(*^ ::[ Information = "This is a Mathematica Notebook file. It contains ASCII text, and can be transferred by email, ftp, or other text-file transfer utility. It should be read or edited using a copy of Mathematica or MathReader. If you received this as email, use your mail application or copy/paste to save everything from the line containing (*^ down to the line containing ^*) into a plain text file. On some systems you may have to give the file a name ending with ".ma" to allow Mathematica to recognize it as a Notebook. 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fontset = special1, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, fullFontName, -9, "-*-times-medium-r-normal-*-*-120-75-75-*-*-*-*", 12, fontName, "times"; fontset = special2, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, fullFontName, -9, "-*-times-medium-r-normal-*-*-120-75-75-*-*-*-*", 12, fontName, "times"; fontset = special3, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, fullFontName, -9, "-*-times-medium-r-normal-*-*-120-75-75-*-*-*-*", 12, fontName, "times"; fontset = special4, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, fullFontName, -9, "-*-times-medium-r-normal-*-*-120-75-75-*-*-*-*", 12, fontName, "times"; fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, fullFontName, -9, "-*-times-medium-r-normal-*-*-120-75-75-*-*-*-*", 12, fontName, "times";paletteColors = 128; currentKernel; ] :[font = title; inactive; preserveAspect; startGroup] Beginning Mathematica :[font = subtitle; inactive; preserveAspect] D.C.M. Burbulla and C.T.J. Dodson :[font = subtitle; inactive; preserveAspect] University of Toronto :[font = text; inactive; preserveAspect] This NoteBook is intended to illustrate some basic operations in Mathematica and can be personalized for saving in your home directory for reference purposes. Note the use of square brackets [ ] to enclose arguments of functions, and note that Mathematica functions are usually standard names but start with capital letters. You can get help about a function by using ? before the function name. Click on the side bars to open interactive cells. :[font = subsection; inactive; Cclosed; preserveAspect; groupLikeInput; startGroup] ELEMENTARY ALGEBRA CALCULUS AND PLOTS :[font = input; Cclosed; preserveAspect; startGroup] ?Expand :[font = print; inactive; preserveAspect; groupLikeInput; endGroup] Expand[expr] expands out products and positive integer powers in expr. Expand[expr, patt] avoids expanding elements of expr which do not contain terms matching the pattern patt. :[font = input; Cclosed; preserveAspect; startGroup] Expand[(x^2-3x)^3] ;[s] 2:0,0;1,1;20,-1; 2:1,0,-9 -*-courier-bold-r-normal-*-*-120-75-75-*-*-*-*,courier,1,12,0,0,0;1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,courier,0,12,0,0,0; :[font = output; output; inactive; preserveAspect; groupLikeInput; endGroup] -27*x^3 + 27*x^4 - 9*x^5 + x^6 ;[o] 3 4 5 6 -27 x + 27 x - 9 x + x :[font = input; Cclosed; preserveAspect; startGroup] ?Solve :[font = print; inactive; preserveAspect; groupLikeInput; endGroup] Solve[eqns, vars] attempts to solve an equation or set of equations for the variables vars. Any variable ineqns but not vars is regarded as a parameter. Solve[eqns] treats all variables encountered as vars above. 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Metetra .549 .658 .908 r .42393 .50581 .44685 .53492 .48651 .5329 .46383 .50344 Metetra .596 .678 .898 r .46383 .50344 .48651 .5329 .52705 .52665 .50466 .49682 Metetra .637 .694 .886 r .50466 .49682 .52705 .52665 .56834 .51608 .54628 .48586 Metetra .672 .707 .874 r .54628 .48586 .56834 .51608 .61022 .50112 .58853 .47048 Metetra .702 .718 .862 r .58853 .47048 .61022 .50112 .65254 .48173 .63127 .45065 Metetra .728 .727 .849 r .63127 .45065 .65254 .48173 .69515 .4579 .67433 .42637 Metetra .751 .734 .837 r .67433 .42637 .69515 .4579 .73786 .42968 .71753 .39767 Metetra .771 .741 .825 r .71753 .39767 .73786 .42968 .78051 .39713 .76069 .36463 Metetra .79 .746 .813 r .76069 .36463 .78051 .39713 .82292 .36036 .80365 .32736 Metetra .806 .751 .802 r .80365 .32736 .82292 .36036 .86493 .31951 .84622 .286 Metetra .18 .506 .911 r .2515 .4511 .27581 .4758 .31092 .48888 .2867 .46396 Metetra .293 .57 .934 r .2867 .46396 .31092 .48888 .34737 .49836 .32328 .47321 Metetra .388 .62 .944 r .32328 .47321 .34737 .49836 .38507 .50407 .36117 .47865 Metetra .468 .658 .944 r .36117 .47865 .38507 .50407 .42393 .50581 .40027 .4801 Metetra .535 .687 .937 r .40027 .4801 .42393 .50581 .46383 .50344 .44046 .4774 Metetra .59 .709 .926 r .44046 .4774 .46383 .50344 .50466 .49682 .48164 .47043 Metetra .636 .727 .914 r .48164 .47043 .50466 .49682 .54628 .48586 .52365 .45908 Metetra .675 .74 .9 r .52365 .45908 .54628 .48586 .58853 .47048 .56634 .44328 Metetra .708 .751 .885 r .56634 .44328 .58853 .47048 .63127 .45065 .60956 .42299 Metetra .736 .759 .871 r .60956 .42299 .63127 .45065 .67433 .42637 .65314 .39822 Metetra .761 .766 .857 r .65314 .39822 .67433 .42637 .71753 .39767 .6969 .36901 Metetra .783 .772 .843 r .6969 .36901 .71753 .39767 .76069 .36463 .74065 .33543 Metetra .802 .777 .829 r .74065 .33543 .76069 .36463 .80365 .32736 .78423 .29759 Metetra .82 .781 .816 r .78423 .29759 .80365 .32736 .84622 .286 .82743 .25565 Metetra .094 .495 .901 r .22617 .42964 .2515 .4511 .2867 .46396 .26148 .44234 Metetra .226 .574 .941 r .26148 .44234 .2867 .46396 .32328 .47321 .29824 .45141 Metetra .339 .636 .962 r .29824 .45141 .32328 .47321 .36117 .47865 .33636 .45663 Metetra .435 .683 .969 r .33636 .45663 .36117 .47865 .40027 .4801 .37574 .45782 Metetra .513 .718 .965 r .37574 .45782 .40027 .4801 .44046 .4774 .41629 .45483 Metetra .578 .743 .955 r .41629 .45483 .44046 .4774 .48164 .47043 .45786 .44752 Metetra .631 .762 .941 r .45786 .44752 .48164 .47043 .52365 .45908 .50033 .43578 Metetra .676 .776 .926 r .50033 .43578 .52365 .45908 .56634 .44328 .54353 .41955 Metetra .713 .787 .91 r .54353 .41955 .56634 .44328 .60956 .42299 .5873 .39879 Metetra .744 .795 .893 r .5873 .39879 .60956 .42299 .65314 .39822 .63148 .3735 Metetra .771 .801 .877 r .63148 .3735 .65314 .39822 .6969 .36901 .67587 .34373 Metetra .795 .806 .861 r .67587 .34373 .6969 .36901 .74065 .33543 .7203 .30955 Metetra .815 .809 .845 r .7203 .30955 .74065 .33543 .78423 .29759 .76457 .27108 Metetra .833 .812 .83 r .76457 .27108 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.02763 .47311 m .03116 .47187 L s P P 0 0 m 1 0 L 1 .82055 L 0 .82055 L closepath clip newpath p .002 w .06024 .26735 m .02494 .49015 L s .02494 .49015 m .40296 .79562 L s .40296 .79562 m .41001 .59401 L s .41001 .59401 m .06024 .26735 L s .67932 .02494 m .94594 .43277 L s .94594 .43277 m .97506 .64585 L s .97506 .64585 m .69286 .25814 L s .69286 .25814 m .67932 .02494 L s .06024 .26735 m .02494 .49015 L s .02494 .49015 m .69286 .25814 L s .69286 .25814 m .67932 .02494 L s .67932 .02494 m .06024 .26735 L s .41001 .59401 m .94594 .43277 L s .94594 .43277 m .97506 .64585 L s .97506 .64585 m .40296 .79562 L s .40296 .79562 m .41001 .59401 L s P p .744 .635 .738 r .0015 w .38203 .74551 .40314 .79053 .44075 .75394 .41986 .70875 Metetra .733 .63 .744 r .41986 .70875 .44075 .75394 .47826 .72146 .45756 .67607 Metetra .719 .624 .75 r .45756 .67607 .47826 .72146 .51581 .69297 .49527 .64734 Metetra .705 .617 .756 r .49527 .64734 .51581 .69297 .55352 .6684 .53313 .6225 Metetra .689 .61 .762 r .53313 .6225 .55352 .6684 .59154 .64769 .57128 .60148 Metetra .672 .601 .768 r .57128 .60148 .59154 .64769 .63001 .63082 .60985 .58426 Metetra .652 .592 .774 r .60985 .58426 .63001 .63082 .66909 .61778 .64901 .57083 Metetra .632 .581 .779 r .64901 .57083 .66909 .61778 .70892 .6086 .6889 .56123 Metetra .609 .569 .784 r .6889 .56123 .70892 .6086 .74968 .60335 .7297 .5555 Metetra .585 .556 .788 r .7297 .5555 .74968 .60335 .79154 .60211 .7716 .55374 Metetra .559 .542 .792 r .7716 .55374 .79154 .60211 .83471 .60502 .81477 .55607 Metetra .532 .526 .794 r .81477 .55607 .83471 .60502 .8794 .61223 .85945 .56265 Metetra .503 .51 .796 r .85945 .56265 .8794 .61223 .92585 .62395 .90588 .57369 Metetra .473 .492 .797 r .90588 .57369 .92585 .62395 .97432 .64043 .95431 .58943 Metetra .753 .65 .747 r .36048 .70409 .38203 .74551 .41986 .70875 .39858 .66711 Metetra .741 .646 .754 r .39858 .66711 .41986 .70875 .45756 .67607 .43652 .63418 Metetra .727 .64 .761 r .43652 .63418 .45756 .67607 .49527 .64734 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.91515 .49845 Metetra .773 .686 .768 r .31573 .63173 .33841 .66618 .37683 .62893 .35453 .59416 Metetra .76 .683 .777 r .35453 .59416 .37683 .62893 .41507 .59571 .39311 .56061 Metetra .746 .678 .786 r .39311 .56061 .41507 .59571 .45326 .56642 .43162 .53098 Metetra .729 .673 .795 r .43162 .53098 .45326 .56642 .49156 .54096 .4702 .50517 Metetra .711 .666 .805 r .4702 .50517 .49156 .54096 .53009 .51928 .50901 .48313 Metetra .69 .658 .814 r .50901 .48313 .53009 .51928 .56902 .50134 .54818 .46482 Metetra .666 .648 .823 r .54818 .46482 .56902 .50134 .60848 .48714 .58788 .45023 Metetra .64 .637 .831 r .58788 .45023 .60848 .48714 .64865 .4767 .62827 .43939 Metetra .611 .623 .839 r .62827 .43939 .64865 .4767 .6897 .47008 .66952 .43234 Metetra .579 .607 .845 r .66952 .43234 .6897 .47008 .73181 .46736 .71181 .42917 Metetra .544 .589 .85 r .71181 .42917 .73181 .46736 .77517 .46866 .75536 .43001 Metetra .506 .569 .853 r .75536 .43001 .77517 .46866 .82002 .47413 .80038 .435 Metetra .466 .547 .855 r .80038 .435 .82002 .47413 .86658 .48398 .84711 .44434 Metetra .424 .522 .854 r .84711 .44434 .86658 .48398 .91515 .49845 .89584 .45827 Metetra .784 .707 .779 r .29234 .60071 .31573 .63173 .35453 .59416 .33156 .56275 Metetra .771 .704 .789 r .33156 .56275 .35453 .59416 .39311 .56061 .37054 .52883 Metetra .756 .701 .8 r .37054 .52883 .39311 .56061 .43162 .53098 .40942 .49882 Metetra .739 .696 .811 r .40942 .49882 .43162 .53098 .4702 .50517 .44835 .47264 Metetra .719 .69 .822 r .44835 .47264 .4702 .50517 .50901 .48313 .48748 .45022 Metetra .696 .682 .832 r .48748 .45022 .50901 .48313 .54818 .46482 .52695 .43152 Metetra .671 .672 .843 r .52695 .43152 .54818 .46482 .58788 .45023 .56694 .41654 Metetra .642 .661 .853 r .56694 .41654 .58788 .45023 .62827 .43939 .60759 .40529 Metetra .61 .646 .861 r .60759 .40529 .62827 .43939 .66952 .43234 .6491 .39784 Metetra .574 .63 .869 r .6491 .39784 .66952 .43234 .71181 .42917 .69164 .39425 Metetra .535 .61 .874 r .69164 .39425 .71181 .42917 .75536 .43001 .73543 .39465 Metetra .493 .587 .877 r .73543 .39465 .75536 .43001 .80038 .435 .78068 .3992 Metetra .448 .562 .878 r .78068 .3992 .80038 .435 .84711 .44434 .82766 .40808 Metetra .4 .535 .876 r .82766 .40808 .84711 .44434 .89584 .45827 .87663 .42155 Metetra .795 .73 .791 r .26811 .57311 .29234 .60071 .33156 .56275 .30782 .53471 Metetra .782 .728 .803 r .30782 .53471 .33156 .56275 .37054 .52883 .34726 .50036 Metetra .766 .726 .815 r .34726 .50036 .37054 .52883 .40942 .49882 .38657 .46994 Metetra .748 .722 .827 r .38657 .46994 .40942 .49882 .44835 .47264 .4259 .44335 Metetra .728 .717 .84 r .4259 .44335 .44835 .47264 .48748 .45022 .4654 .42052 Metetra .703 .709 .853 r .4654 .42052 .48748 .45022 .52695 .43152 .50523 .40142 Metetra .676 .7 .865 r .50523 .40142 .52695 .43152 .56694 .41654 .54556 .38604 Metetra .644 .688 .876 r .54556 .38604 .56694 .41654 .60759 .40529 .58654 .3744 Metetra .608 .673 .886 r .58654 .3744 .60759 .40529 .6491 .39784 .62836 .36655 Metetra .568 .655 .894 r .62836 .36655 .6491 .39784 .69164 .39425 .67121 .36256 Metetra .524 .633 .9 r .67121 .36256 .69164 .39425 .73543 .39465 .7153 .36256 Metetra .476 .608 .903 r .7153 .36256 .73543 .39465 .78068 .3992 .76085 .3667 Metetra .424 .579 .903 r .76085 .3667 .78068 .3992 .82766 .40808 .80814 .37519 Metetra .37 .547 .898 r .80814 .37519 .82766 .40808 .87663 .42155 .85743 .38826 Metetra .807 .755 .804 r .24293 .54895 .26811 .57311 .30782 .53471 .2832 .51005 Metetra .794 .754 .817 r .2832 .51005 .30782 .53471 .34726 .50036 .32315 .47522 Metetra .778 .753 .831 r .32315 .47522 .34726 .50036 .38657 .46994 .36295 .44434 Metetra .759 .75 .845 r .36295 .44434 .38657 .46994 .4259 .44335 .40274 .4173 Metetra .736 .746 .86 r .40274 .4173 .4259 .44335 .4654 .42052 .44268 .39405 Metetra .71 .74 .874 r .44268 .39405 .4654 .42052 .50523 .40142 .48292 .37453 Metetra .68 .731 .889 r .48292 .37453 .50523 .40142 .54556 .38604 .52365 .35874 Metetra .645 .718 .902 r .52365 .35874 .54556 .38604 .58654 .3744 .56501 .34671 Metetra .604 .702 .913 r .56501 .34671 .58654 .3744 .62836 .36655 .6072 .33846 Metetra .559 .683 .922 r .6072 .33846 .62836 .36655 .67121 .36256 .65042 .3341 Metetra .508 .658 .928 r .65042 .3341 .67121 .36256 .7153 .36256 .69488 .33373 Metetra .453 .629 .929 r .69488 .33373 .7153 .36256 .76085 .3667 .7408 .33751 Metetra .394 .596 .927 r .7408 .33751 .76085 .3667 .80814 .37519 .78846 .34565 Metetra .331 .559 .919 r .78846 .34565 .80814 .37519 .85743 .38826 .83815 .35839 Metetra .819 .781 .817 r .21667 .52827 .24293 .54895 .2832 .51005 .25755 .4888 Metetra .805 .783 .832 r .25755 .4888 .2832 .51005 .32315 .47522 .29809 .45344 Metetra .789 .783 .847 r .29809 .45344 .32315 .47522 .36295 .44434 .33844 .42205 Metetra .769 .782 .864 r .33844 .42205 .36295 .44434 .40274 .4173 .37875 .39453 Metetra .745 .779 .881 r .37875 .39453 .40274 .4173 .44268 .39405 .41919 .37082 Metetra .716 .773 .897 r .41919 .37082 .44268 .39405 .48292 .37453 .45991 .35087 Metetra .682 .764 .913 r .45991 .35087 .48292 .37453 .52365 .35874 .50109 .33467 Metetra .643 .752 .928 r .50109 .33467 .52365 .35874 .56501 .34671 .5429 .32223 Metetra .597 .735 .94 r .5429 .32223 .56501 .34671 .6072 .33846 .58553 .31361 Metetra .545 .713 .95 r .58553 .31361 .6072 .33846 .65042 .3341 .62918 .30889 Metetra .487 .685 .954 r .62918 .30889 .65042 .3341 .69488 .33373 .67407 .30818 Metetra .423 .651 .954 r .67407 .30818 .69488 .33373 .7408 .33751 .72044 .31165 Metetra .354 .612 .948 r .72044 .31165 .7408 .33751 .78846 .34565 .76855 .31949 Metetra .283 .569 .935 r .76855 .31949 .78846 .34565 .83815 .35839 .81871 .33197 Metetra .831 .81 .83 r .18915 .51116 .21667 .52827 .25755 .4888 .23074 .47105 Metetra .817 .813 .846 r .23074 .47105 .25755 .4888 .29809 .45344 .27193 .43508 Metetra .799 .815 .864 r .27193 .43508 .29809 .45344 .33844 .42205 .3129 .40314 Metetra .778 .815 .883 r .3129 .40314 .33844 .42205 .37875 .39453 .3538 .3751 Metetra .752 .814 .901 r .3538 .3751 .37875 .39453 .41919 .37082 .3948 .3509 Metetra .72 .809 .92 r .3948 .3509 .41919 .37082 .45991 .35087 .43606 .33049 Metetra .682 .801 .938 r .43606 .33049 .45991 .35087 .50109 .33467 .47776 .31386 Metetra .637 .788 .954 r .47776 .31386 .50109 .33467 .5429 .32223 .52009 .30103 Metetra .585 .769 .967 r .52009 .30103 .5429 .32223 .58553 .31361 .56322 .29204 Metetra .524 .743 .975 r .56322 .29204 .58553 .31361 .62918 .30889 .60737 .28698 Metetra .456 .711 .977 r .60737 .28698 .62918 .30889 .67407 .30818 .65276 .28596 Metetra .382 .671 .973 r .65276 .28596 .67407 .30818 .72044 .31165 .69964 .28916 Metetra .303 .625 .96 r .69964 .28916 .72044 .31165 .76855 .31949 .74829 .29677 Metetra .222 .573 .941 r .74829 .29677 .76855 .31949 .81871 .33197 .79901 .30905 Metetra .843 .84 .842 r .16022 .49773 .18915 .51116 .23074 .47105 .20258 .45689 Metetra .828 .845 .86 r .20258 .45689 .23074 .47105 .27193 .43508 .24451 .42026 Metetra .809 .849 .88 r .24451 .42026 .27193 .43508 .3129 .40314 .28617 .3877 Metetra .785 .851 .9 r .28617 .3877 .3129 .40314 .3538 .3751 .32774 .35909 Metetra .756 .851 .921 r .32774 .35909 .3538 .3751 .3948 .3509 .36937 .33437 Metetra .721 .847 .942 r .36937 .33437 .3948 .3509 .43606 .33049 .41124 .31348 Metetra .678 .839 .961 r .41124 .31348 .43606 .33049 .47776 .31386 .45354 .29642 Metetra .626 .824 .977 r .45354 .29642 .47776 .31386 .52009 .30103 .49644 .28318 Metetra .564 .802 .988 r .49644 .28318 .52009 .30103 .56322 .29204 .54014 .27383 Metetra .494 .772 .994 r .54014 .27383 .56322 .29204 .60737 .28698 .58487 .26846 Metetra .415 .732 .991 r .58487 .26846 .60737 .28698 .65276 .28596 .63084 .26717 Metetra .329 .684 .979 r .63084 .26717 .65276 .28596 .69964 .28916 .67832 .27014 Metetra .239 .628 .958 r .67832 .27014 .69964 .28916 .74829 .29677 .72758 .27757 Metetra .148 .567 .929 r .72758 .27757 .74829 .29677 .79901 .30905 .77895 .28975 Metetra .853 .871 .853 r .12967 .48812 .16022 .49773 .20258 .45689 .1729 .44647 Metetra .836 .878 .873 r .1729 .44647 .20258 .45689 .24451 .42026 .21565 .4091 Metetra .816 .884 .894 r .21565 .4091 .24451 .42026 .28617 .3877 .25809 .37586 Metetra .789 .887 .916 r .25809 .37586 .28617 .3877 .32774 .35909 .30039 .34664 Metetra .757 .888 .938 r .30039 .34664 .32774 .35909 .36937 .33437 .34273 .32136 Metetra .716 .884 .959 r .34273 .32136 .36937 .33437 .41124 .31348 .38529 .29997 Metetra .665 .875 .977 r .38529 .29997 .41124 .31348 .45354 .29642 .42825 .28245 Metetra .605 .857 .991 r .42825 .28245 .45354 .29642 .49644 .28318 .47181 .26881 Metetra .533 .83 .999 r .47181 .26881 .49644 .28318 .54014 .27383 .51616 .25911 Metetra .45 .792 .998 r .51616 .25911 .54014 .27383 .58487 .26846 .56153 .25344 Metetra .358 .744 .987 r .56153 .25344 .58487 .26846 .63084 .26717 .60817 .25191 Metetra .261 .685 .965 r .60817 .25191 .63084 .26717 .67832 .27014 .65632 .2547 Metetra .161 .619 .933 r .65632 .2547 .67832 .27014 .72758 .27757 .70629 .26204 Metetra .063 .548 .893 r .70629 .26204 .72758 .27757 .77895 .28975 .75841 .27419 Metetra .86 .902 .862 r .09727 .48253 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Metetra .114 .663 .811 r .48485 .23127 .51172 .23461 .55994 .23273 .53401 .22929 Metetra .002 .576 .755 r .53401 .22929 .55994 .23273 .60973 .23535 .58478 .23192 Metetra .102 0 0 r .58478 .23192 .60973 .23535 .66142 .2427 .63748 .23942 Metetra .194 0 0 r .63748 .23942 .66142 .2427 .71536 .25509 .69251 .25211 Metetra P p .002 w .67932 .02494 m .94594 .43277 L s .94594 .43277 m .97506 .64585 L s .97506 .64585 m .69286 .25814 L s .69286 .25814 m .67932 .02494 L s .06024 .26735 m .02494 .49015 L s .02494 .49015 m .69286 .25814 L s .69286 .25814 m .67932 .02494 L s .67932 .02494 m .06024 .26735 L s P p p .002 w .06024 .26735 m .67932 .02494 L s P p .002 w .06024 .26735 m .0648 .2716 L s P [(-1)] .05113 .25884 1 .93395 Mshowa p .002 w .20126 .21213 m .2056 .21661 L s P [(-0.5)] .1926 .20316 .96648 1 Mshowa p .002 w .35089 .15354 m .35496 .15826 L s P [(0)] .34275 .1441 .86223 1 Mshowa p .002 w .50994 .09126 m .5137 .09623 L s P [(0.5)] .50241 .08133 .75799 1 Mshowa p .002 w .67932 .02494 m .68274 .03015 L s P [(1)] .6725 .0145 .65374 1 Mshowa p .001 w .08779 .25656 m .0905 .25914 L s P p .001 w .11567 .24565 m .11835 .24825 L s P p .001 w .14386 .2346 m .14652 .23724 L s P p .001 w .17239 .22343 m .17502 .22609 L s P p .001 w .23048 .20069 m .23305 .20341 L s P p .001 w .26004 .18911 m .26258 .19186 L s P p .001 w .28996 .1774 m .29247 .18017 L s P p .001 w .32024 .16554 m .32272 .16835 L s P p .001 w .38192 .14139 m .38433 .14425 L s P p .001 w .41333 .12909 m .4157 .13198 L s P p .001 w .44513 .11664 m .44747 .11956 L s P p .001 w .47733 .10403 m .47963 .10698 L s P p .001 w .54296 .07833 m .54518 .08134 L s P p .001 w .5764 .06524 m .57857 .06828 L s P p .001 w .61027 .05198 m .6124 .05505 L s P p .001 w .64457 .03854 m .64666 .04164 L s P P % End of Graphics MathPictureEnd :[font = input; Cclosed; preserveAspect; startGroup] ParametricPlot[{2 Cos[t],3 Sin[t]},{t,0,2Pi}, AspectRatio->Automatic]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; 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I*3^(1/2))*(2160 + 432*33^(1/2))^(1/3))/(24*2^(1/3))} } ;[o] 1/3 1/3 -6 2 (2160 + 432 Sqrt[33]) {{x -> ------------------------ + ------------------------}, 1/3 1/3 (2160 + 432 Sqrt[33]) 12 2 1/3 3 2 (1 + I Sqrt[3]) {x -> ------------------------ - 1/3 (2160 + 432 Sqrt[33]) 1/3 (1 - I Sqrt[3]) (2160 + 432 Sqrt[33]) ----------------------------------------}, 1/3 24 2 1/3 3 2 (1 - I Sqrt[3]) {x -> ------------------------ - 1/3 (2160 + 432 Sqrt[33]) 1/3 (1 + I Sqrt[3]) (2160 + 432 Sqrt[33]) ----------------------------------------}} 1/3 24 2 :[font = input; Cclosed; preserveAspect; startGroup] N[Solve[f'[x]==0,x]] (* Finds horizontal tangents *) ;[s] 1:0,0;55,-1; 1:1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,Courier,0,12,0,0,0; :[font = output; output; inactive; preserveAspect; endGroup] {{x -> 0.650135488419564}, {x -> -0.325067744209782 + 1.347964062939702*I}, {x -> -0.325067744209782 - 1.347964062939702*I}} ;[o] {{x -> 0.650135}, {x -> -0.325068 + 1.34796 I}, {x -> -0.325068 - 1.34796 I}} :[font = input; Cclosed; preserveAspect; startGroup] Plot[f[x],{x,0.6,0.7}]; (* Plots f[x] near real minimum *) ;[s] 1:0,0;59,-1; 1:1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,Courier,0,12,0,0,0; :[font = postscript; PostScript; formatAsPostScript; output; inactive; closed; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174; endGroup; endGroup] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations -5.69048 9.52381 33.592 41.7632 [ [(0.62)] .21429 .18151 0 2 Msboxa [(0.64)] .40476 .18151 0 2 Msboxa [(0.66)] .59524 .18151 0 2 Msboxa [(0.68)] .78571 .18151 0 2 Msboxa [(0.7)] .97619 .18151 0 2 Msboxa [(-0.804)] .01131 .01446 1 0 Msboxa [(-0.802)] .01131 .09798 1 0 Msboxa [(-0.798)] .01131 .26504 1 0 Msboxa [(-0.796)] .01131 .34856 1 0 Msboxa [(-0.794)] .01131 .43209 1 0 Msboxa [(-0.792)] .01131 .51562 1 0 Msboxa [(-0.79)] .01131 .59914 1 0 Msboxa [ 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.01559 L .5248 .01613 L .52976 .01678 L .53968 .01848 L .5496 .02068 L .55952 .02338 L .57937 .03031 L .59921 .03927 L .61905 .05027 L .65873 .07839 L .69841 .11473 L .7381 .15934 L .77778 .21226 L Mistroke .81746 .27355 L .85714 .34325 L .89683 .42141 L .93651 .50808 L .97619 .60332 L Mfstroke P P % End of Graphics MathPictureEnd :[font = subsubtitle; inactive; Cclosed; preserveAspect; startGroup] INTEGRATION :[font = input; Cclosed; preserveAspect; startGroup] ?Integrate :[font = print; inactive; preserveAspect] Integrate[f,x] gives the indefinite integral of f with respect to x. Integrate[f,{x,xmin,xmax}] gives the definite integral. Integrate[f,{x,xmin,xmax},{y,ymin,ymax}] gives a multiple integral. :[font = input; Cclosed; preserveAspect; startGroup] Integrate[x^2, x] :[font = output; output; inactive; preserveAspect; endGroup] x^3/3 ;[o] 3 x -- 3 :[font = input; Cclosed; preserveAspect; startGroup] Integrate[Sin[x],x] :[font = output; output; inactive; preserveAspect; endGroup] -Cos[x] ;[o] -Cos[x] :[font = input; Cclosed; preserveAspect; startGroup] Integrate[Sin[x],{x,0,Pi/15}] :[font = output; output; inactive; preserveAspect; endGroup; endGroup; endGroup] 1 - Cos[Pi/15] ;[o] Pi 1 - Cos[--] 15 :[font = title; inactive; Cclosed; preserveAspect; startGroup] NUMERICAL INTEGRATION ;[s] 2:0,0;1,1;23,-1; 2:1,0,-9 -*-times-bold-r-normal-*-*-240-75-75-*-*-*-*,Times,1,24,0,0,0;1,0,-9 -*-times-medium-i-normal-*-*-140-75-75-*-*-*-*,Times,2,14,0,0,0; :[font = input; Cclosed; preserveAspect; startGroup] NIntegrate[Sin[x],{x,0,Pi/15}] ;[s] 2:0,0;1,1;31,-1; 2:1,0,-9 -*-courier-bold-r-normal-*-*-120-75-75-*-*-*-*,Courier,1,12,0,0,0;1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,Courier,0,12,0,0,0; :[font = output; output; inactive; preserveAspect; endGroup] 0.02185239926619436 ;[o] 0.0218524 :[font = input; Cclosed; preserveAspect; startGroup] NIntegrate[E^(-x^2),{x,0,1}] ;[s] 2:0,0;1,1;29,-1; 2:1,0,-9 -*-courier-bold-r-normal-*-*-120-75-75-*-*-*-*,Courier,1,12,0,0,0;1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,Courier,0,12,0,0,0; :[font = output; output; inactive; preserveAspect; endGroup] 0.7468241328124272 ;[o] 0.746824 :[font = input; preserveAspect] er[y_]:= NIntegrate[E^(-x^2),{x,0,y}] (* FUNCTION AS INTEGRAL *) ;[s] 1:0,0;65,-1; 1:1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,Courier,0,12,0,0,0; :[font = input; Cclosed; preserveAspect; startGroup] Plot[er[y],{y,0,2}]; ;[s] 2:0,0;1,1;21,-1; 2:1,0,-9 -*-courier-bold-r-normal-*-*-120-75-75-*-*-*-*,Courier,1,12,0,0,0;1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,Courier,0,12,0,0,0; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 281.75; pictureHeight = 174.125; endGroup] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.47619 0.0147151 0.66729 [ [(0.5)] .2619 .01472 0 2 Msboxa [(1)] .5 .01472 0 2 Msboxa [(1.5)] .7381 .01472 0 2 Msboxa [(2)] .97619 .01472 0 2 Msboxa [(0.2)] .01131 .14817 1 0 Msboxa [(0.4)] .01131 .28163 1 0 Msboxa [(0.6)] .01131 .41509 1 0 Msboxa [(0.8)] .01131 .54855 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .2619 .01472 m .2619 .02097 L s P [(0.5)] .2619 .01472 0 2 Mshowa p .002 w .5 .01472 m .5 .02097 L s P [(1)] .5 .01472 0 2 Mshowa p .002 w .7381 .01472 m .7381 .02097 L s P [(1.5)] .7381 .01472 0 2 Mshowa p .002 w .97619 .01472 m .97619 .02097 L s P [(2)] .97619 .01472 0 2 Mshowa p .001 w .07143 .01472 m .07143 .01847 L s P p .001 w .11905 .01472 m .11905 .01847 L s P p .001 w .16667 .01472 m .16667 .01847 L s P p .001 w .21429 .01472 m .21429 .01847 L s P p .001 w .30952 .01472 m .30952 .01847 L s P p .001 w .35714 .01472 m .35714 .01847 L s P p .001 w .40476 .01472 m .40476 .01847 L s P p .001 w .45238 .01472 m .45238 .01847 L s P p .001 w .54762 .01472 m .54762 .01847 L s P p .001 w .59524 .01472 m .59524 .01847 L s P p .001 w .64286 .01472 m .64286 .01847 L s P p .001 w .69048 .01472 m .69048 .01847 L s P p .001 w .78571 .01472 m .78571 .01847 L s P p .001 w .83333 .01472 m .83333 .01847 L s P p .001 w .88095 .01472 m .88095 .01847 L s P p .001 w .92857 .01472 m .92857 .01847 L s P p .002 w 0 .01472 m 1 .01472 L s P p .002 w .02381 .14817 m .03006 .14817 L s P [(0.2)] .01131 .14817 1 0 Mshowa p .002 w .02381 .28163 m .03006 .28163 L s P [(0.4)] .01131 .28163 1 0 Mshowa p .002 w .02381 .41509 m .03006 .41509 L s P [(0.6)] .01131 .41509 1 0 Mshowa p .002 w .02381 .54855 m .03006 .54855 L s P [(0.8)] .01131 .54855 1 0 Mshowa p .001 w .02381 .04141 m .02756 .04141 L s P p .001 w .02381 .0681 m .02756 .0681 L s P p .001 w .02381 .09479 m .02756 .09479 L s P p .001 w .02381 .12148 m .02756 .12148 L s P p .001 w .02381 .17486 m .02756 .17486 L s P p .001 w .02381 .20156 m .02756 .20156 L s P p .001 w .02381 .22825 m .02756 .22825 L s P p .001 w .02381 .25494 m .02756 .25494 L s P p .001 w .02381 .30832 m .02756 .30832 L s P p .001 w .02381 .33501 m .02756 .33501 L s P p .001 w .02381 .36171 m .02756 .36171 L s P p .001 w .02381 .3884 m .02756 .3884 L s P p .001 w .02381 .44178 m .02756 .44178 L s P p .001 w .02381 .46847 m .02756 .46847 L s P p .001 w .02381 .49516 m .02756 .49516 L s P p .001 w .02381 .52186 m .02756 .52186 L s P p .001 w .02381 .57524 m .02756 .57524 L s P p .001 w .02381 .60193 m .02756 .60193 L s P p .002 w .02381 0 m .02381 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p p .004 w .02381 .01472 m .06349 .07019 L .10317 .12491 L .14286 .17813 L .18254 .22917 L .22222 .27747 L .2619 .32252 L .30159 .36398 L .34127 .4016 L .38095 .43527 L .42063 .46499 L .46032 .49086 L .5 .51306 L .53968 .53186 L .57937 .54756 L .61905 .56049 L .65873 .57099 L .69841 .5794 L .7381 .58604 L .77778 .59122 L .81746 .59519 L .85714 .5982 L .89683 .60045 L .93651 .60211 L .97619 .60332 L s P P % End of Graphics MathPictureEnd :[font = input; Cclosed; preserveAspect; startGroup] erdata=Table[{y,er[y]},{y,0,2,0.1}]; TableForm[erdata] (*TABULATE FUNCTION *) ;[s] 1:0,0;78,-1; 1:1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,Courier,0,12,0,0,0; :[font = output; output; inactive; preserveAspect; endGroup; endGroup; endGroup] TableForm[{{0, 0}, {0.1, 0.0996676642903364}, {0.2, 0.1973650309263709}, {0.3, 0.2912378826569657}, {0.4, 0.3796528397004753}, {0.5, 0.4612810064127925}, {0.6, 0.5351535268080788}, {0.7, 0.6006856680827443}, {0.8, 0.6576698563283956}, {0.9, 0.7062415149635398}, {1., 0.7468241328124271}, {1.1, 0.7800614325765695}, {1.2, 0.806744758123119}, {1.3, 0.827742989393204}, {1.4, 0.843940713841665}, {1.5, 0.856188393624903}, {1.6, 0.865266225942295}, {1.7, 0.871861593406661}, {1.8, 0.87655863425916}, {1.900000000000001, 0.879837609720086}, {2., 0.882081390762422}}] ;[o] 0 0 0.1 0.0996677 0.2 0.197365 0.3 0.291238 0.4 0.379653 0.5 0.461281 0.6 0.535154 0.7 0.600686 0.8 0.65767 0.9 0.706242 1. 0.746824 1.1 0.780061 1.2 0.806745 1.3 0.827743 1.4 0.843941 1.5 0.856188 1.6 0.865266 1.7 0.871862 1.8 0.876559 1.9 0.879838 2. 0.882081 :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] MULTIPLE INTEGRALS :[font = input; Cclosed; preserveAspect; startGroup] Integrate[E^y + Sin[x], {y,0,1},{x,0,Pi/2}] ;[s] 2:0,0;1,1;45,-1; 2:1,0,-9 -*-courier-bold-r-normal-*-*-120-75-75-*-*-*-*,Courier,1,12,0,0,0;1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,Courier,0,12,0,0,0; :[font = output; output; inactive; preserveAspect; endGroup] 1 - Pi/2 + (E*Pi)/2 ;[o] Pi E Pi 1 - -- + ---- 2 2 :[font = input; preserveAspect] f1[a_]:= Integrate[E^y + Sin[x], {y,0,a},{x,0,Pi/2}] (*FUNCTION AS INTEGRAL*) ;[s] 2:0,0;1,1;79,-1; 2:1,0,-9 -*-courier-bold-r-normal-*-*-120-75-75-*-*-*-*,Courier,1,12,0,0,0;1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,Courier,0,12,0,0,0; :[font = input; Cclosed; preserveAspect; startGroup] Plot[f1[a],{a,0,1}]; (* PLOT FUNCTION *) ;[s] 1:0,0;43,-1; 1:1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,Courier,0,12,0,0,0; :[font = postscript; PostScript; formatAsPostScript; output; inactive; closed; 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Cclosed; preserveAspect; startGroup] t=Table[{a,N[f1[a]]},{a,0,1,0.2}] (*TABULATE IN 0.2 STEPS FROM 0 TO 1*) ;[s] 2:0,0;1,1;79,-1; 2:1,0,-9 -*-courier-bold-r-normal-*-*-120-75-75-*-*-*-*,Courier,1,12,0,0,0;1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,Courier,0,12,0,0,0; :[font = output; output; inactive; preserveAspect; endGroup] {{0, 0}, {0.2, 0.5477786392602537}, {0.4, 1.172556428481918}, {0.6000000000000001, 1.891381191842438}, {0.8, 2.725075188812775}, {1., 3.699070784541886}} ;[o] {{0, 0}, {0.2, 0.547779}, {0.4, 1.17256}, {0.6, 1.89138}, {0.8, 2.72508}, {1., 3.69907}} :[font = input; Cclosed; preserveAspect; startGroup] TableForm[t] (* TABLE FORM FOR DATA *) ;[s] 2:0,0;1,1;45,-1; 2:1,0,-9 -*-courier-bold-r-normal-*-*-120-75-75-*-*-*-*,Courier,1,12,0,0,0;1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,Courier,0,12,0,0,0; :[font = output; output; inactive; preserveAspect] TableForm[{{0, 0}, {0.2, 0.5477786392602537}, {0.4, 1.172556428481918}, {0.6000000000000001, 1.891381191842438}, {0.8, 2.725075188812775}, {1., 3.699070784541886}}] ;[o] 0 0 0.2 0.547779 0.4 1.17256 0.6 1.89138 0.8 2.72508 1. 3.69907 :[font = output; output; inactive; closed; preserveAspect; endGroup] TableForm[{{0, 0}, {0.1, 0.0996676642903364}, {0.2, 0.1973650309263709}, {0.3, 0.2912378826569657}, {0.4, 0.3796528397004753}, {0.5, 0.4612810064127925}, {0.6, 0.5351535268080788}, {0.7, 0.6006856680827443}, {0.8, 0.6576698563283956}, {0.9, 0.7062415149635398}, {1., 0.7468241328124271}, {1.1, 0.7800614325765695}, {1.2, 0.806744758123119}, {1.3, 0.827742989393204}, {1.4, 0.843940713841665}, {1.5, 0.856188393624903}, {1.6, 0.865266225942295}, {1.7, 0.871861593406661}, {1.8, 0.87655863425916}, {1.900000000000001, 0.879837609720086}, {2., 0.882081390762422}}] ;[o] 0 0 0.1 0.0996677 0.2 0.197365 0.3 0.291238 0.4 0.379653 0.5 0.461281 0.6 0.535154 0.7 0.600686 0.8 0.65767 0.9 0.706242 1. 0.746824 1.1 0.780061 1.2 0.806745 1.3 0.827743 1.4 0.843941 1.5 0.856188 1.6 0.865266 1.7 0.871862 1.8 0.876559 1.9 0.879838 2. 0.882081 :[font = input; preserveAspect] PLOT OF NUMERICAL SOLUTION OF ODE ;[s] 3:0,0;1,1;2,2;36,-1; 3:1,0,-9 -*-courier-bold-r-normal-*-*-120-75-75-*-*-*-*,Courier,1,12,0,0,0;1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,Courier,0,12,0,0,0;1,0,-9 -*-times-medium-i-normal-*-*-140-75-75-*-*-*-*,Times,2,14,0,0,0; 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inactive; Cclosed; preserveAspect; startGroup] ORDINARY DIFFERENTIAL EQUATIONS :[font = text; inactive; preserveAspect] You can think of a first order differential equation as a way to specify a vector field in the x-y plane. Thus, dy/dx is the size of the y-component when the x-component has value 1. Then a solution to the equation is a curve that follows the vector field flow. Mathematica can show this as follows. The equation dy/dx =x has as solution (x^2)/2 + c for any constant c. The vector field has components (1,x) and the solution curves are y(x) = (x^2)/2 + c. We show these as follows using Mathematica's special package :[font = input; preserveAspect] < {{Thickness[0.02],RGBColor[1,0,0]}}]; :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174; endGroup] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.47619 0.0147151 1.17721 [ [(-1)] .02381 .01472 0 2 Msboxa [(-0.5)] .2619 .01472 0 2 Msboxa [(0.5)] .7381 .01472 0 2 Msboxa [(1)] .97619 .01472 0 2 Msboxa [(0.1)] .4875 .13244 1 0 Msboxa [(0.2)] .4875 .25016 1 0 Msboxa [(0.3)] .4875 .36788 1 0 Msboxa [(0.4)] .4875 .4856 1 0 Msboxa [(0.5)] .4875 .60332 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .02381 .01472 m .02381 .02097 L s P [(-1)] .02381 .01472 0 2 Mshowa p .002 w .2619 .01472 m .2619 .02097 L s P 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0.857143 1.10102 0.958158 MAarrowhead1 % End of user PostScript P p .004 w .9304 .57632 m .97619 .60332 L s % Start of user PostScript 1. 1. 1.10102 1.10102 MAarrowhead1 % End of user PostScript P P P % End of Graphics MathPictureEnd :[font = text; inactive; preserveAspect] Mathematica has a special package to help solving differential equations. :[font = input; preserveAspect] < x^2/2}} ;[o] 2 x {{y[x] -> --}} 2 :[font = input; Cclosed; preserveAspect; startGroup] DSolve[{y'[x] == 1 - y[x], y[0] == 4}, y[x], x] ;[s] 2:0,0;1,1;48,-1; 2:1,0,-9 -*-courier-bold-r-normal-*-*-120-75-75-*-*-*-*,Courier,1,12,0,0,0;1,0,-9 -*-courier-medium-r-normal-*-*-120-75-75-*-*-*-*,Courier,0,12,0,0,0; :[font = output; output; inactive; preserveAspect; endGroup] {{y[x] -> 1 + 3/E^x}} ;[o] 3 {{y[x] -> 1 + --}} x E :[font = text; inactive; preserveAspect] The following example concerns the logistic growth equation which has differential equation DE: x'(t) = k x(t) (1 - x(t)/m) In the following example we shall take k = 1 and m = 4. :[font = input; Cclosed; preserveAspect; startGroup] ?NDSolve :[font = print; inactive; preserveAspect; endGroup] NDSolve[eqns, y, {x, xmin, xmax}] finds a numerical solution to the differential equations eqns for the function y with the independent variable x in the range xmin to xmax. NDSolve[eqns, {y1, y2, ...}, {x, xmin, xmax}] finds numerical solutions for the functions yi. NDSolve[eqns, y, {x, x1, x2, ...}] forces a function evaluation at each of x1, x2, ... The range of numerical integration is from Min[x1, x2, ...] to Max[x1, x2, ...]. :[font = text; inactive; preserveAspect] The following function is defined to produce a numerical solution to the above differential equation with initial condition, x = a if t = 0 : :[font = input; preserveAspect] curve[a_] := NDSolve[{x'[t] == x[t] (1 - x[t]/4), x[0] == a}, x, {t, 0, 6}] :[font = input; Cclosed; preserveAspect; startGroup] curve[1] :[font = output; output; inactive; preserveAspect; endGroup] {{x -> InterpolatingFunction[{0., 6.}, {{0., 0., {1., 0}, {0}}, {0.002436830796320685, 0., {1.001829850572221, 0.7509140868474389}, {0}}, {0.004873661592641371, 0.002436830796320685, {1.003661927245428, 0.7518276098503134}, {0}}, {0.05004822306891955, 0.004873661592641371, {1.038005190122613, 0.76864151059564, 0.1860992137594504}, {0, 0}}, {0.0952227845451977, 0.05004822306891955, {1.073102590676646, 0.7852153241835362, 0.1834418868304846}, {0, 0}}, {0.1403973460214759, 0.0952227845451977, 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That is, in the long run the value of x approaches 4. (If x = 0, when t = 0 then x(t) = 0 , for all values of t.) What happens if x = 4 when t = 0 ? :[font = section; inactive; Cclosed; preserveAspect; startGroup] MORE ADVANCED TOPICS :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] VECTOR ANALYSIS ;[s] 2:0,0;1,1;17,-1; 2:1,0,-9 -*-times-bold-r-normal-*-*-140-75-75-*-*-*-*,Times,1,14,0,0,0;1,0,-9 -*-times-medium-i-normal-*-*-140-75-75-*-*-*-*,Times,2,14,0,0,0; :[font = text; inactive; preserveAspect] Note that Mathematica has a special package Calculus`VectorAnalysis` which contains functions to compute Jacobians, change coordinates among most of the standard systems, provide arclength functions,as well as the operator functions Div, Grad, Curl and Laplacian. 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SAVING A GRAPHIC OUTPUT TO A POSTSCRIPT FILE *) Display["/yourpathname/graphic1.ps",Out[3]] (* use your home directory path *) (* YOU NEED TO CONVERT GRAPHIC graphic1.ps TO EPS FORM USING* ) psfix /yourpathname/graphic1.ps > /yourpathname/graphic1.eps (* NOW YOU CAN DELETE /yourpathname/graphic1.ps AND KEEP THE eps VERSION *) (* USE ghostview TO VIEW eps FILES *) ^*)