1nf.mac is from the book "Perturbation Methods, Bifurcation Theory and 2Computer Algebra" by Rand & Armbruster (Springer 1987) 3 4This maxima run applies normal forms to a system of equations. The 5functions NF, GEN, DECOMPOSE, HOPF2, HOPF3 and TRANSFORM are 6used. 7 8The example is from p55-62. It is presented without discussion. 9maxima-5.9.0 cvs reproduces the results from the book. 10 11(C1) load("./nf.mac"); 12(D1) ./nf.mac 13(C2) load("./transfor.mac"); 14(D2) ./transfor.mac 15(C3) nf(); 16DO YOU WANT TO ENTER NEW VARIABLE NAMES (Y/N)? 17y; 18HOW MANY EQS 192; 20SYMBOL FOR OLD X[ 1 ] 21x; 22SYMBOL FOR OLD X[ 2 ] 23y; 24SYMBOL FOR NEW X[ 1 ] 25u; 26SYMBOL FOR NEW X[ 2 ] 27v; 28DO YOU WANT TO ENTER NEW D.E.'S (Y/N)? 29y; 30ENTER RHS OF EQ. NO. 1 ,D x /DT = 31-y+fxx/2*x^2+fxy*x*y+fyy/2*y^2+fxxx/6*x^3+fxxy/2*x^2*y+fxyy/2*x*y^2+fyyy/6*y^3; 32 3 2 2 2 3 33 fyyy y fxyy x y fyy y fxxy x y fxxx x 34D x /DT = ------- + --------- + ------ + --------- + fxy x y - y + ------- 35 6 2 2 2 6 36 37 2 38 fxx x 39 + ------ 40 2 41ENTER RHS OF EQ. NO. 2 ,D y /DT = 42x+gxx/2*x^2+gxy*x*y+gyy/2*y^2+gxxx/6*x^3+gxxy/2*x^2*y+gxyy/2*x*y^2+gyyy/6*y^3; 43 3 2 2 2 3 2 44 gyyy y gxyy x y gyy y gxxy x y gxxx x gxx x 45D y /DT = ------- + --------- + ------ + --------- + gxy x y + ------- + ------ 46 6 2 2 2 6 2 47 48 + x 49INPUT NEAR-IDENTITY TRANSFORMATION 50 51(USE PREV FOR PREVIOUS TRANSFORMATION) 52x = u + ? 53gen(2); 54 2 2 55x = A v + A u v + A u + u 56 1, [0, 2] 1, [1, 1] 1, [2, 0] 57y = v + ? 58gen(2); 59 2 2 60y = A v + A u v + v + A u 61 2, [0, 2] 2, [1, 1] 2, [2, 0] 62ENTER TRUNCATION ORDER (HIGHEST ORDER TERMS TO BE KEPT) 632; 64du 2 65-- = - v - ((2 A + 2 A - fxx) u 66dT 2, [2, 0] 1, [1, 1] 67 68 + (2 A - 4 A + 4 A - 2 fxy) v u 69 2, [1, 1] 1, [2, 0] 1, [0, 2] 70 71 2 72 + (2 A - 2 A - fyy) v )/2 + . . . 73 2, [0, 2] 1, [1, 1] 74dv 2 75-- = u - ((2 A - 2 A - gxx) u 76dT 2, [1, 1] 1, [2, 0] 77 78 + (- 4 A + 4 A - 2 A - 2 gxy) v u 79 2, [2, 0] 2, [0, 2] 1, [1, 1] 80 81 2 82 + (- 2 A - 2 A - gyy) v )/2 + . . . 83 2, [1, 1] 1, [0, 2] 84DO YOU WANT TO ENTER ANOTHER TRANSFORMATION (Y/N) 85n; 86 du 2 87(D3)/T/ [-- = - v - ((2 A + 2 A - fxx) u 88 dT 2, [2, 0] 1, [1, 1] 89 90 + (2 A - 4 A + 4 A - 2 fxy) v u 91 2, [1, 1] 1, [2, 0] 1, [0, 2] 92 93 2 94 + (2 A - 2 A - fyy) v )/2 + . . ., 95 2, [0, 2] 1, [1, 1] 96 97dv 2 98-- = u - ((2 A - 2 A - gxx) u 99dT 2, [1, 1] 1, [2, 0] 100 101 + (- 4 A + 4 A - 2 A - 2 gxy) v u 102 2, [2, 0] 2, [0, 2] 1, [1, 1] 103 104 2 105 + (- 2 A - 2 A - gyy) v )/2 + . . .] 106 2, [1, 1] 1, [0, 2] 107 108 109(C4) hopf2(); 110 2 gyy + gxx + 2 fxy - 2 gxy + 2 fyy + fxx 111(D4) [[A = - -------------------, A = ---------------------, 112 1, [2, 0] 6 2, [2, 0] 6 113 114 gxy - fyy + fxx gyy - gxx + fxy 115A = ---------------, A = - ---------------, 116 1, [1, 1] 3 2, [1, 1] 3 117 118 - gyy - 2 gxx + 2 fxy 2 gxy + fyy + 2 fxx 119A = ---------------------, A = -------------------]] 120 1, [0, 2] 6 2, [0, 2] 6 121 122 123 124(C5) nf(); 125DO YOU WANT TO ENTER NEW VARIABLE NAMES (Y/N)? 126N; 127DO YOU WANT TO ENTER NEW D.E.'S (Y/N)? 128N; 129INPUT NEAR-IDENTITY TRANSFORMATION 130 131(USE PREV FOR PREVIOUS TRANSFORMATION) 132x = u + ? 133ev(prev,%); 134 2 135 (- gyy - 2 gxx + 2 fxy) v (gxy - fyy + fxx) u v 136x = -------------------------- + --------------------- 137 6 3 138 139 2 140 (2 gyy + gxx + 2 fxy) u 141 - ------------------------ + u 142 6 143y = v + ? 144ev(prev,%); 145 2 146 (2 gxy + fyy + 2 fxx) v (gyy - gxx + fxy) u v 147y = ------------------------ - --------------------- + v 148 6 3 149 150 2 151 (- 2 gxy + 2 fyy + fxx) u 152 + -------------------------- 153 6 154ENTER TRUNCATION ORDER (HIGHEST ORDER TERMS TO BE KEPT) 1552; 156du 157-- = - v + . . . 158dT 159dv 160-- = u + . . . 161dT 162DO YOU WANT TO ENTER ANOTHER TRANSFORMATION (Y/N) 163Y; 164INPUT NEAR-IDENTITY TRANSFORMATION 165 166(USE PREV FOR PREVIOUS TRANSFORMATION) 167x = u + ? 168prev+gen(3); 169 2 170 3 2 (- gyy - 2 gxx + 2 fxy) v 171x = A v + A u v + -------------------------- 172 1, [0, 3] 1, [1, 2] 6 173 174 2 (gxy - fyy + fxx) u v 3 175 + A u v + --------------------- + A u 176 1, [2, 1] 3 1, [3, 0] 177 178 2 179 (2 gyy + gxx + 2 fxy) u 180 - ------------------------ + u 181 6 182y = v + ? 183prev+gen(3); 184 2 185 3 2 (2 gxy + fyy + 2 fxx) v 186y = A v + A u v + ------------------------ 187 2, [0, 3] 2, [1, 2] 6 188 189 2 (gyy - gxx + fxy) u v 3 190 + A u v - --------------------- + v + A u 191 2, [2, 1] 3 2, [3, 0] 192 193 2 194 (- 2 gxy + 2 fyy + fxx) u 195 + -------------------------- 196 6 197ENTER TRUNCATION ORDER (HIGHEST ORDER TERMS TO BE KEPT) 1983; 199du 200-- = - v - (((fxy + gxx + 2 gyy) fxx - 2 fxy fyy + 2 fxy gxy + 6 A 201dT 2, [3, 0] 202 203 3 2 2 204 + 6 A - fxxx) u + (- 2 fxx + (fyy - 2 gxy) fxx - 2 fyy + 2 gxy fyy 205 1, [2, 1] 206 207 2 208 + 4 fxy + (- gxx + 4 gyy) fxy + 6 A - 18 A + 12 A 209 2, [2, 1] 1, [3, 0] 1, [1, 2] 210 211 2 212 - 3 fxxy) v u + ((- 6 fxy + 2 gxx + gyy) fxx + (3 fxy - 2 gxx + 2 gyy) fyy 213 214 2 215 - 4 fxy gxy + 6 A - 12 A + 18 A - 3 fxyy) v u 216 2, [1, 2] 1, [2, 1] 1, [0, 3] 217 218 2 2 219 + (- 2 fyy fxx - fyy - 2 gxy fyy - 2 fxy + (2 gxx + gyy) fxy + 6 A 220 2, [0, 3] 221 222 3 223 - 6 A - fyyy) v )/6 + . . . 224 1, [1, 2] 225dv 2 2 226-- = u + ((gxy fxx + 2 gxy fyy - 2 gxy - 2 gxx fxy - gxx - 2 gyy gxx 227dT 228 229 3 230 - 6 A + 6 A + gxxx) u 231 2, [2, 1] 1, [3, 0] 232 233 + ((2 gxx + gyy) fxx + (- 2 gxx + 2 gyy) fyy + (- 4 fxy + 3 gxx - 6 gyy) gxy 234 235 2 236 + 18 A - 12 A + 6 A + 3 gxxy) v u 237 2, [3, 0] 2, [1, 2] 1, [2, 1] 238 239 2 2 240 + (4 gxy fxx - gxy fyy + 4 gxy + (2 gxx - 2 gyy) fxy - 2 gxx + gyy gxx 241 242 2 2 243 - 2 gyy + 12 A - 18 A + 6 A + 3 gxyy) v u 244 2, [2, 1] 2, [0, 3] 1, [1, 2] 245 246 + (2 gyy fxx + gyy fyy + (2 fxy - 2 gxx + gyy) gxy + 6 A 247 2, [1, 2] 248 249 3 250 + 6 A + gyyy) v )/6 + . . . 251 1, [0, 3] 252DO YOU WANT TO ENTER ANOTHER TRANSFORMATION (Y/N) 253N; 254 du 255(D5)/T/ [-- = - v - (((fxy + gxx + 2 gyy) fxx - 2 fxy fyy + 2 fxy gxy 256 dT 257 258 3 259 + 6 A + 6 A - fxxx) u 260 2, [3, 0] 1, [2, 1] 261 262 2 2 2 263 + (- 2 fxx + (fyy - 2 gxy) fxx - 2 fyy + 2 gxy fyy + 4 fxy 264 265 + (- gxx + 4 gyy) fxy + 6 A - 18 A + 12 A - 3 fxxy) 266 2, [2, 1] 1, [3, 0] 1, [1, 2] 267 268 2 269 v u + ((- 6 fxy + 2 gxx + gyy) fxx + (3 fxy - 2 gxx + 2 gyy) fyy - 4 fxy gxy 270 271 2 272 + 6 A - 12 A + 18 A - 3 fxyy) v u 273 2, [1, 2] 1, [2, 1] 1, [0, 3] 274 275 2 2 276 + (- 2 fyy fxx - fyy - 2 gxy fyy - 2 fxy + (2 gxx + gyy) fxy + 6 A 277 2, [0, 3] 278 279 3 280 - 6 A - fyyy) v )/6 + . . ., 281 1, [1, 2] 282 283dv 2 2 284-- = u + ((gxy fxx + 2 gxy fyy - 2 gxy - 2 gxx fxy - gxx - 2 gyy gxx 285dT 286 287 3 288 - 6 A + 6 A + gxxx) u 289 2, [2, 1] 1, [3, 0] 290 291 + ((2 gxx + gyy) fxx + (- 2 gxx + 2 gyy) fyy + (- 4 fxy + 3 gxx - 6 gyy) gxy 292 293 2 294 + 18 A - 12 A + 6 A + 3 gxxy) v u 295 2, [3, 0] 2, [1, 2] 1, [2, 1] 296 297 2 2 298 + (4 gxy fxx - gxy fyy + 4 gxy + (2 gxx - 2 gyy) fxy - 2 gxx + gyy gxx 299 300 2 2 301 - 2 gyy + 12 A - 18 A + 6 A + 3 gxyy) v u 302 2, [2, 1] 2, [0, 3] 1, [1, 2] 303 304 + (2 gyy fxx + gyy fyy + (2 fxy - 2 gxx + gyy) gxy + 6 A 305 2, [1, 2] 306 307 3 308 + 6 A + gyyy) v )/6 + . . .] 309 1, [0, 3] 310 311 312(C6) hopf3(); 313 2 2 314(D6) [[A = (2 gyy + (gxx + 7 fxy) gyy - 3 gxyy - 2 gxy 315 1, [3, 0] 316 317 2 2 318 + (- fyy - 7 fxx) gxy - gxxx + 3 gxx + fxy gxx - fyyy - 3 fyy - fxx fyy 319 320 2 2 321 + 2 fxy - 3 fxxy - 2 fxx + 24 %R1)/24, A = %R2, 322 2, [3, 0] 323 324A = (- 3 gyyy + (3 gxy - 3 fyy - 16 fxx) gyy + (3 gxx - 16 fxy) gxy 325 1, [2, 1] 326 327 - 3 gxxy - 5 fxx gxx + 13 fxy fyy - 3 fxyy - 11 fxx fxy + 5 fxxx - 48 %R2) 328 329 2 2 330/48, A = (6 gyy + (9 fxy - 9 gxx) gyy - 9 gxyy - 18 gxy 331 2, [2, 1] 332 333 2 2 334 + (13 fyy - 11 fxx) gxy + 3 gxxx + 3 gxx - 15 fxy gxx + fyyy - fyy 335 336 2 2 337 + 3 fxx fyy + 6 fxy - 3 fxxy - 2 fxx + 48 %R1)/48, 338 339 2 2 340A = (2 gyy + (5 gxx + 3 fxy) gyy - 3 gxyy + 2 gxy 341 1, [1, 2] 342 343 2 2 344 + (- 17 fyy - 5 fxx) gxy - 3 gxxx + 5 gxx + 15 fxy gxx - 5 fyyy - 3 fyy 345 346 2 2 347 - 11 fxx fyy - 14 fxy + 3 fxxy + 2 fxx + 48 %R1)/48, 348 349A = - (3 gyyy + (21 gxy - 5 fyy + 4 fxx) gyy + (24 fxy - 15 gxx) gxy 350 2, [1, 2] 351 352 - 9 gxxy + (8 fyy - 7 fxx) gxx - 5 fxy fyy + 3 fxyy + 7 fxx fxy - fxxx 353 354 - 48 %R2)/48, A = (- gyyy + (5 gxy - 5 fyy - 6 fxx) gyy 355 1, [0, 3] 356 357 + (4 fxy - gxx) gxy - 3 gxxy + (4 fyy - 5 fxx) gxx - fxy fyy + 3 fxyy 358 359 + 5 fxx fxy + fxxx - 24 %R2)/24, A = %R1]] 360 2, [0, 3] 361 362 363(C7) expand(ev(newdes,%)); 364 2 3 3 3 3 2 3 365 du gyy v 5 gxx gyy v 5 fxy gyy v gxyy v gxy v 366(D7) [-- = ------- + ------------ - ------------ - ------- + ------- 367 dT 24 48 48 16 24 368 369 3 3 3 2 3 3 3 370 fyy gxy v 5 fxx gxy v gxxx v 5 gxx v fxy gxx v fyyy v 371 - ---------- - ------------ - ------- + --------- - ---------- + ------- 372 48 48 16 48 48 16 373 374 2 3 3 2 3 3 2 3 2 375 5 fyy v 5 fxx fyy v fxy v fxxy v fxx v gyyy u v 376 + --------- + ------------ + ------- + ------- + ------- + --------- 377 48 48 24 16 24 16 378 379 2 2 2 2 2 380 gxy gyy u v fyy gyy u v gxx gxy u v gxxy u v fxx gxx u v 381 - ------------ + ------------ - ------------ + --------- - ------------ 382 16 16 16 16 16 383 384 2 2 2 2 2 2 385 fxy fyy u v fxyy u v fxx fxy u v fxxx u v gyy u v 386 + ------------ + --------- + ------------ + --------- + --------- 387 16 16 16 16 24 388 389 2 2 2 2 2 2 390 5 gxx gyy u v 5 fxy gyy u v gxyy u v gxy u v fyy gxy u v 391 + -------------- - -------------- - --------- + --------- - ------------ 392 48 48 16 24 48 393 394 2 2 2 2 2 2 395 5 fxx gxy u v gxxx u v 5 gxx u v fxy gxx u v fyyy u v 396 - -------------- - --------- + ----------- - ------------ + --------- 397 48 16 48 48 16 398 399 2 2 2 2 2 2 2 2 400 5 fyy u v 5 fxx fyy u v fxy u v fxxy u v fxx u v 401 + ----------- + -------------- + --------- + --------- + --------- - v 402 48 48 24 16 24 403 404 3 3 3 3 3 3 405 gyyy u gxy gyy u fyy gyy u gxx gxy u gxxy u fxx gxx u 406 + ------- - ---------- + ---------- - ---------- + ------- - ---------- 407 16 16 16 16 16 16 408 409 3 3 3 3 410 fxy fyy u fxyy u fxx fxy u fxxx u 411 + ---------- + ------- + ---------- + -------, 412 16 16 16 16 413 414 3 3 3 3 3 3 415dv gyyy v gxy gyy v fyy gyy v gxx gxy v gxxy v fxx gxx v 416-- = ------- - ---------- + ---------- - ---------- + ------- - ---------- 417dT 16 16 16 16 16 16 418 419 3 3 3 3 2 2 2 420 fxy fyy v fxyy v fxx fxy v fxxx v gyy u v 5 gxx gyy u v 421 + ---------- + ------- + ---------- + ------- - --------- - -------------- 422 16 16 16 16 24 48 423 424 2 2 2 2 2 2 425 5 fxy gyy u v gxyy u v gxy u v fyy gxy u v 5 fxx gxy u v 426 + -------------- + --------- - --------- + ------------ + -------------- 427 48 16 24 48 48 428 429 2 2 2 2 2 2 2 430 gxxx u v 5 gxx u v fxy gxx u v fyyy u v 5 fyy u v 431 + --------- - ----------- + ------------ - --------- - ----------- 432 16 48 48 16 48 433 434 2 2 2 2 2 2 2 435 5 fxx fyy u v fxy u v fxxy u v fxx u v gyyy u v 436 - -------------- - --------- - --------- - --------- + --------- 437 48 24 16 24 16 438 439 2 2 2 2 2 440 gxy gyy u v fyy gyy u v gxx gxy u v gxxy u v fxx gxx u v 441 - ------------ + ------------ - ------------ + --------- - ------------ 442 16 16 16 16 16 443 444 2 2 2 2 2 3 3 445 fxy fyy u v fxyy u v fxx fxy u v fxxx u v gyy u 5 gxx gyy u 446 + ------------ + --------- + ------------ + --------- - ------- - ------------ 447 16 16 16 16 24 48 448 449 3 3 2 3 3 3 3 450 5 fxy gyy u gxyy u gxy u fyy gxy u 5 fxx gxy u gxxx u 451 + ------------ + ------- - ------- + ---------- + ------------ + ------- 452 48 16 24 48 48 16 453 454 2 3 3 3 2 3 3 2 3 455 5 gxx u fxy gxx u fyyy u 5 fyy u 5 fxx fyy u fxy u 456 - --------- + ---------- - ------- - --------- - ------------ - ------- 457 48 48 16 48 48 24 458 459 3 2 3 460 fxxy u fxx u 461 - ------- - ------- + u] 462 16 24 463 464(C8) transform(); 465ENTER NUMBER OF EQUATIONS 4662; 467ENTER SYMBOL FOR ORIGINAL VARIABLE 1 468u; 469ENTER SYMBOL FOR ORIGINAL VARIABLE 2 470v; 471ENTER SYMBOL FOR TRANSFORMED VARIABLE 1 472r; 473ENTER SYMBOL FOR TRANSFORMED VARIABLE 2 474theta; 475THE RHS'S OF THE D.E.'S ARE FUNCTIONS OF THE ORIGINAL VARIABLES: 476ENTER RHS OF u D.E. 477D u /DT = 478rhs(part(%,1)); 479 2 3 3 3 3 2 3 480 gyy v 5 gxx gyy v 5 fxy gyy v gxyy v gxy v 481D u /DT = ------- + ------------ - ------------ - ------- + ------- 482 24 48 48 16 24 483 484 3 3 3 2 3 3 3 485 fyy gxy v 5 fxx gxy v gxxx v 5 gxx v fxy gxx v fyyy v 486 - ---------- - ------------ - ------- + --------- - ---------- + ------- 487 48 48 16 48 48 16 488 489 2 3 3 2 3 3 2 3 2 490 5 fyy v 5 fxx fyy v fxy v fxxy v fxx v gyyy u v 491 + --------- + ------------ + ------- + ------- + ------- + --------- 492 48 48 24 16 24 16 493 494 2 2 2 2 2 495 gxy gyy u v fyy gyy u v gxx gxy u v gxxy u v fxx gxx u v 496 - ------------ + ------------ - ------------ + --------- - ------------ 497 16 16 16 16 16 498 499 2 2 2 2 2 2 500 fxy fyy u v fxyy u v fxx fxy u v fxxx u v gyy u v 501 + ------------ + --------- + ------------ + --------- + --------- 502 16 16 16 16 24 503 504 2 2 2 2 2 2 505 5 gxx gyy u v 5 fxy gyy u v gxyy u v gxy u v fyy gxy u v 506 + -------------- - -------------- - --------- + --------- - ------------ 507 48 48 16 24 48 508 509 2 2 2 2 2 2 510 5 fxx gxy u v gxxx u v 5 gxx u v fxy gxx u v fyyy u v 511 - -------------- - --------- + ----------- - ------------ + --------- 512 48 16 48 48 16 513 514 2 2 2 2 2 2 2 2 515 5 fyy u v 5 fxx fyy u v fxy u v fxxy u v fxx u v 516 + ----------- + -------------- + --------- + --------- + --------- - v 517 48 48 24 16 24 518 519 3 3 3 3 3 3 520 gyyy u gxy gyy u fyy gyy u gxx gxy u gxxy u fxx gxx u 521 + ------- - ---------- + ---------- - ---------- + ------- - ---------- 522 16 16 16 16 16 16 523 524 3 3 3 3 525 fxy fyy u fxyy u fxx fxy u fxxx u 526 + ---------- + ------- + ---------- + ------- 527 16 16 16 16 528ENTER RHS OF v D.E. 529D v /DT = 530rhs(part(%,2)); 531 3 3 3 3 3 3 532 gyyy v gxy gyy v fyy gyy v gxx gxy v gxxy v fxx gxx v 533D v /DT = ------- - ---------- + ---------- - ---------- + ------- - ---------- 534 16 16 16 16 16 16 535 536 3 3 3 3 2 2 2 537 fxy fyy v fxyy v fxx fxy v fxxx v gyy u v 5 gxx gyy u v 538 + ---------- + ------- + ---------- + ------- - --------- - -------------- 539 16 16 16 16 24 48 540 541 2 2 2 2 2 2 542 5 fxy gyy u v gxyy u v gxy u v fyy gxy u v 5 fxx gxy u v 543 + -------------- + --------- - --------- + ------------ + -------------- 544 48 16 24 48 48 545 546 2 2 2 2 2 2 2 547 gxxx u v 5 gxx u v fxy gxx u v fyyy u v 5 fyy u v 548 + --------- - ----------- + ------------ - --------- - ----------- 549 16 48 48 16 48 550 551 2 2 2 2 2 2 2 552 5 fxx fyy u v fxy u v fxxy u v fxx u v gyyy u v 553 - -------------- - --------- - --------- - --------- + --------- 554 48 24 16 24 16 555 556 2 2 2 2 2 557 gxy gyy u v fyy gyy u v gxx gxy u v gxxy u v fxx gxx u v 558 - ------------ + ------------ - ------------ + --------- - ------------ 559 16 16 16 16 16 560 561 2 2 2 2 2 3 3 562 fxy fyy u v fxyy u v fxx fxy u v fxxx u v gyy u 5 gxx gyy u 563 + ------------ + --------- + ------------ + --------- - ------- - ------------ 564 16 16 16 16 24 48 565 566 3 3 2 3 3 3 3 567 5 fxy gyy u gxyy u gxy u fyy gxy u 5 fxx gxy u gxxx u 568 + ------------ + ------- - ------- + ---------- + ------------ + ------- 569 48 16 24 48 48 16 570 571 2 3 3 3 2 3 3 2 3 572 5 gxx u fxy gxx u fyyy u 5 fyy u 5 fxx fyy u fxy u 573 - --------- + ---------- - ------- - --------- - ------------ - ------- 574 48 48 16 48 48 24 575 576 3 2 3 577 fxxy u fxx u 578 - ------- - ------- + u 579 16 24 580THE TRANSFORMATION IS ENTERED NEXT: 581ENTER u AS A FUNCTION OF THE NEW VARIABLES 582u = 583r*cos(theta); 584u = r COS(THETA) 585ENTER v AS A FUNCTION OF THE NEW VARIABLES 586v = 587r*sin(theta); 588v = r SIN(THETA) 589 dr 590(D8) [[-- = ((gyyy + (fyy - gxy) gyy - gxx gxy + gxxy - fxx gxx + fxy fyy 591 dT 592 593 3 2 594 + fxyy + fxx fxy + fxxx) r SIN (THETA) 595 596 + (gyyy + (fyy - gxy) gyy - gxx gxy + gxxy - fxx gxx + fxy fyy + fxyy 597 598 3 2 599 + fxx fxy + fxxx) r COS (THETA))/16, 600 601dTHETA 2 2 602------ = - ((2 gyy + (5 gxx - 5 fxy) gyy - 3 gxyy + 2 gxy 603 dT 604 605 2 2 606 + (- fyy - 5 fxx) gxy - 3 gxxx + 5 gxx - fxy gxx + 3 fyyy + 5 fyy 607 608 2 2 2 2 609 + 5 fxx fyy + 2 fxy + 3 fxxy + 2 fxx ) r SIN (THETA) 610 611 2 2 612 + (2 gyy + (5 gxx - 5 fxy) gyy - 3 gxyy + 2 gxy + (- fyy - 5 fxx) gxy 613 614 2 2 2 615 - 3 gxxx + 5 gxx - fxy gxx + 3 fyyy + 5 fyy + 5 fxx fyy + 2 fxy + 3 fxxy 616 617 2 2 2 618 + 2 fxx ) r COS (THETA) - 48)/48]] 619 620 621(C9) trigsimp(%); 622 dr 623(D9) [[-- = (gyyy + (fyy - gxy) gyy - gxx gxy + gxxy - fxx gxx + fxy fyy + fxyy 624 dT 625 626 3 dTHETA 2 627 + fxx fxy + fxxx) r /16, ------ = - ((2 gyy + (5 gxx - 5 fxy) gyy - 3 gxyy 628 dT 629 630 2 2 2 631 + 2 gxy + (- fyy - 5 fxx) gxy - 3 gxxx + 5 gxx - fxy gxx + 3 fyyy + 5 fyy 632 633 2 2 2 634 + 5 fxx fyy + 2 fxy + 3 fxxy + 2 fxx ) r - 48)/48]] 635 636 637 638Local Variables: *** 639mode: Text *** 640End: ***