*theta]*r[x] + F[theta,theta]*theta[x])*theta[x] + F[theta ]*theta[xx];\nfyy := (F[rr]*r[y]+F[r*theta]*theta[y])*r[y] + F[r]*r[yy ] + (F[r*theta]*r[y] + F[theta,theta]*theta[y])*theta[y] + F[theta]*th eta[yy];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "where we have utilized the equality of th e derivatives " }{XPPEDIT 18 0 "F[r theta]" "6#&%\"FG6#*&%\"rG\"\"\"%& thetaGF(" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "F[theta r]" "6#&%\"FG6#* &%&thetaG\"\"\"%\"rGF(" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "Then, the derivatives " }{XPPEDIT 18 0 "r[x],r[xx],r[y],r[yy],theta[x],theta[xx],theta[y],theta[yy]" "6* &%\"rG6#%\"xG&F$6#%#xxG&F$6#%\"yG&F$6#%#yyG&%&thetaG6#F&&F16#F)&F16#F, &F16#F/" }{TEXT -1 79 " must all be computed and expressed in terms of the polar coordinate variables " }{XPPEDIT 18 0 "r" "6#%\"rG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 55 ". Thus , the derivatives are computed and converted via" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 463 "rx : = simplify(subs(x=X,y=Y,diff(R,x)), symbolic);\nrxx := simplify( subs(x=X,y=Y,diff(R,x,x)),symbolic);\nry := simplify(subs(x=X,y= Y,diff(R,y)), symbolic);\nryy := simplify(subs(x=X,y=Y,diff(R,y, y)),symbolic);\ntheta_x := simplify(subs(x=X,y=Y,diff(T,x)), symboli c);\ntheta_xx := simplify(subs(x=X,y=Y,diff(T,x,x)),symbolic);\ntheta_ y := simplify(subs(x=X,y=Y,diff(T,y)), symbolic);\ntheta_yy := simpl ify(subs(x=X,y=Y,diff(T,y,y)),symbolic);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "Making these \+ substitutions into the sum " }{XPPEDIT 18 0 "f[xx]+f[yy]" "6#,&&%\"fG6 #%#xxG\"\"\"&F%6#%#yyGF(" }{TEXT -1 8 ", we get" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "expand(sim plify(subs(r[x]=rx, r[xx]=rxx, r[y]=ry, r[yy]=ryy, theta[x]=theta_x, t heta[xx]=theta_xx, theta[y]=theta_y, theta[yy]=theta_yy, fxx+fyy)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 39 "which we recognize as the equivalent of" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "unassi gn('f');\nexpand(laplacian(f(r,theta),[r,theta], coords=polar));" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}}{MARK "1" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }