趣味小程序Jenn3D:带你进入神奇的超球面空间

  

官方网站:http://www.math.cmu.edu/~fho/jenn/
Windows版下载:http://www.math.cmu.edu/~fho/jenn/jenn3d_win_2008_01_15.zip

   想知道各种几何模型在超球面(四维球的球面)上的样子吗?这个程序可以把各种几何体映射到超球面上,然后用三维的方式展示出来。你会发现几何体的棱和面都是弯的,这是因为这些几何体是在四维球面中的。就像三维球表面上的赤道和两根经线组成的“三角形”一样,每条边都是弯的。

    当然,最神奇的还是在这样的空间里下围棋!
    Windows版超球面围棋程序下载:http://www.math.cmu.edu/~fho/jenn/jenngo_win.zip
    双击左键下黑子,双击右键下白子;左键拖动旋转,右键拖动遍历第四维。
    你会发现,这个空间在边界处与自身相交。

SAGE:一个新的开源多平台数学软件

    SAGE是一个新的开源数学软件,和Mathematica、Maxima等软件一样可以进行各种复杂的数学运算。SAGE系统基于Python语言,如果你曾经用过Python,使用SAGE会感觉格外顺手。SAGE包含有一个在线版本,注册后你可以在线制作自己的数学文档,方便教学或自己学习。

主页:http://www.sagemath.org/
截屏:http://www.sagemath.org/screen_shots/
下载:http://www.sagemath.org/download.html
教程:http://www.sagemath.org/doc/html/tut/index.html

空间想象力大挑战!Smale球面外翻问题

      

    Smale球面外翻问题(Smale's Sphere Eversion Paradox)是微分拓扑学中的一个非常有趣的问题:在允许与自身相交的情况下,是否有可能无损地、平滑地、不留折痕地把一个球面的内侧翻到外面来。答案是肯定的,并且球面外翻的方法不只一种。上面这段有趣的动画里就演示了球面外翻问题的一种常见解法。你能看出这是怎么变的吗?你能把整个变换过程的每个细节都想清楚吗?你是否能在头脑里清晰地想象出整个过程?你又如何给别人解释这一过程?
    这个小程序可以帮助你观察这个球面外翻过程。你可以拉进拉远,从任意角度观察任一时刻该球面的形状。程序提供了球面透明、只查看半球等实用功能便于你一步一步进行分析。

YouTube链接:http://www.youtube.com/watch?v=R_w4HYXuo9M
了解更多:http://torus.math.uiuc.edu/jms/Papers/isama/color/opt2.htm

Linux下的数学工具Maxima 简明教程(下)

三角运算
(%i1) trigexpand(sin(10*x+y));
(%o1)                 cos(10 x) sin(y) + sin(10 x) cos(y)
(%i2) trigexpand(sin(2*x));
(%o2)                           2 cos(x) sin(x)
(%i3) trigsimp(2*cos(x)^2+sin(x)^2);
                                     2
(%o3)                             cos (x) + 1
(%i4) trigreduce(-sin(x)^2+3*cos(x)^2+x);
                      cos(2 x)      cos(2 x)   1        1
(%o4)                 -------- + 3 (-------- + -) + x - -
                         2             2       2        2

代数推理
(%i1) assume(x>0,y<-1,z>=0);
(%o1)                      [x > 0, y < - 1, z >= 0]
(%i2) assume(a<b and b<c);
(%o2)                           [b > a, c > b]
(%i3) facts();
(%o3)               [x > 0, - 1 > y, z >= 0, b > a, c > b]
(%i4) is(a>c);
(%o4)                                false
(%i5) is(z-y>0);
(%o5)                                true
(%i6) is(z-x>0);

Maxima was unable to evaluate the predicate:
z - x > 0
-- an error.  Quitting.  To debug this try debugmode(true);
(%i7) prederror:false;
(%o7)                                false
(%i8) is(z-x>0);
(%o8)                               unknown
(%i9) forget(a<b);
(%o9)                               [b > a]
(%i10) is(a>c);
(%o10)                              unknown

级数计算
(%i1) sum(i,i,1,5);
(%o1)                                 15
(%i2) sum(i^2,i,1,5);
(%o2)                                 55
(%i3) sum(1/2^i,i,1,inf);
                                   inf
                                   ====
                                        1
(%o3)                               >    --
                                   /      i
                                   ====  2
                                   i = 1
(%i4) sum(1/2^i,i,1,inf),simpsum;
(%o4)                                  1
(%i5) sum(1/i^2,i,1,inf),simpsum;
                                        2
                                     %pi
(%o5)                                ----
                                      6
(%i6) sum(1/i,i,1,inf),simpsum;
(%o6)                                 inf

微积分
(%i1) limit(1/x,x,inf);
(%o1)                                  0
(%i2) limit(sin(x)/x,x,0);
(%o2)                                  1
(%i3) limit(sin(x),x,inf);
(%o3)                            &n
bsp;    ind
(%i4) diff(3*x^2+x+5/x,x);
                                       5
(%o4)                            6 x - -- + 1
                                        2
                                       x
(%i5) diff(sin(x)*tan(x),x);
                                           2
(%o5)                   cos(x) tan(x) + sec (x) sin(x)
(%i6) diff(%e^(a*x),x);
                                        a x
(%o6)                               a %e
(%i7) integrate(sin(x)^3,x);
                                  3
                               cos (x)
(%o7)                          ------- - cos(x)
                                  3
(%i8) integrate(x^3,x,1,3);
(%o8)                                 20
(%i9) taylor(%e^x,x,0,3);
                                     2    3
                                    x    x
(%o9)/T/                    1 + x + -- + -- + . . .
                                    2    6
(%i10) taylor(sin(x),x,0,5);
                                  3    5
                                 x    x
(%o10)/T/                    x - -- + --- + . . .
                                 6    120
(%i11) taylor(sqrt(x+1),x,1,3);
                                                     2                  3
                    sqrt(2) (x - 1)   sqrt(2) (x - 1)    sqrt(2) (x - 1)
(%o11)/T/ sqrt(2) + --------------- - ---------------- + ----------------
                           4                 32                128
                                                                        + . . .
(%i12) ratsimp(%);
                      3              2
             sqrt(2) x  - 7 sqrt(2) x  + 43 sqrt(2) x + 91 sqrt(2)
(%o12)       -----------------------------------------------------
                                      128

矩阵运算
(%i1) f[i,j]:=i+j;
(%o1)                           f     := i + j
                                 i, j
(%i2) genmatrix(f,3,3);
                                  [ 2  3  4 ]
                                  [         ]
(%o2)                       &nbs
p;     [ 3  4  5 ]
                                  [         ]
                                  [ 4  5  6 ]
(%i3) g[i,j]:=i-2^j;
                                              j
(%o3)                           g     := i - 2
                                 i, j
(%i4) genmatrix(g,3,3);
                               [ - 1  - 3  - 7 ]
                               [               ]
(%o4)                          [  0   - 2  - 6 ]
                               [               ]
                               [  1   - 1  - 5 ]
(%i5) %o2+%o4;
                                 [ 1  0  - 3 ]
                                 [           ]
(%o5)                            [ 3  2  - 1 ]
                                 [           ]
                                 [ 5  4   1  ]
(%i6) %o2.%o4;
                               [ 2  - 16  - 52 ]
                               [               ]
(%o6)                          [ 2  - 22  - 70 ]
                               [               ]
                               [ 2  - 28  - 88 ]
(%i7) %o2^^3;
                               [ 360  474  588 ]
                               [               ]
(%o7)                          [ 474  624  774 ]
                               [               ]
                               [ 588  774  960 ]
(%i8) x:matrix([17, 3],[-8, 11]);
                                  [ 17   3  ]
(%o8)                             [         ]
                                  [ - 8  11 ]
(%i9) x^^-1;
                                [ 11      3  ]
                                [ ---  - --- ]
                                [ 211    211 ]
(%o9)                           [            ]
                                [  8    17   ]
            &n
bsp;                   [ ---   ---  ]
                                [ 211   211  ]

想了解更多请阅读官方文档:
http://maxima.sourceforge.net/docs/manual/en/maxima.html

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