share/pdiff/pdiff-doc.tex

   1 % TeX code generated by batTeX. Don't edit this file; edit the
   2 % input file pdiff.tex instead.
   3 \documentclass[12pt]{article}
   4 \usepackage{mathptm}
   5 \usepackage{battex}
   6 \usepackage{color}
   7
   8 \title{A positional derivative package for Maxima}
   9  \author{Barton Willis \\
  10    University of Nebraska at Kearney \\
  11    Kearney Nebraska}
  12 \date{September 10, 2002\footnote{With minor updates November, 2006.}}
  13
  14 \begin{document}
  15
  16 \maketitle
  17
  18 \subsection*{Introduction}
  19
  20 \noindent  Working with derivatives of unknown functions\footnote{By
  21 {\em unknown  function\/}, we mean a function that isn't bound to a formula and
  22 that has a derivative that isn't known to Maxima.} can be  cumbersome in Maxima.
  23 If we want, for example, the first order  Taylor polynomial of
  24 $f(x + x^2)$ about $x = 1$, we get
  25
  26
  27 \begin{mcline}{c1}
  28 taylor(f(x + x^2),x,1,1);
  29 \end{mcline}
  30
  31
  32
  33 \begin{mdline}{d1}
  34    f\left(2\right)+\left(\left.{{d}\over{d\,x}}\,f\left(x^2+x\right)
  35  \right|_{x=1}\right)\,\left(x-1\right)+\cdots
  36 \end{mdline}
  37
  38 \noindent To ``simplify'' the Taylor polynomial, we must assign a gradient to $f$
  39
  40 \begin{mcline}{c2}
  41 gradef(f(x),df(x))$
  42 \end{mcline}
  43
  44 \begin{mcline}{c3}
  45 taylor(f(x+x^2),x,1,1);
  46 \end{mcline}
  47
  48 \begin{mdline}{d3}
  49    f\left(2\right)+3\,df\left(2\right)\,\left(x-1\right)+\cdots
  50 \end{mdline}
  51
  52 This method works well for simple problems, but it
  53 is tedious for functions of several variables or high order
  54 derivatives. The positional derivative package {\tt pdiff} gives
  55 an alternative to using {\tt gradef} when working with derivatives
  56 of unknown functions.
  57
  58 \subsection*{Usage}
  59
  60 To use the positional derivative package, first load it from the Maxima input prompt.
  61
  62 \begin{mcline}{c1}
  63      load(pdiff)$
  64 \end{mcline}
  65
  66
  67 \noindent Loading
  68 {\tt pdiff.lisp} sets the option variable {\tt use\_pdiff} to true;
  69 when {\tt use\_diff} is true, Maxima will
  70 indicate derivatives of  unknown functions positionally. To illustrate,
  71 the first three  derivatives of $f$ are
  72
  73
  74 \begin{mcline}{c2}
  75 [diff(f(x),x),
  76   diff(f(x),x,2),
  77   diff(f(x),x,3)];
  78 \end{mcline}
  79
  80
  81
  82 \begin{mdline}{d2}
  83    \left[ f_{\left(1\right)}(x),f_{\left(2\right)}(x),f_{\left(3
  84  \right)}(x) \right]
  85 \end{mdline}
  86
  87 The subscript indicates the order of the derivative;  since $f$ is a function of
  88 one variable, the subscript has only one index.  When a function has more
  89 than one variable, the subscript has an index for each variable
  90
  91
  92 \begin{mcline}{c3}
  93      [diff(f(x,y),x,0,y,1), diff(f(y,x),x,0,y,1)];
  94 \end{mcline}
  95
  96 \begin{mdline}{d3}
  97    \left[ f_{\left(0,1\right)}(x,y),f_{\left(1,0\right)}(y,x)
  98   \right]
  99 \end{mdline}
 100
 101 \noindent Setting {\tt use\_pdiff} to false  (either locally or globally) inhibits
 102 derivatives from begin computed positionally
 103
 104
 105 \begin{mcline}{c4}
 106       diff(f(x,x^2),x), use_pdiff : false;
 107 \end{mcline}
 108
 109
 110
 111 \begin{mdline}{d4}
 112    {{d}\over{d\,x}}\,f\left(x,x^2\right)
 113 \end{mdline}
 114
 115 \begin{mcline}{c5}
 116    diff(f(x,x^2),x), use_pdiff : true;
 117 \end{mcline}
 118
 119
 120
 121 \begin{mdline}{d5}
 122    f_{\left(1,0\right)}(x,x^2)+2\,x\,f_{\left(0,1\right)}(x,x^2)
 123 \end{mdline}
 124
 125
 126 Taylor polynomials of unknown functions can be found without using
 127 {\tt gradef}. An example
 128
 129
 130 \begin{mcline}{c6}
 131      taylor(f(x+x^2),x,1,2);
 132 \end{mcline}
 133
 134
 135
 136 \begin{mdline}{d6}
 137    f\left(2\right)+3\,f_{\left(1\right)}(2)\,\left(x-1\right)+{{
 138  \left(2\,f_{\left(1\right)}(2)+9\,f_{\left(2\right)}(2)\right)\,
 139  \left(x-1\right)^2}\over{2}}+\cdots
 140 \end{mdline}
 141
 142 \noindent Additionally, we can verify that $ y = f(x-c \, t) + g(x+c \,t)$ is
 143 a solution to a wave equation without using {\tt gradef}
 144
 145
 146 \begin{mcline}{c7}
 147      y : f(x-c*t) + g(x+c*t)$
 148 \end{mcline}
 149
 150 \begin{mcline}{c8}
 151    ratsimp(diff(y,t,2) - c^2 * diff(y,x,2));
 152 \end{mcline}
 153
 154
 155
 156 \begin{mdline}{d8}
 157    0
 158 \end{mdline}
 159
 160 \begin{mcline}{c9}
 161    remvalue(y)$
 162 \end{mcline}
 163
 164 \noindent Expressions involving  positional derivatives can be differentiated
 165
 166 \begin{mcline}{c10}
 167      e : diff(f(x,y),x);
 168 \end{mcline}
 169
 170
 171 \begin{mdline}{d10}
 172    f_{\left(1,0\right)}(x,y)
 173 \end{mdline}
 174
 175 \begin{mcline}{c11}
 176    diff(e,y);
 177 \end{mcline}
 178
 179
 180
 181 \begin{mdline}{d11}
 182    f_{\left(1,1\right)}(x,y)
 183 \end{mdline}
 184
 185
 186 \noindent The chain rule is applied when needed
 187
 188
 189 \begin{mcline}{c12}
 190      [diff(f(x^2),x), diff(f(g(x)),x)];
 191 \end{mcline}
 192
 193
 194
 195 \begin{mdline}{d12}
 196    \left[ 2\,x\,f_{\left(1\right)}(x^2),g_{\left(1\right)}(x)\,f_{
 197  \left(1\right)}(g\left(x\right)) \right]
 198 \end{mdline}
 199
 200
 201 The positional derivative package doesn't alter the way known functions
 202 are differentiated
 203
 204
 205 \begin{mcline}{c13}
 206      diff(exp(-x^2),x);
 207 \end{mcline}
 208
 209
 210
 211 \begin{mdline}{d13}
 212    -2\,x\,e^ {- x^2 }
 213 \end{mdline}
 214
 215
 216 \noindent To convert positional derivatives to standard Maxima derivatives,
 217 use {\tt convert\_to\_diff}
 218
 219
 220 \begin{mcline}{c14}
 221      e : [diff(f(x),x), diff(f(x,y),x,1,y,1)];
 222 \end{mcline}
 223
 224
 225
 226 \begin{mdline}{d14}
 227    \left[ f_{\left(1\right)}(x),f_{\left(1,1\right)}(x,y) \right]
 228 \end{mdline}
 229
 230 \begin{mcline}{c15}
 231    e : convert_to_diff(e);
 232 \end{mcline}
 233
 234
 235
 236 \begin{mdline}{d15}
 237    \left[ {{d}\over{d\,x}}\,f\left(x\right),{{d^2}\over{d\,y\,d\,x}}
 238  \,f\left(x,y\right) \right]
 239 \end{mdline}
 240
 241 \noindent To convert back to a positional derivative, use {\tt ev} with {\tt diff}
 242 as an argument
 243
 244
 245 \begin{mcline}{c16}
 246      ev(e,diff);
 247 \end{mcline}
 248
 249
 250
 251 \begin{mdline}{d16}
 252    \left[ f_{\left(1\right)}(x),f_{\left(1,1\right)}(x,y) \right]
 253 \end{mdline}
 254
 255 \noindent Conversion to standard derivatives  sometimes requires the
 256 introduction of  a dummy variable. Here's an example
 257
 258
 259 \begin{mcline}{c17}
 260      e : diff(f(x,y),x,1,y,1);
 261 \end{mcline}
 262
 263
 264
 265 \begin{mdline}{d17}
 266    f_{\left(1,1\right)}(x,y)
 267 \end{mdline}
 268
 269 \begin{mcline}{c18}
 270    e : subst(p(s),y,e);
 271 \end{mcline}
 272
 273
 274
 275 \begin{mdline}{d18}
 276    f_{\left(1,1\right)}(x,p\left(s\right))
 277 \end{mdline}
 278
 279 \begin{mcline}{c19}
 280    e : convert_to_diff(e);
 281 \end{mcline}
 282
 283
 284
 285 \begin{mdline}{d19}
 286    \left.{{d^2}\over{d\,\%x_0\,d\,x}}\,f\left(x,\%x_0\right)\right|_{
 287  \left[ \%x_0=p\left(s\right) \right] }
 288 \end{mdline}
 289
 290 Dummy variables have the form ci, where i=0,1,2\dots and c is the
 291 value of the option variable {\tt dummy\_char}. The default value
 292 for {\tt dummy\_char} is {\tt \%x}. If a user variable conflicts with a
 293 dummy variable,  the conversion process can give an
 294 incorrect value. To convert the previous expression back to a positional derivative,
 295 use {\tt ev} with {\tt diff} and {\tt at} as arguments
 296
 297
 298 \begin{mcline}{c20}
 299       ev(e,diff,at);
 300 \end{mcline}
 301
 302
 303
 304 \begin{mdline}{d20}
 305    f_{\left(1,1\right)}(x,p\left(s\right))
 306 \end{mdline}
 307
 308
 309 Maxima correctly evaluates expressions involving positional derivatives
 310 if a formula is  later given to the function.  (Thus converting an unknown
 311 function into a known one.)  Here is an example;  let
 312
 313
 314 \begin{mcline}{c21}
 315      e : diff(f(x^2),x);
 316 \end{mcline}
 317
 318
 319
 320 \begin{mdline}{d21}
 321    2\,x\,f_{\left(1\right)}(x^2)
 322 \end{mdline}
 323
 324 \noindent Now, give $f$ a formula
 325
 326
 327 \begin{mcline}{c22}
 328      f(x) := x^5;
 329 \end{mcline}
 330
 331
 332
 333 \begin{mdline}{d22}
 334    f\left(x\right):=x^5
 335 \end{mdline}
 336
 337 \noindent and evaluate {\tt e}
 338
 339
 340 \begin{mcline}{c23}
 341      ev(e);
 342 \end{mcline}
 343
 344
 345
 346 \begin{mdline}{d23}
 347    10\,x^9
 348 \end{mdline}
 349
 350
 351 \noindent This result is the same as
 352
 353
 354
 355 \begin{mcline}{c24}
 356      diff(f(x^2),x);
 357 \end{mcline}
 358
 359
 360
 361 \begin{mdline}{d24}
 362    10\,x^9
 363 \end{mdline}
 364
 365 \noindent In this calculation, Maxima first evaluates $f(x)$ to $x^{10}$ and then
 366 does the derivative.  Additionally, we can substitute a value for $x$
 367 before evaluating
 368
 369
 370
 371 \begin{mcline}{c25}
 372      ev(subst(2,x,e));
 373 \end{mcline}
 374
 375
 376
 377 \begin{mdline}{d25}
 378    5120
 379 \end{mdline}
 380
 381
 382 \noindent We can duplicate this with
 383
 384
 385
 386 \begin{mcline}{c26}
 387      subst(2,x,diff(f(x^2),x));
 388 \end{mcline}
 389
 390
 391
 392 \begin{mdline}{d26}
 393    5120
 394 \end{mdline}
 395
 396 \begin{mcline}{c27}
 397    remfunction(f);
 398 \end{mcline}
 399
 400
 401
 402 \begin{mdline}{d27}
 403    \left[ f \right]
 404 \end{mdline}
 405
 406
 407 \noindent We can also evaluate a positional derivative using a local
 408 function definition
 409
 410
 411 \begin{mcline}{c28}
 412      e : diff(g(x),x);
 413 \end{mcline}
 414
 415
 416
 417 \begin{mdline}{d28}
 418    g_{\left(1\right)}(x)
 419 \end{mdline}
 420
 421 \begin{mcline}{c29}
 422    e, g(x) := sqrt(x);
 423 \end{mcline}
 424
 425
 426
 427 \begin{mdline}{d29}
 428    {{1}\over{2\,\sqrt{x}}}
 429 \end{mdline}
 430
 431 \begin{mcline}{c30}
 432    e, g = sqrt;
 433 \end{mcline}
 434
 435
 436
 437 \begin{mdline}{d30}
 438    {{1}\over{2\,\sqrt{x}}}
 439 \end{mdline}
 440
 441 \begin{mcline}{c31}
 442    e, g = h;
 443 \end{mcline}
 444
 445
 446
 447 \begin{mdline}{d31}
 448    h_{\left(1\right)}(x)
 449 \end{mdline}
 450
 451 \begin{mcline}{c32}
 452    e, g = lambda([t],t^2);
 453 \end{mcline}
 454
 455
 456
 457 \begin{mdline}{d32}
 458    2\,x
 459 \end{mdline}
 460
 461
 462 \subsection*{The {\tt pderivop} function}
 463
 464 If $F$ is an atom and $i_1, i_2, \dots i_n$ are nonnegative
 465 integers, then $ \mbox{pderivop}(F,i_1,i_2, \dots i_n)$, is the
 466 function that has the formula
 467 \[
 468    \frac{\partial^{i_1 + i_2 + \cdots + i_n}}{\partial x_1^{i_1}
 469      \partial x_2^{i_2}  \cdots \partial x_n^{i_n}} F(x_1,x_2, \dots x_n).
 470 \]
 471 If any of the derivative arguments are not nonnegative integers,
 472 we'll get an error
 473
 474
 475 \begin{mcline}{c33}
 476      pderivop(f,2,-1);
 477 \end{mcline}
 478
 479 Each derivative order must be a nonnegative integer
 480
 481
 482 \noindent The {\tt pderivop} function can be composed with itself
 483
 484 \begin{mcline}{c34}
 485      pderivop(pderivop(f,3,4),1,2);
 486 \end{mcline}
 487
 488
 489
 490 \begin{mdline}{d34}
 491    f_{\left(4,6\right)}
 492 \end{mdline}
 493 \noindent If the number of derivative arguments between two calls
 494 to {\tt pderivop} isn't the same, Maxima gives an error
 495
 496 \begin{mcline}{c35}
 497      pderivop(pderivop(f,3,4),1);
 498 \end{mcline}
 499
 500 The function f expected 2 derivative argument(s), but it received 1
 501
 502
 503 When {\tt pderivop} is applied to a known function, the result is
 504 a lambda form\footnote{If you repeat theses calculations, you
 505 may get a different prefix for the {\tt gensym} variables.}
 506
 507 \begin{mcline}{c37}
 508       f(x) := x^2;
 509 \end{mcline}
 510
 511
 512
 513 \begin{mdline}{d37}
 514    f\left(x\right):=x^2
 515 \end{mdline}
 516
 517 \begin{mcline}{c38}
 518    df : pderivop(f,1);
 519 \end{mcline}
 520
 521
 522
 523 \begin{mdline}{d38}
 524    \lambda\left(\left[ Q_{1253} \right] ,2\,Q_{1253}\right)
 525 \end{mdline}
 526
 527 \begin{mcline}{c39}
 528    apply(df,[z]);
 529 \end{mcline}
 530
 531
 532
 533 \begin{mdline}{d39}
 534    2\,z
 535 \end{mdline}
 536
 537 \begin{mcline}{c40}
 538    ddf : pderivop(f,2);
 539 \end{mcline}
 540
 541
 542
 543 \begin{mdline}{d40}
 544    \lambda\left(\left[ Q_{1254} \right] ,2\right)
 545 \end{mdline}
 546
 547 \begin{mcline}{c41}
 548    apply(ddf,[10]);
 549 \end{mcline}
 550
 551
 552
 553 \begin{mdline}{d41}
 554    2
 555 \end{mdline}
 556
 557 \begin{mcline}{c42}
 558    remfunction(f);
 559 \end{mcline}
 560
 561
 562
 563 \begin{mdline}{d42}
 564    \left[ f \right]
 565 \end{mdline}
 566
 567
 568 If the first argument to {\tt pderivop} is a lambda form, the
 569 result is another lambda form
 570
 571
 572
 573 \begin{mcline}{c43}
 574      f : pderivop(lambda([x],x^2),1);
 575 \end{mcline}
 576
 577
 578
 579 \begin{mdline}{d43}
 580    \lambda\left(\left[ Q_{1255} \right] ,2\,Q_{1255}\right)
 581 \end{mdline}
 582
 583 \begin{mcline}{c44}
 584    apply(f,[a]);
 585 \end{mcline}
 586
 587
 588
 589 \begin{mdline}{d44}
 590    2\,a
 591 \end{mdline}
 592
 593 \begin{mcline}{c45}
 594    f : pderivop(lambda([x],x^2),2);
 595 \end{mcline}
 596
 597
 598
 599 \begin{mdline}{d45}
 600    \lambda\left(\left[ Q_{1256} \right] ,2\right)
 601 \end{mdline}
 602
 603 \begin{mcline}{c46}
 604    apply(f,[a]);
 605 \end{mcline}
 606
 607
 608
 609 \begin{mdline}{d46}
 610    2
 611 \end{mdline}
 612
 613 \begin{mcline}{c47}
 614    f : pderivop(lambda([x],x^2),3);
 615 \end{mcline}
 616
 617
 618
 619 \begin{mdline}{d47}
 620    \lambda\left(\left[ Q_{1257} \right] ,0\right)
 621 \end{mdline}
 622
 623 \begin{mcline}{c48}
 624    apply(f,[a]);
 625 \end{mcline}
 626
 627
 628
 629 \begin{mdline}{d48}
 630    0
 631 \end{mdline}
 632
 633 \begin{mcline}{c49}
 634    remvalue(f)$
 635 \end{mcline}
 636
 637
 638 If the first argument to {\tt pderivop} isn't an atom or
 639 a lambda form, Maxima will signal an error
 640
 641
 642
 643 \begin{mcline}{c50}
 644      pderivop(f+g,1);
 645 \end{mcline}
 646
 647 Non-atom g+f used as a function
 648
 649
 650 You may use {\tt tellsimpafter} together with {\tt pderivop} to
 651 give a value to a derivative of a function at a point; an
 652 example
 653
 654
 655 \begin{mcline}{c51}
 656 tellsimpafter(pderivop(f,1)(1),1)$
 657 \end{mcline}
 658
 659 \begin{mcline}{c52}
 660 tellsimpafter(pderivop(f,2)(1),2)$
 661 \end{mcline}
 662
 663 \begin{mcline}{c53}
 664 diff(f(x),x,2) + diff(f(x),x)$
 665 \end{mcline}
 666
 667 \begin{mcline}{c54}
 668 subst(1,x,%);
 669 \end{mcline}
 670
 671 \begin{mdline}{d54}
 672    3
 673 \end{mdline}
 674
 675 This technique works for functions of several variables as well
 676
 677 \begin{mcline}{c55}
 678 kill(rules)$
 679 \end{mcline}
 680
 681
 682 \begin{mcline}{c56}
 683 tellsimpafter(pderivop(f,1,0)(0,0),a)$
 684 \end{mcline}
 685
 686 \begin{mcline}{c57}
 687 tellsimpafter(pderivop(f,0,1)(0,0),b)$
 688 \end{mcline}
 689
 690 \begin{mcline}{c58}
 691 sublis([x = 0, y = 0], diff(f(x,y),x) + diff(f(x,y),y));
 692 \end{mcline}
 693
 694
 695
 696 \begin{mdline}{d58}
 697    b+a
 698 \end{mdline}
 699
 700
 701 \subsection*{\TeX{}-ing positional derivatives}
 702
 703 Several option variables control how positional derivatives
 704 are converted to \TeX{}. When the option variable
 705 {\tt tex\_uses\_prime\_for\_derivatives} is true (default false),
 706 makes functions of one variable \TeX{} as superscripted primes
 707
 708
 709 \begin{mcline}{c59}
 710 tex_uses_prime_for_derivatives : true$
 711 \end{mcline}
 712
 713 \begin{mcline}{c60}
 714 tex(makelist(diff(f(x),x,i),i,1,3))$
 715 \end{mcline}
 716
 717 \begin{mdline}{d60}
 718 \left[ f^{\prime}(x),f^{\prime\prime}(x),f^{\prime\prime\prime}(x)
 719   \right]
 720 \end{mdline}
 721
 722 \begin{mcline}{c61}
 723    tex(makelist(pderivop(f,i),i,1,3))$
 724 \end{mcline}
 725
 726 \[
 727 \left[ f^{\prime},f^{\prime\prime},f^{\prime\prime\prime} \right]
 728 \]
 729
 730 When the derivative order exceeds the value of the option
 731 variable {\tt tex\_prime\_limit}, (default 3)
 732 derivatives are indicated with parenthesis delimited superscripts
 733
 734
 735 \begin{mcline}{c62}
 736    tex(makelist(pderivop(f,i),i,1,5)), tex_prime_limit : 0$
 737 \end{mcline}
 738
 739 \[
 740 \left[ f^{(1)},f^{(2)},f^{(3)},f^{(4)},f^{(5)} \right]
 741 \]
 742
 743 \begin{mcline}{c63}
 744    tex(makelist(pderivop(f,i),i,1,5)), tex_prime_limit : 5$
 745 \end{mcline}
 746
 747
 748 \[
 749 \left[ f^{\prime},f^{\prime\prime},f^{\prime\prime\prime},f^{\prime
 750  \prime\prime\prime},f^{\prime\prime\prime\prime\prime} \right]
 751 \]
 752
 753 The value of {\tt tex\_uses\_prime\_for\_derivatives} doesn't change the way
 754 functions of two or more variables are converted to \TeX{}.
 755
 756
 757 \begin{mcline}{c64}
 758      tex(pderivop(f,2,1));
 759 \end{mcline}
 760
 761 \[
 762 f_{\left(2,1\right)}
 763 \]
 764
 765
 766
 767 When the option variable {\tt tex\_uses\_named\_subscripts\_for\_derivatives}
 768 (default false) is true, a derivative with respect to the i-th argument is
 769 indicated by a subscript that is the i-th element of the option variable
 770 {\tt tex\_diff\_var\_names}.  An example is the clearest way to
 771 describe this.
 772
 773
 774 \begin{mcline}{c65}
 775      tex_uses_named_subscripts_for_derivatives : true$
 776 \end{mcline}
 777
 778 \begin{mcline}{c66}
 779    tex_diff_var_names;
 780 \end{mcline}
 781
 782
 783
 784 \begin{mdline}{d66}
 785    \left[ x,y,z \right]
 786 \end{mdline}
 787
 788 \begin{mcline}{c67}
 789
 790   tex([pderivop(f,1,0), pderivop(f,0,1), pderivop(f,1,1), pderivop(f,2,0)]);
 791 \end{mcline}
 792
 793 $$\left[ f_{x},f_{y},f_{xy},f_{xx} \right] $$
 794
 795
 796 \begin{mcline}{c68}
 797    tex_diff_var_names : [a,b];
 798 \end{mcline}
 799
 800
 801
 802 \begin{mdline}{d68}
 803    \left[ a,b \right]
 804 \end{mdline}
 805
 806 \begin{mcline}{c69}
 807
 808   tex([pderivop(f,1,0), pderivop(f,0,1), pderivop(f,1,1), pderivop(f,2,0)]);
 809 \end{mcline}
 810
 811 $$\left[ f_{a},f_{b},f_{ab},f_{aa} \right] $$
 812
 813
 814
 815 \begin{mcline}{c70}
 816    tex_diff_var_names : [x,y,z];
 817 \end{mcline}
 818
 819
 820
 821 \begin{mdline}{d70}
 822    \left[ x,y,z \right]
 823 \end{mdline}
 824
 825 \begin{mcline}{c71}
 826    tex([diff(f(x,y),x), diff(f(y,x),y)]);
 827 \end{mcline}
 828
 829 $$\left[ f_{x}(x,y),f_{x}(y,x) \right] $$
 830
 831
 832
 833 When the derivative order exceeds {tt tex\_prime\_limit}, revert to the default
 834 method for converting to \TeX{}
 835
 836
 837 \begin{mcline}{c72}
 838      tex(diff(f(x,y,z),x,1,y,1,z,1)), tex_prime_limit : 4$
 839 \end{mcline}
 840
 841 \[
 842 f_{xyz}(x,y,z)
 843 \]
 844
 845 \begin{mcline}{c73}
 846 tex(diff(f(x,y,z),x,1,y,1,z,1)), tex_prime_limit : 1$
 847 \end{mcline}
 848
 849 $$f_{\left(1,1,1\right)}(x,y,z)$$
 850 \subsection*{A longer example}
 851
 852 We'll use the positional derivative package to change the independent
 853 variable of the differential equation
 854
 855
 856 \begin{mcline}{c74}
 857       de :  4*x^2*'diff(y,x,2) + 4*x*'diff(y,x,1) + (x-1)*y = 0;
 858 \end{mcline}
 859
 860
 861
 862 \begin{mdline}{d74}
 863    4\,x^2\,\left({{d^2}\over{d\,x^2}}\,y\right)+4\,x\,\left({{d
 864  }\over{d\,x}}\,y\right)+\left(x-1\right)\,y=0
 865 \end{mdline}
 866
 867 With malice aforethought, we'll assume a solution of the form
 868 $ y = g(x^n)$, where $n$ is a number.  Substituting $y \rightarrow g(x^n)$
 869 in the  differential equation  gives
 870
 871
 872 \begin{mcline}{c75}
 873      de : subst(g(x^n),y,de);
 874 \end{mcline}
 875
 876
 877
 878 \begin{mdline}{d75}
 879    4\,x^2\,\left({{d^2}\over{d\,x^2}}\,g\left(x^{n}\right)\right)+4\,
 880  x\,\left({{d}\over{d\,x}}\,g\left(x^{n}\right)\right)+\left(x-1
 881  \right)\,g\left(x^{n}\right)=0
 882 \end{mdline}
 883
 884 \begin{mcline}{c76}
 885    de : ev(de, diff);
 886 \end{mcline}
 887
 888
 889
 890 \begin{mdline}{d76}
 891    4\,x^2\,\left(n^2\,x^{2\,n-2}\,g^{\prime\prime}(x^{n})+\left(n-1
 892  \right)\,n\,x^{n-2}\,g^{\prime}(x^{n})\right)+4\,n\,x^{n}\,g^{\prime
 893  }(x^{n})+\left(x-1\right)\,g\left(x^{n}\right)=0
 894 \end{mdline}
 895
 896 Now let $x \rightarrow t^{1/n}$
 897
 898
 899 \begin{mcline}{c77}
 900      de : radcan(subst(x^(1/n),x, de));
 901 \end{mcline}
 902
 903
 904
 905 \begin{mdline}{d77}
 906    4\,n^2\,x^2\,g^{\prime\prime}(x)+4\,n^2\,x\,g^{\prime}(x)+\left(x
 907  ^{{{1}\over{n}}}-1\right)\,g\left(x\right)=0
 908 \end{mdline}
 909
 910 Setting $n \rightarrow 1/2$, we recognize that $g$ is the order 1 Bessel
 911 equation
 912
 913
 914 \begin{mcline}{c78}
 915      subst(1/2,n, de);
 916 \end{mcline}
 917
 918
 919
 920 \begin{mdline}{d78}
 921    x^2\,g^{\prime\prime}(x)+x\,g^{\prime}(x)+\left(x^2-1\right)\,g
 922  \left(x\right)=0
 923 \end{mdline}
 924
 925
 926 \subsection*{Limitations}
 927
 928 \begin{itemize}
 929   \item Positional derivatives of subscripted functions are not allowed.
 930
 931   \item Derivatives of unknown functions with symbolic orders are not
 932    computed positionally.
 933
 934   \item The {\tt pdiff.lisp} code alters the Maxima functions {\tt mqapply}
 935    and {\tt sdiffgrad} Although I'm unaware  of any problems associated
 936    with these altered functions, there may be some.  Setting {\tt use\_pdiff} to
 937    false should restore {\tt mqapply} and {\tt sdiffgrad} to their
 938    original functioning.
 939
 940 \end{itemize}
 941
 942 \subsection*{Conclusion}
 943
 944 The {\tt pdiff} package provides a simple way of working with
 945 derivatives of unknown functions.  If you find a bug in the package,
 946 or if you have a comment or a question, please send it to
 947 {\tt willisb@unk.edu}.
 948
 949 The {\tt pdiff} package could serve as a basis for a Maxima package
 950 differential and integral operators.
 951
 952 \end{document}