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(*^ ::[paletteColors = 128; fontset = title, "New York", 18, L2, center, bold, nohscroll; fontset = subtitle, "New York", 14, L2, center, bold, nohscroll; fontset = subsubtitle, "New York", 12, L2, center, bold, nohscroll; fontset = section, "New York", 14, L2, bold, nohscroll, grayBox; fontset = subsection, "New York", 12, L2, bold, nohscroll, blackBox; fontset = subsubsection, "New York", 10, L2, bold, nohscroll, whiteBox; fontset = text, "New York", 10, L2, nohscroll; fontset = smalltext, "New York", 10, L2, nohscroll; fontset = input, "Courier", 12, L2, bold, B65535, nowordwrap; fontset = output, "Courier", 12, L2, R65535, B65535, nowordwrap; fontset = message, "Courier", 12, L2, R65535, nowordwrap; fontset = print, "Courier", 12, L2, nowordwrap; fontset = info, "Courier", 12, L2, nowordwrap; fontset = postscript, "Courier", 12, L2, nowordwrap; fontset = name, "Geneva", 10, L2, italic, B65535, nohscroll; fontset = header, "New York", 10, L2, nohscroll; fontset = footer, "New York", 12, L2, center, nohscroll; fontset = help, "Geneva", 10, L2, nohscroll; fontset = clipboard, "New York", 12, L2; fontset = completions, "New York", 12, L2; fontset = network, "Courier", 10, L2, nowordwrap; fontset = graphlabel, "Courier", 12, L2; fontset = special1, "New York", 12, L2; fontset = special2, "New York", 12, L2, center; fontset = special3, "New York", 12, L2, right; fontset = special4, "New York", 12, L2; fontset = special5, "New York", 12, L2;] :[inactive; startGroup; font = title; ] A Sampling of Mathematica™ ;[s] 4:0,2;14,1;25,2;27,0;28,-1; 4:1,23,17,New York,1,18,0,0,0;1,21,17,New York,3,18,65535,0,0;2,23,17,New York,1,18,65535,0,0;0,21,17,New York,0,18,65535,0,0; :[inactive; font = text; ] This file is a Mathematica Notebook that gives some examples of what Mathematica can do. ;[s] 6:0,0;15,1;26,0;69,1;80,0;90,0;91,-1; 2:4,14,10,New York,0,10,0,0,0;2,12,10,New York,2,10,0,0,0; :[inactive; font = text; ] For information on how to read this Notebook, see the file Read This First! ;[s] 3:0,0;59,1;75,0;76,-1; 2:2,14,10,New York,0,10,0,0,0;1,12,10,New York,1,10,0,0,0; :[inactive; font = text; ] This file is loosely based on Chapter 0 of The Mathematica Book: "Mathematica: A System for Doing Mathematics by Computer", by Stephen Wolfram. This book was published by Addison-Wesley in 1988, and is available at most bookstores. ;[s] 6:0,0;47,1;58,0;66,1;77,0;232,0;233,-1; 2:4,14,10,New York,0,10,0,0,0;2,12,10,New York,2,10,0,0,0; :[inactive; font = text; ] For information on how to obtain copies of Mathematica itself, see the section "Buying Mathematica" in the file Read This First! ;[s] 7:0,0;43,1;54,0;88,1;99,0;113,2;129,0;130,-1; 3:4,14,10,New York,0,10,0,0,0;2,12,10,New York,2,10,0,0,0;1,12,10,New York,1,10,0,0,0; :[inactive; startGroup; Cclosed; font = section; ] Numerical Calculations ;[s] 2:0,1;22,0;23,-1; 2:1,19,14,New York,1,14,0,0,0;1,19,14,New York,1,14,65535,0,0; :[inactive; font = text; ] You can use Mathematica as an enhanced scientific calculator. Let's start with a simple example. ;[s] 6:0,0;12,1;23,0;81,2;82,0;98,0;99,-1; 3:4,14,10,New York,0,10,0,0,0;1,12,10,New York,2,10,0,0,0;1,14,10,New York,0,10,65535,0,65535; :[startGroup; font = input; ] 45 + 78 ;[s] 2:0,1;7,0;8,-1; 2:1,14,10,Courier,1,12,0,0,65535;1,14,10,Courier,1,12,0,0,65535; :[inactive; output; endGroup; font = output; ] 123 ;[o] 123 :[inactive; font = text; ] The first line here is what you type into Mathematica. The second line is the result Mathematica gives. If you are reading this Notebook on a color system, the input is blue and the output purple. ;[s] 5:0,0;42,1;53,0;86,1;97,0;200,-1; 2:3,14,10,New York,0,10,0,0,0;2,12,10,New York,2,10,0,0,0; :[inactive; font = text; ] Now let's try something more difficult. :[startGroup; font = input; ] 3^100 :[inactive; output; endGroup; font = output; ] 515377520732011331036461129765621272702107522001 ;[o] 515377520732011331036461129765621272702107522001 :[inactive; font = text; ] Unlike a calculator, Mathematica gives an exact answer for 3 raised to the power 100. ;[s] 3:0,0;21,1;32,0;87,-1; 2:2,14,10,New York,0,10,0,0,0;1,12,10,New York,2,10,0,0,0; :[inactive; font = text; ] Now let's really test out Mathematica. Here is 3 raised to the power 1000. ;[s] 3:0,0;26,2;37,0;76,-1; 3:2,14,10,New York,0,10,0,0,0;0,12,10,New York,3,10,0,0,0;1,12,10,New York,2,10,0,0,0; :[startGroup; font = input; ] 3^1000 :[inactive; output; endGroup; font = output; ] 13220708194808066368904552597521443659654220327521481676649\ 2036822682859734670489954077831385060806196390977769687258\ 2355950954582100618911865342725257953674027620225198320803\ 8780147742289648412743904001175886180411289478156230944380\ 6156617305408667449050617812548034440554705439703889581746\ 5368254916136220830268563778582290228416398307887896918556\ 4040848989376093732421718463599386955167650189405881090604\ 2608967143886410281435038564874716583201061436613217310276\ 8902855220001 ;[o] 13220708194808066368904552597521443659654220327521481676649\ 2036822682859734670489954077831385060806196390977769687258\ 2355950954582100618911865342725257953674027620225198320803\ 8780147742289648412743904001175886180411289478156230944380\ 6156617305408667449050617812548034440554705439703889581746\ 5368254916136220830268563778582290228416398307887896918556\ 4040848989376093732421718463599386955167650189405881090604\ 2608967143886410281435038564874716583201061436613217310276\ 8902855220001 :[inactive; font = text; ] This took about half a second on a Macintosh II. :[inactive; font = text; ] Here's the result in the form you might get on a calculator. :[startGroup; font = input; ] N[%] :[inactive; output; endGroup; font = output; ] "1.32207"*10^"477" ;[o] 477 1.32207 10 :[inactive; font = text; ] Here is the value of pi to two hundred decimal places. ;[s] 3:0,0;25,2;54,0;55,-1; 5:2,14,10,New York,0,10,0,0,0;0,12,10,Chicago,1,10,0,0,0;1,14,10,New York,0,10,0,0,0;0,11,9,Symbol,1,10,0,0,0;0,15,12,Symbol,1,14,0,0,0; :[startGroup; font = input; ] N[Pi, 200] :[inactive; output; endGroup; font = output; ] 3.141592653589793238462643383279502884197169399375105820974\ 9445923078164062862089986280348253421170679821480865132823\ 0664709384460955058223172535940812848111745028410270193852\ 1105559644622948954930382 ;[o] 3.141592653589793238462643383279502884197169399375105820974\ 9445923078164062862089986280348253421170679821480865132823\ 0664709384460955058223172535940812848111745028410270193852\ 1105559644622948954930382 :[inactive; font = text; ] Mathematica knows about a big collection of mathematical functions — most of those you would find in any book of mathematical tables. ;[s] 3:0,1;11,0;134,0;135,-1; 2:2,14,10,New York,0,10,0,0,0;1,12,10,New York,2,10,0,0,0; :[startGroup; font = input; ] BesselJ[5, 34.6] :[inactive; output; endGroup; font = output; ] 0.0511826 ;[o] 0.0511826 :[startGroup; font = input; ] Log[4.5 + 2I] :[inactive; output; endGroup; font = output; ] 1.59421 + 0.418224*I ;[o] 1.59421 + 0.418224 I :[startGroup; font = input; ] Zeta[1/2 + 14.3 I] :[inactive; output; endGroup; endGroup; font = output; ] -0.0119878 + 0.132231*I ;[o] -0.0119878 + 0.132231 I :[inactive; startGroup; Cclosed; font = section; ] Algebraic Calculations ;[s] 2:0,1;22,0;23,-1; 2:1,19,14,New York,1,14,0,0,0;1,19,14,New York,1,14,65535,0,0; :[inactive; font = text; ] One of the most important features of Mathematica is its ability to deal with mathematical formulas in algebraic form. ;[s] 3:0,0;38,1;49,0;120,-1; 2:2,14,10,New York,0,10,0,0,0;1,12,10,New York,2,10,0,0,0; :[startGroup; font = input; ] (1 + x)^3 :[inactive; output; endGroup; font = output; ] (1 + x)^3 ;[o] 3 (1 + x) :[inactive; font = text; ] This is what Mathematica does if you type in a simple algebraic expression. You can expand out the expression like this: ;[s] 3:0,0;13,1;24,0;121,-1; 2:2,14,10,New York,0,10,0,0,0;1,12,10,New York,2,10,0,0,0; :[startGroup; font = input; ] Expand[%] :[inactive; output; endGroup; font = output; ] 1 + 3*x + 3*x^2 + x^3 ;[o] 2 3 1 + 3 x + 3 x + x :[inactive; font = text; ] Mathematica gives an explicit formula for the result. You can factor this result to get back to what you started from. % always stands for the last result that Mathematica gave you. ;[s] 6:0,1;11,0;119,2;120,0;160,1;171,0;182,-1; 3:3,14,10,New York,0,10,0,0,0;2,12,10,New York,2,10,0,0,0;1,11,9,Courier,0,10,0,0,0; :[startGroup; font = input; ] Factor[%] :[inactive; output; endGroup; font = output; ] (1 + x)^3 ;[o] 3 (1 + x) :[inactive; font = text; ] Now let's try a more complicated example. :[startGroup; font = input; ] (1 + 2x + 5y)^7 :[inactive; output; endGroup; font = output; ] (1 + 2*x + 5*y)^7 ;[o] 7 (1 + 2 x + 5 y) :[startGroup; font = input; ] Expand[%] :[inactive; output; endGroup; font = output; ] 1 + 14*x + 84*x^2 + 280*x^3 + 560*x^4 + 672*x^5 + 448*x^6 + 128*x^7 + 35*y + 420*x*y + 2100*x^2*y + 5600*x^3*y + 8400*x^4*y + 6720*x^5*y + 2240*x^6*y + 525*y^2 + 5250*x*y^2 + 21000*x^2*y^2 + 42000*x^3*y^2 + 42000*x^4*y^2 + 16800*x^5*y^2 + 4375*y^3 + 35000*x*y^3 + 105000*x^2*y^3 + 140000*x^3*y^3 + 70000*x^4*y^3 + 21875*y^4 + 131250*x*y^4 + 262500*x^2*y^4 + 175000*x^3*y^4 + 65625*y^5 + 262500*x*y^5 + 262500*x^2*y^5 + 109375*y^6 + 218750*x*y^6 + 78125*y^7 ;[o] 2 3 4 5 6 1 + 14 x + 84 x + 280 x + 560 x + 672 x + 448 x + 7 2 3 128 x + 35 y + 420 x y + 2100 x y + 5600 x y + 4 5 6 2 2 8400 x y + 6720 x y + 2240 x y + 525 y + 5250 x y + 2 2 3 2 4 2 5 2 21000 x y + 42000 x y + 42000 x y + 16800 x y + 3 3 2 3 3 3 4375 y + 35000 x y + 105000 x y + 140000 x y + 4 3 4 4 2 4 70000 x y + 21875 y + 131250 x y + 262500 x y + 3 4 5 5 2 5 175000 x y + 65625 y + 262500 x y + 262500 x y + 6 6 7 109375 y + 218750 x y + 78125 y :[startGroup; font = input; ] Factor[%] :[inactive; output; endGroup; font = output; ] (1 + 2*x + 5*y)^7 ;[o] 7 (1 + 2 x + 5 y) :[inactive; startGroup; Cclosed; font = subsection; ] Calculus :[inactive; font = text; ] You can use Mathematica to do calculus. Here's a simple integral. ;[s] 3:0,0;12,1;23,0;66,-1; 2:2,14,10,New York,0,10,0,0,0;1,12,10,New York,2,10,0,0,0; :[startGroup; font = input; ] Integrate[x^n, x] :[inactive; output; endGroup; font = output; ] x^(1 + n)/(1 + n) ;[o] 1 + n x ------ 1 + n :[inactive; font = text; ] Here's a more complicated example. :[startGroup; font = input; ] Integrate[x/(x^3-1), x] :[inactive; output; endGroup; font = output; ] (3^(1/2)*ArcTan[(1 + 2*x)/3^(1/2)])/3 + Log[1 - x]/3 - Log[1 + x + x^2]/6 ;[o] 1 + 2 x Sqrt[3] ArcTan[-------] 2 Sqrt[3] Log[1 - x] Log[1 + x + x ] ----------------------- + ---------- - --------------- 3 3 6 :[inactive; font = text; ] Now let's try differentiating again. :[startGroup; font = input; ] D[%, x] :[inactive; output; endGroup; font = output; ] -1/(3*(1 - x)) + 2/(3*(1 + (1 + 2*x)^2/3)) - (1 + 2*x)/(6*(1 + x + x^2)) ;[o] -1 2 1 + 2 x --------- + ------------------ - -------------- 3 (1 - x) 2 2 (1 + 2 x) 6 (1 + x + x ) 3 (1 + ----------) 3 :[inactive; font = text; ] This gives the expression in a different algebraic form. We can get back our original form using Simplify. ;[s] 3:0,0;97,2;105,0;107,-1; 3:2,14,10,New York,0,10,0,0,0;0,11,9,Courier,0,10,0,0,0;1,11,9,Courier,1,10,0,0,0; :[startGroup; font = input; ] Simplify[%] :[inactive; output; endGroup; endGroup; font = output; ] x/(-1 + x^3) ;[o] x ------- 3 -1 + x :[inactive; startGroup; Cclosed; font = subsection; ] Solving Equations :[inactive; font = text; ] This is how you solve a quadratic equation in Mathematica. ;[s] 3:0,0;46,1;57,0;59,-1; 2:2,14,10,New York,0,10,0,0,0;1,12,10,New York,2,10,0,0,0; :[startGroup; font = input; ] Solve[x^2 + 2 a x + 1 == 0, x] :[inactive; output; endGroup; font = output; ] {{x -> (-2*a + (-4 + 4*a^2)^(1/2))/2}, {x -> (-2*a - (-4 + 4*a^2)^(1/2))/2}} ;[o] 2 -2 a + Sqrt[-4 + 4 a ] {{x -> ----------------------}, 2 2 -2 a - Sqrt[-4 + 4 a ] {x -> ----------------------}} 2 :[inactive; font = text; ] Here's a more complicated example. :[startGroup; font = input; ] Solve[x^5 + 3x + 1 == 0, x] :[inactive; output; endGroup; font = output; ] {ToRules[Roots[3*x + x^5 == -1, x]]} ;[o] 5 {ToRules[Roots[3 x + x == -1, x]]} :[inactive; font = text; ] It is a fact of mathematics that there is no way to get an exact formula for the solutions of a quintic equation like this. You can nevertheless ask Mathematica to give you numerical results. You get the five complex number roots to the equation. ;[s] 3:0,0;150,1;161,0;249,-1; 2:2,14,10,New York,0,10,0,0,0;1,12,10,New York,2,10,0,0,0; :[startGroup; font = input; ] N[%] :[inactive; output; endGroup; endGroup; endGroup; font = output; ] {{x -> -0.839072 - 0.943852*I}, {x -> -0.839072 + 0.943852*I}, {x -> -0.331989}, {x -> 1.00507 - 0.937259*I}, {x -> 1.00507 + 0.937259*I}} ;[o] {{x -> -0.839072 - 0.943852 I}, {x -> -0.839072 + 0.943852 I}, {x -> -0.331989}, {x -> 1.00507 - 0.937259 I}, {x -> 1.00507 + 0.937259 I}} :[inactive; startGroup; Cclosed; font = section; ] Graphics ;[s] 2:0,1;8,0;9,-1; 2:1,19,14,New York,1,14,0,0,0;1,19,14,New York,1,14,65535,0,0; :[startGroup; font = input; ] Plot[Sin[x], {x, 0, 2Pi}] :[inactive; PostScript; output; endGroup; pictureLeft = 61; pictureWidth = 282; pictureHeight = 174; preserveAspect; font = postscript; ] %! %%Creator: Mathematica %%AspectRatio: 0.61803 MathPictureStart % Scaling calculations 0.02381 0.15158 0.30902 0.2943 [ [(1.)] 0.17539 0.29652 0 1 Msboxa [(2.)] 0.32696 0.29652 0 1 Msboxa [(3.)] 0.47854 0.29652 0 1 Msboxa [(4.)] 0.63011 0.29652 0 1 Msboxa [(5.)] 0.78169 0.29652 0 1 Msboxa [(6.)] 0.93327 0.29652 0 1 Msboxa [(-1.)] 0.01131 0.01472 1 0 Msboxa [(-0.5)] 0.01131 0.16187 1 0 Msboxa [(0.5)] 0.01131 0.45617 1 0 Msboxa [(1.)] 0.01131 0.60332 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 0.61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath %%Object: Graphics [ ] 0 setdash 0 setgray gsave 0.002 setlinewidth 0 0.30902 moveto 1 0.30902 lineto stroke 0.17539 0.30277 moveto 0.17539 0.31527 lineto stroke [(1.)] 0.17539 0.29652 0 1 Mshowa 0.32696 0.30277 moveto 0.32696 0.31527 lineto stroke [(2.)] 0.32696 0.29652 0 1 Mshowa 0.47854 0.30277 moveto 0.47854 0.31527 lineto stroke [(3.)] 0.47854 0.29652 0 1 Mshowa 0.63011 0.30277 moveto 0.63011 0.31527 lineto stroke [(4.)] 0.63011 0.29652 0 1 Mshowa 0.78169 0.30277 moveto 0.78169 0.31527 lineto stroke [(5.)] 0.78169 0.29652 0 1 Mshowa 0.93327 0.30277 moveto 0.93327 0.31527 lineto stroke [(6.)] 0.93327 0.29652 0 1 Mshowa 0.02381 0 moveto 0.02381 0.61803 lineto stroke 0.01756 0.01472 moveto 0.03006 0.01472 lineto stroke [(-1.)] 0.01131 0.01472 1 0 Mshowa 0.01756 0.16187 moveto 0.03006 0.16187 lineto stroke [(-0.5)] 0.01131 0.16187 1 0 Mshowa 0.01756 0.45617 moveto 0.03006 0.45617 lineto stroke [(0.5)] 0.01131 0.45617 1 0 Mshowa 0.01756 0.60332 moveto 0.03006 0.60332 lineto stroke [(1.)] 0.01131 0.60332 1 0 Mshowa grestore 0 0 moveto 1 0 lineto 1 0.618034 lineto 0 0.618034 lineto closepath clip newpath gsave 0 setgray gsave 0.004 setlinewidth 0.02381 0.30902 moveto 0.06349 0.38519 lineto 0.10317 0.45617 lineto 0.14286 0.51712 lineto 0.1627 0.5425 lineto 0.18254 0.56389 lineto 0.20238 0.58092 lineto 0.2123 0.5877 lineto 0.22222 0.59329 lineto 0.23214 0.59766 lineto 0.2371 0.59939 lineto 0.24206 0.6008 lineto 0.24702 0.6019 lineto 0.2495 0.60233 lineto 0.25198 0.60269 lineto 0.25446 0.60296 lineto 0.2557 0.60307 lineto 0.25694 0.60316 lineto 0.25818 0.60323 lineto 0.25942 0.60328 lineto 0.26066 0.60331 lineto 0.2619 0.60332 lineto 0.26314 0.60331 lineto 0.26438 0.60328 lineto 0.26563 0.60323 lineto 0.26687 0.60316 lineto 0.26935 0.60296 lineto 0.27183 0.60269 lineto 0.27431 0.60233 lineto 0.27679 0.6019 lineto 0.28175 0.6008 lineto 0.28671 0.59939 lineto 0.29167 0.59766 lineto 0.30159 0.59329 lineto 0.32143 0.58092 lineto 0.34127 0.56389 lineto 0.38095 0.51712 lineto 0.42063 0.45617 lineto 0.46032 0.38519 lineto 0.5 0.30902 lineto 0.53968 0.23285 lineto 0.57937 0.16187 lineto 0.61905 0.10091 lineto 0.63889 0.07553 lineto 0.65873 0.05414 lineto 0.67857 0.03712 lineto 0.68849 0.03033 lineto 0.69841 0.02474 lineto 0.70833 0.02037 lineto 0.71329 0.01865 lineto Mistroke 0.71825 0.01723 lineto 0.72321 0.01613 lineto 0.72569 0.0157 lineto 0.72817 0.01535 lineto 0.73065 0.01507 lineto 0.73189 0.01496 lineto 0.73313 0.01487 lineto 0.73438 0.0148 lineto 0.73562 0.01475 lineto 0.73686 0.01472 lineto 0.7381 0.01472 lineto 0.73934 0.01472 lineto 0.74058 0.01475 lineto 0.74182 0.0148 lineto 0.74306 0.01487 lineto 0.74554 0.01507 lineto 0.74802 0.01535 lineto 0.7505 0.0157 lineto 0.75298 0.01613 lineto 0.75794 0.01723 lineto 0.7629 0.01865 lineto 0.76786 0.02037 lineto 0.77778 0.02474 lineto 0.79762 0.03712 lineto 0.81746 0.05414 lineto 0.85714 0.10091 lineto 0.89683 0.16187 lineto 0.93651 0.23285 lineto 0.97619 0.30902 lineto Mfstroke grestore grestore % End of Graphics MathPictureEnd :[inactive; font = text; ] Here is a simple Mathematica plot. ;[s] 4:0,0;17,1;28,0;35,0;36,-1; 2:3,14,10,New York,0,10,0,0,0;1,12,10,New York,2,10,0,0,0; :[inactive; font = text; ] Now for some three-dimensional graphics. :[startGroup; font = input; ] Plot3D[Sin[x y], {x, 0, 3}, {y, 0, 3}] :[inactive; PostScript; output; endGroup; pictureLeft = 61; pictureWidth = 282; pictureHeight = 282; preserveAspect; font = postscript; ] %! %%Creator: Mathematica %%AspectRatio: 1 MathPictureStart % Scaling calculations 0.02381 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0.03248 0.52913 lineto stroke 0.03248 0.52913 moveto 0.65863 0.31163 lineto stroke 0.03248 0.52913 moveto 0.38687 0.8155 lineto stroke 0.92319 0.67509 moveto 0.38687 0.8155 lineto stroke 0.92319 0.67509 moveto 0.65863 0.31163 lineto stroke % End of Graphics MathPictureEnd :[inactive; endGroup; font = text; ] Mathematica generates all graphics in PostScript, so that you can resize pictures, and make use of the resolution available on different types of printers. ;[s] 2:0,1;11,0;157,-1; 3:1,14,10,New York,0,10,0,0,0;1,12,10,New York,2,10,0,0,0;0,14,10,New York,0,10,0,0,0; :[inactive; startGroup; Cclosed; font = section; ] Animated Graphics ;[s] 2:0,1;18,0;19,-1; 2:1,19,14,New York,1,14,0,0,0;1,19,14,New York,1,14,65535,0,0; :[inactive; font = text; ] You can use sequences of graphics cells in a Notebook as frames in a "movie". To show a movie, you select its sequence of cells (by clicking the bracket that encloses all of them), then type Y. The movie appears in the first graphics cell you have selected, so be sure it is visible in the window before starting the animation. The movie is produced by showing in rapid succession the graphics cells in the selected sequence. A click anywhere will stop the animation. ;[s] 3:0,0;191,1;193,0;472,-1; 2:2,14,10,New York,0,10,0,0,0;1,15,12,Chicago,0,12,0,0,0; :[inactive; font = text; ] For information on other controls for viewing Mathematica movies, see the Notebook Read This First! ;[s] 5:0,0;46,1;57,0;84,2;100,0;101,-1; 3:3,14,10,New York,0,10,0,0,0;1,12,10,New York,2,10,0,0,0;1,12,10,New York,1,10,0,0,0; :[inactive; font = text; ] This Notebook contains a sample of animation in Mathematica. Due to disk space limitations, this is a very simple two-dimensional example. Look in the "Animations" folder on your disk for other examples of animated graphics. ;[s] 4:0,0;48,1;59,0;226,0;227,-1; 6:3,14,10,New York,0,10,0,0,0;1,12,10,New York,2,10,0,0,0;0,12,10,New York,2,10,0,0,0;0,14,10,New York,0,10,0,0,0;0,12,10,New York,2,10,0,0,0;0,14,10,New York,0,10,0,0,0; :[inactive; font = text; ] This animation shows how the curve of a Bessel function changes with the order of the Bessel function. (The movie looks best when run fairly slowly, with the frames shown cyclically.) :[font = input; ] <<Animation.m :[font = input; ] MoviePlot[BesselJ[n, x], {x, 0, 15}, {n, 1, 7, 0.2}, PlotRange->{-0.5, 1.0}] :[inactive; PICT; pictureID = 13084; output; startGroup; Cclosed; pictureLeft = 76; pictureWidth = 282; pictureHeight = 171; preserveAspect; font = postscript; ] :[inactive; PICT; pictureID = 12426; output; pictureLeft = 76; pictureWidth = 282; pictureHeight = 171; preserveAspect; font = postscript; ] :[inactive; PICT; pictureID = 26137; output; pictureLeft = 76; pictureWidth = 282; pictureHeight = 171; preserveAspect; font = postscript; ] :[inactive; PICT; pictureID = 12558; output; pictureLeft = 76; pictureWidth = 282; pictureHeight = 171; preserveAspect; font = postscript; ] :[inactive; PICT; pictureID = 14240; output; pictureLeft = 76; pictureWidth = 282; pictureHeight = 171; preserveAspect; font = postscript; ] :[inactive; PICT; pictureID = 9142; output; pictureLeft = 76; pictureWidth = 282; pictureHeight = 171; preserveAspect; font = postscript; ] :[inactive; PICT; pictureID = 22065; output; pictureLeft = 76; pictureWidth = 282; pictureHeight = 171; preserveAspect; font = postscript; ] :[inactive; PICT; pictureID = 910; output; pictureLeft = 76; pictureWidth = 282; pictureHeight = 171; preserveAspect; font = postscript; ] :[inactive; PICT; pictureID = 19887; output; pictureLeft = 76; pictureWidth = 282; pictureHeight = 171; preserveAspect; font = postscript; ] :[inactive; PICT; pictureID = 28709; output; pictureLeft = 76; pictureWidth = 282; pictureHeight = 171; preserveAspect; font = postscript; ] :[inactive; PICT; pictureID = 9183; output; pictureLeft = 76; pictureWidth = 282; pictureHeight = 171; preserveAspect; font = postscript; ] :[inactive; PICT; pictureID = 20703; output; pictureLeft = 76; pictureWidth = 282; pictureHeight = 171; preserveAspect; font = postscript; ] :[inactive; PICT; pictureID = 11535; output; pictureLeft = 76; pictureWidth = 282; pictureHeight = 171; preserveAspect; font = postscript; ] :[inactive; PICT; pictureID = 30976; output; pictureLeft = 76; pictureWidth = 282; pictureHeight = 171; preserveAspect; font = postscript; ] :[inactive; PICT; pictureID = 9455; output; pictureLeft = 76; pictureWidth = 282; pictureHeight = 171; preserveAspect; font = postscript; ] :[inactive; PICT; pictureID = 29084; output; pictureLeft = 76; pictureWidth = 282; pictureHeight = 171; preserveAspect; font = postscript; ] :[inactive; PICT; pictureID = 30681; output; pictureLeft = 76; pictureWidth = 282; pictureHeight = 171; preserveAspect; font = postscript; ] :[inactive; PICT; pictureID = 31488; output; pictureLeft = 76; pictureWidth = 282; pictureHeight = 171; preserveAspect; font = postscript; ] :[inactive; PICT; pictureID = 4847; output; pictureLeft = 76; pictureWidth = 282; pictureHeight = 171; preserveAspect; font = postscript; ] :[inactive; PICT; pictureID = 17950; output; pictureLeft = 76; pictureWidth = 282; pictureHeight = 171; preserveAspect; font = postscript; ] :[inactive; PICT; pictureID = 18992; output; pictureLeft = 76; pictureWidth = 282; pictureHeight = 171; preserveAspect; font = postscript; ] :[inactive; PICT; pictureID = 18147; output; pictureLeft = 76; pictureWidth = 282; pictureHeight = 171; preserveAspect; font = postscript; ] :[inactive; PICT; pictureID = 8348; output; pictureLeft = 76; pictureWidth = 282; pictureHeight = 171; preserveAspect; font = postscript; ] :[inactive; PICT; pictureID = 24678; output; pictureLeft = 76; pictureWidth = 282; pictureHeight = 171; preserveAspect; font = postscript; ] :[inactive; PICT; pictureID = 8016; output; pictureLeft = 76; pictureWidth = 282; pictureHeight = 171; preserveAspect; font = postscript; ] :[inactive; PICT; pictureID = 19083; output; pictureLeft = 76; pictureWidth = 282; pictureHeight = 171; preserveAspect; font = postscript; ] :[inactive; PICT; pictureID = 15287; output; pictureLeft = 76; pictureWidth = 282; pictureHeight = 171; preserveAspect; font = postscript; ] :[inactive; PICT; pictureID = 18030; output; pictureLeft = 76; pictureWidth = 282; pictureHeight = 171; preserveAspect; font = postscript; ] :[inactive; PICT; pictureID = 26621; output; pictureLeft = 76; pictureWidth = 282; pictureHeight = 171; preserveAspect; font = postscript; ] :[inactive; PICT; pictureID = 1088; output; pictureLeft = 76; pictureWidth = 282; pictureHeight = 171; preserveAspect; font = postscript; ] :[inactive; PICT; pictureID = 30762; output; endGroup; endGroup; pictureLeft = 76; pictureWidth = 282; pictureHeight = 171; preserveAspect; font = postscript; ] :[inactive; startGroup; Cclosed; font = section; ] Programming in Mathematica ;[s] 4:0,2;15,1;26,2;27,0;28,-1; 3:1,19,14,New York,1,14,0,0,0;1,17,14,New York,3,14,65535,0,0;2,19,14,New York,1,14,65535,0,0; :[inactive; font = text; ] You can use Mathematica not only as a "calculator", but also as a full symbolic programming language. Many application packages covering specific areas have been or are being written in the Mathematica language. ;[s] 6:0,1;0,2;12,1;23,0;191,1;202,0;213,-1; 3:2,14,10,New York,0,10,0,0,0;3,12,10,New York,2,10,0,0,0;1,14,10,New York,0,10,0,0,0; :[inactive; startGroup; Cclosed; font = subsubsection; ] A Graphics Application Package :[font = input; ] <<Polyhedra.m :[inactive; font = text; ] This loads in a package that defines properties of polyhedra. The package defines, among other things, the geometry of a dodecahedron. :[startGroup; font = input; ] Dodecahedron[ ] // Short :[inactive; output; endGroup; font = output; ] The Unformatted text for this cell was not generated. Use options in the Actions Settings dialog box to control when Unformatted text is generated. ;[o] {Polygon[<<1>>], <<10>>, Polygon[{<<5>>}]} :[inactive; font = text; ] The Dodecahedron function gives the coordinates for the faces of a dodecahedron, shown here in shortened form. ;[s] 5:0,0;4,3;16,1;17,2;110,0;111,-1; 4:2,14,10,New York,0,10,0,0,0;1,11,9,Courier,0,10,0,0,0;1,14,10,New York,0,10,0,0,0;1,12,10,New York,1,10,0,0,0; :[startGroup; font = input; ] Show[Graphics3D[ % ]] :[inactive; PostScript; output; pictureLeft = 50; pictureWidth = 282; pictureHeight = 277; preserveAspect; font = postscript; ] %! %%Creator: Mathematica Mpstart % Start of picture % Scaling calculations [ [ 0 0 0 0 ] [ 1 1 0 0 ] ] Mscale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 0 moveto 1 0 lineto 1 1 lineto 0 1 lineto closepath clip newpath %%Object: Graphics3D gsave 0.002 setlinewidth 0.11508 0.25083 moveto 0.41985 0.55718 lineto stroke 0.11508 0.25083 moveto 0.04222 0.71406 lineto stroke 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Use options in the Actions Settings dialog box to control when Unformatted text is generated. ;[o] -Graphics3D- :[inactive; startGroup; Cclosed; font = subsubsection; ] More on Programming :[inactive; font = text; ] There are several styles of programming in Mathematica. One of them is procedural programming, as you would find in a standard structured programming language such as C or Pascal. Another is "rule-based programming". The idea is to give transformation rules which specify how Mathematica should transform expressions it receives as input. You can give rules that mimic the formulas you might find in a mathematics textbook. Here is an example of how you might teach Mathematica about a new form of logarithm function, called nlog. ;[s] 9:0,0;43,1;54,0;279,1;290,0;472,1;483,0;531,2;535,0;537,-1; 4:5,14,10,New York,0,10,0,0,0;3,12,10,New York,2,10,0,0,0;1,11,9,Courier,0,10,0,0,0;0,11,9,Courier,1,10,0,0,0; :[startGroup; font = input; ] nlog[a b c d^2] :[inactive; output; endGroup; font = output; ] nlog[a*b*c*d^2] ;[o] 2 nlog[a b c d ] :[inactive; font = text; ] Mathematica initially knows nothing about our new function, so it does nothing to expressions involving nlogs. ;[s] 2:0,1;11,0;111,-1; 2:1,14,10,New York,0,10,0,0,0;1,12,10,New York,2,10,0,0,0; :[font = input; ] nlog[x_ y_] := nlog[x] + nlog[y] :[inactive; font = text; ] This tells Mathematica how to expand out logarithms of products. ;[s] 3:0,0;11,1;22,0;65,-1; 2:2,14,10,New York,0,10,0,0,0;1,12,10,New York,2,10,0,0,0; :[startGroup; font = input; ] nlog[a b c d^2] :[inactive; output; endGroup; font = output; ] nlog[a] + nlog[b] + nlog[c] + nlog[d^2] ;[o] 2 nlog[a] + nlog[b] + nlog[c] + nlog[d ] :[inactive; font = text; ] Now Mathematica can expand the nlog out. ;[s] 5:0,0;4,1;15,0;31,2;35,0;42,-1; 3:3,14,10,New York,0,10,0,0,0;1,12,10,New York,2,10,0,0,0;1,11,9,Courier,0,10,0,0,0; :[font = input; ] nlog[x_^n_] := n nlog[x] :[inactive; font = text; ] This gives a rule for nlog of a power. ;[s] 3:0,0;22,1;26,0;39,-1; 2:2,14,10,New York,0,10,0,0,0;1,11,9,Courier,0,10,0,0,0; :[startGroup; font = input; ] nlog[a b c d^2] :[inactive; output; endGroup; font = output; ] nlog[a] + nlog[b] + nlog[c] + 2*nlog[d] ;[o] nlog[a] + nlog[b] + nlog[c] + 2 nlog[d] :[inactive; endGroup; endGroup; font = text; ] Now Mathematica can expand the expression out completely. ;[s] 3:0,0;4,1;15,0;58,-1; 2:2,14,10,New York,0,10,0,0,0;1,12,10,New York,2,10,0,0,0; :[inactive; startGroup; Cclosed; font = section; ] Mathematica's User Interface ;[s] 3:0,1;11,2;28,0;29,-1; 3:1,19,14,New York,1,14,0,0,0;1,17,14,New York,3,14,65535,0,0;1,19,14,New York,1,14,65535,0,0; :[inactive; font = text; ] Mathematica consists of two parts — the "kernel", which actually does computations, and the "front end", which deals with interaction with the user. The kernel of Mathematica is essentially the same on all computers that support Mathematica. The front end, on the other hand, works differently on different kinds of computer. On the Macintosh, Mathematica has a sophisticated front end that takes advantage of the Macintosh's unique user interface capabilities. (You can actually use the Macintosh front end even if you are using a "remote kernel", say on a supercomputer connected through a network.) ;[s] 8:0,1;11,0;164,1;175,0;230,1;241,0;348,1;359,0;607,-1; 3:4,14,10,New York,0,10,0,0,0;4,12,10,New York,2,10,0,0,0;0,14,10,New York,0,10,0,0,0; :[inactive; startGroup; Cclosed; font = subsubsection; ] Notebooks :[inactive; font = text; ] One of the most important aspects of the Macintosh front end is its ability to support Mathematica "Notebooks". This file is an example of a Notebook. Notebooks consist of a mixture of text, graphics, and Mathematica input. Notebooks can be used like "interactive textbooks" — you read the text in the Notebook, then use the Mathematica input in the Notebook to perform calculations. ;[s] 7:0,0;87,1;98,0;209,1;220,0;330,1;341,0;389,-1; 4:4,14,10,New York,0,10,0,0,0;3,12,10,New York,2,10,0,0,0;0,11,9,Helvetica,1,10,0,0,0;0,12,10,Geneva,1,10,0,0,0; :[inactive; font = text; ] Here is a sample Mathematica Notebook called Point Plots and Space Curves. ;[s] 5:0,0;17,1;28,0;45,2;73,0;77,-1; 3:3,14,10,New York,0,10,0,0,0;1,12,10,New York,2,10,0,0,0;1,12,10,New York,4,10,0,0,0; :[inactive; startGroup; Cclosed; font = title; ] Point Plots and Space Curves by Theodore W. Gray ;[s] 4:0,1;28,0;29,2;48,0;49,-1; 4:2,23,17,New York,1,18,0,0,0;1,17,14,New York,1,14,0,0,0;1,15,12,New York,2,12,0,0,0;0,12,10,New York,1,10,0,0,0; :[inactive; font = text; ] This Notebook defines the functions PointPlot, PointPlot3D, PointParamPlot3D, SpaceCurve, and PointSpaceCurve. These functions let you make discrete point plots in two and three dimensions. The SpaceCurve and PointSpaceCurve functions let you make three-dimensional functions of one parameter (lines or points in 3D). (Note: This demonstration copy does not include the actual definitions of the functions, only examples of their use.) ;[s] 7:0,0;65,1;70,0;216,2;231,0;327,3;332,0;444,-1; 4:4,14,10,New York,0,10,0,0,0;1,14,10,New York,0,10,0,0,0;1,14,10,New York,0,10,0,0,0;1,12,10,New York,1,10,65535,0,0; :[inactive; startGroup; Cclosed; font = section; ] Examples :[inactive; font = text; ] Each of the functions defined in this Notebook is a variation of either Plot, Plot3D, or ParametricPlot3D. The arguments are quite similar to these standard functions. Following are descriptions of each of the functions. :[inactive; startGroup; Cclosed; font = subsection; ] PointPlot :[inactive; font = text; ] PointPlot[ f, {x, min, max, (step)}] produces a plot of f(x) vs. x. Here is an example: ;[s] 3:0,0;66,1;69,0;94,-1; 2:2,14,10,New York,0,10,0,0,0;1,12,10,New York,2,10,0,0,0; :[startGroup; font = input; ] PointPlot[Sin[x], {x, 0, 2 Pi, 0.2}] :[inactive; PICT; pictureID = 20909; output; pictureLeft = 67; pictureWidth = 243; pictureHeight = 147; preserveAspect; font = postscript; ] :[inactive; output; endGroup; endGroup; font = output; ] The Unformatted text for this cell was not generated. Use options in the Actions Settings dialog box to control when Unformatted text is generated. ;[o] -Graphics- :[inactive; startGroup; Cclosed; font = subsection; ] PointPlot3D :[inactive; font = text; ] PointPlot3D[ f, {x, min, max, (step)}, {y, min, max, (step)}] produces a plot of f(x, y), plotted with points instead of surfaces. ;[s] 5:0,0;42,1;65,0;93,3;135,0;136,-1; 6:3,14,10,New York,0,10,0,0,0;1,14,10,New York,0,10,0,0,0;0,14,10,New York,0,10,0,0,0;1,14,10,New York,0,10,0,0,0;0,14,10,New York,0,10,0,0,0;0,14,10,New York,0,10,0,0,0; :[startGroup; font = input; ] PointPlot3D[x y, {x, -4, 4, 0.5}, {y, -4, 4, 0.5}, BoxRatios->{1,1,1}] ;[s] 3:0,0;32,1;73,0;75,-1; 2:2,14,10,Courier,1,12,0,0,65535;1,14,10,Courier,1,12,0,0,65535; :[inactive; PICT; pictureID = 12547; output; pictureLeft = 107; pictureWidth = 202; pictureHeight = 198; preserveAspect; font = postscript; ] :[inactive; output; endGroup; endGroup; font = output; ] The Unformatted text for this cell was not generated. Use options in the Actions Settings dialog box to control when Unformatted text is generated. ;[o] -Graphics3D- :[inactive; startGroup; Cclosed; font = subsection; ] PointParamPlot3D :[inactive; font = text; ] PointParamPlot3D[{x,y,z}, {u, min, max, (step)}, {v, min, max, (step)}] produces a parametric plot of (x(u,v), y(u,v), z(u,v)), plotted with points instead of surfaces. ;[s] 8:0,0;52,1;75,0;116,2;126,3;133,4;172,3;173,0;174,-1; 5:3,14,10,New York,0,10,0,0,0;1,14,10,New York,0,10,0,0,0;1,14,10,New York,0,10,0,0,0;2,14,10,New York,0,10,0,0,0;1,14,10,New York,0,10,0,0,0; :[startGroup; font = input; ] PointParamPlot3D[{Sin[u] Cos[v], Sin[u] Sin[v], Cos[u]}, {u, 0, Pi, Pi/15}, {v, 0, 2 Pi, Pi/15}, BoxRatios->{1,1,1}] :[inactive; PICT; pictureID = 23646; output; pictureLeft = 113; pictureWidth = 205; pictureHeight = 201; preserveAspect; font = postscript; ] :[inactive; output; endGroup; endGroup; font = output; ] The Unformatted text for this cell was not generated. Use options in the Actions Settings dialog box to control when Unformatted text is generated. ;[o] -Graphics3D- :[inactive; startGroup; Cclosed; font = subsection; ] SpaceCurve :[inactive; font = text; ] SpaceCurve[{x,y,z}, {u, min, max, (step)}] produces a parametric plot of (x(u), y(u), z(u)), with the calculated points connected by straight lines. ;[s] 6:0,0;15,0;46,0;85,2;93,3;153,0;154,-1; 5:4,14,10,New York,0,10,0,0,0;0,14,10,New York,0,10,0,0,0;1,14,10,New York,0,10,0,0,0;1,14,10,New York,0,10,0,0,0;0,14,10,New York,0,10,0,0,0; :[startGroup; font = input; ] SpaceCurve[{u Sin[u], u Cos[u], u}, {u, 0, 15, 0.15}, BoxRatios->{1,1,1}] :[inactive; PICT; pictureID = 32679; output; pictureLeft = 107; pictureWidth = 202; pictureHeight = 198; preserveAspect; font = postscript; ] :[inactive; output; endGroup; endGroup; font = output; ] The Unformatted text for this cell was not generated. Use options in the Actions Settings dialog box to control when Unformatted text is generated. ;[o] -Graphics3D- :[inactive; startGroup; Cclosed; font = subsection; ] PointSpaceCurve :[inactive; font = text; ] PointSpaceCurve[{x,y,z}, {u, min, max, (step)}] produces a parametric plot of (x(u), y(u), z(u)), with the calculated points shown as dots. ;[s] 6:0,0;19,0;50,0;89,2;97,3;143,0;144,-1; 5:4,14,10,New York,0,10,0,0,0;0,14,10,New York,0,10,0,0,0;1,14,10,New York,0,10,0,0,0;1,14,10,New York,0,10,0,0,0;0,14,10,New York,0,10,0,0,0; :[startGroup; font = input; ] PointSpaceCurve[{u Sin[u], u Cos[u], u^2}, {u, 0, 15, 0.30}, BoxRatios->{1,1,1}] :[inactive; PICT; pictureID = 13244; output; pictureLeft = 114; pictureWidth = 198; pictureHeight = 194; preserveAspect; font = postscript; ] :[inactive; output; endGroup; endGroup; endGroup; endGroup; endGroup; endGroup; endGroup; font = output; ] The Unformatted text for this cell was not generated. Use options in the Actions Settings dialog box to control when Unformatted text is generated. ;[o] -Graphics3D- ^*)