Mupad 应用实践

Mupad学习记录1

Matlab自带的帮助文件是最好的教程，若想全部学习mupad用法，建议从头到尾看一下mupad帮助文档，打开办法：在Matlab命令栏输入 doc mupad 在弹出的窗口中找到mupad一栏

关于如何打开mupad

1）创建函数句柄

nb = allMuPADNotebooks

2）创建空白文档

nb = mupad或mupad

3) 打开欢迎界面

mupadwelcome

4) 在matlab主界面的“应用程序”中打开

Mupad的基本操作

1.执行命令 Enter

2.暂缓显示结果命令 :

3.换行命令 shift+enter 或 ctrl+enter

4.变量变更的作用范围修改 在mupad界面的notebook中点击关于evaluate的若干命令

5.自动填写 ctrl+space

6.命令窗口 View>Command Bar

7.命令窗口中的#symbols是等待修改的变量，用快捷键tab可以快速切换

8.关于帮助

1) info(#)

2) ?# 直接弹出相应的帮助文档，类似于matlab中的doc命令

3) 光标停留在相应位置

关于数值转化与复数

float(sqrt(56))

float(1.0/3*exp(1)*sin(2))

DIGITS:=20: float(sqrt(56))

DIGITS:=30: float(PI); float(E); delete DIGITS

rectform(1/(sqrt(2) + I))

conjugate(1/(2^(1/2) + I)) 复数转化命令

Re(1/(2^(1/2) + I))

Im(1/(2^(1/2) + I)) 复数的显示问题

abs(1/(2^(1/2) + I))

arg(1/(2^(1/2) + I)) 计算模和极角

关于微分

f := y^2 + 4*x + 6*x^2 + 4*x^3 + x^4: diff(f, y) 求(偏)导

diff(diff(diff(sin(x), x), x), x)

diff(sin(x), x, x, x)

diff(sin(x), x $ 3)

diff(x^2*y^2 + 4*x^2*y + 6*x*y^2, y, x) 高阶（偏）导

其它命令

D D(sin) 注意D(sin)(x^2) 结果显示cos(x^2) 改进方法是自定义函数f := x -> sin(x^2): D(f)

' sin'

关于积分

int((cos(x))^3, x) 不定积分

int((cos(x))^3, x = 0..PI/4) 定积分

int(sin(x)/x, x = -infinity..infinity) 广义积分

当结果不收敛时返回积分公式

numeric::int(sin(x^2)^2, x = -1..1)

float(int(sin(x^2)^2,(x = -1..1))) 常见数值转化公式

关于线性代数

A := matrix([[1, 2], [3, 4], [5, 6]])

B := matrix([[1, 2, 3], [4, 5, 6]])

C := matrix(3, 3, [[-1, -2, -3], [-4, -5, -6], [-7, -8, -9]]) 矩阵的创建

F := matrix(3, 3, [[1, -1, 0], [2, -2]]) 空余位置以0补齐,多余位置返回错误

G := matrix(4, 4, [1, 2, 3, 4], Diagonal) 建立对角矩阵

A*B 乘法运算

C + 10 加上10乘以单位矩阵

det(G) 行列式

1/G 逆矩阵

linalg::eigenvalues(F) 直接求出矩阵F的特征值

关于方程

solve(x^5 + 3*x^4 - 23*x^3 - 51*x^2 + 94*x + 120 = 0, x) 单变量方程

solve(a*x^2 + b*x + c = 0, x) 参数方程

solve([x^2 + x*y + y^2 = 1, x^2 - y^2 = 0], [x, y]) 解方程组

solve([x^2 + y^2 = a, x^2 - y^2 = b], [x, y]) 解含参方程组

o := ode(x^2*diff(y(x), x, x) + 2*x*diff(y(x), x) + x, y(x)):

solve(o) 解常微分方程

solve(x^4 >= 5, x) 解不等式

assume(x in R_); solve(x^4 >= 5, x)

solve(x^4 >= 5, x) assuming x > 0 一定假设下的解方程，否则结果分类讨论十分繁琐

solve(a*x^2 + b*x + c = 0, x) assuming a > 0

solve(x^(5/2) = 1, x) assuming x in R_

solve(x^4 - a = 0, x) assuming a = 16 and x in R_

关于多项式

factor(x^12 - 1) 因式分解

expand((x - 1)*(x + 1)*(x^2 + x + 1)*(x^2 + 1)*(x^2 - x + 1)*(x^4 - x^2 + 1)) 合并运算

collect((x - a)^4 + a*x^3 + b^2*x + b*x + 10*a^4 +(b + a*x)^2, x) 定义未定元的多项式整理

partfrac((7*x^2 + 7*x + 6)/(x^3 + 2*x^2 + 2*x + 1)) 分式的整理

simplify((x - 1)*(x + 1)*(x^2 + x + 1)*(x^2 + 1)*(x^2 - x + 1)*(x^4 - x^2 + 1))

Simplify((x - 1)*(x + 1)*(x^2 + x + 1)*(x^2 + 1)*(x^2 - x + 1)*(x^4 - x^2 + 1)) 简化命令

expand(sin(5*x)) 三角函数的分解

simplify(sqrt(x^2 + 2*x + 1) + sqrt(x^2 - 2*x + 1) + sqrt(x^2 + 4*x + 4) + sqrt(x^2 - 4*x + 4)) assuming x > 2 假设条件下的化简

注意

进行完一步计算之后我们需要删除命令

delete f:

Mupad学习记录2

这一节着重学习关于绘图及其动画控制

绘图

1)一般绘图

plot(sin(x)*cos(3*x)) 绘制二维图

plot(sin(x)*sin(y), #3D) 绘制三维图（#3D替代二维动画）

plot(sin(x), cos(x), tan(x), cot(x) 多个图形在同一张纸里

plot(sin(k*x) $ k = 1..3) 使用$符号使得一系列图画在一张纸上（二维）

plot(-sqrt(r^2 - x^2 - y^2) $ r = 1..5, #3D) 使用$符号使得一系列图画在一张纸上（三维）

plot(sin(x^3)*exp(x), x = 3..5) 使用x=#..#确定作图范围（二维）

plot(sin(x)*sin(y), x = 0..3, y = 1..3, #3D) 使用x=#..#,y=#..#确定作图范围（三维）

plot(sin(k*x) $ k = 1..5, x = 0..2*PI) 在关于$的系列图中设定范围

plot({sin(k*x), k*t^2} $ k = 1..5, x = 0..2*PI, t = -1..1) 多幅系列图的画法及其范围

plot(piecewise([x < - 2, - 1], [-1 < x and x < 0, x^2], [0 < x and x < 1, -x^2], [x > 1, 1])) 分段函数的画法

通过界面直接修改图形属性，而非使用编码修改，如果点击目标对象无反应，选择菜单View > Object Browser

在图形界面，可以自由旋转，缩放，以及自动旋转等，相关命令在图形窗口界面，自行尝试，概不赘述

动画

1)关于参数的动画

plot(exp(x)*sin(a^2*x), x = 1..2, a = 2..6) 二维直角坐标动画

plot(sin(a*x^2 + a*y^2), x = -2..2, y = -2..2, a = 0.1..2, #3D) 三维直角坐标动画

例子1

plot(

plot::Function2d(a*sin(x), x = 0..a*PI, a = 0.5..1),

plot::Rectangle(0..a*PI, 0..a, a = 0.5..1,

LineColor = RGB::Black),

plot::Line2d([0, 0], [PI*a, a], a = 0.5 ..1,

LineColor = RGB::Black)

//其中的plot::Rectangle和plot::Line2d分别是矩形作图和直线作图

例子2

plot(plot::Arc2d(1 + a, [0, 0], AngleRange = 0..a*PI, a = 0..1)):

//其中的plot::Arc2d是指plot::Arc2d（半径，中心，弧线角度范围，动画参数a）

例子3

plot(plot::Function2d(4*a*x, x = 0..1, a = 0..1),

plot::Function2d(b*x^2, x = 0..1, b = 1..4)):

//这是将两幅参数动画画到同一幅图片中，参数的名称不同，注意参数名称是全局变量，必须不同

例子4

f := (x, y) -> 4 - (x - a)^2 - (y - a)^2:

mycolor := proc(x, y, z, a)

local t;

begin

t := sqrt((x - a)^2 + (y - a)^2):

if t < 0.1 then

return(RGB::Red)

elif t < 0.4 then

return(RGB::Orange)

elif t < 0.7 then

return(RGB::Green)

else return(RGB::Blue)

end_if;

end:

plot(plot::Function3d(f, x = -1..1, y = -1..1, a = -1..1,

FillColorFunction = mycolor)):

//这个例子展示了如何用全局变量t定义调色函数myclolor,其中proc是定义程序的命令(procedure的缩写)，local等也是相应配套命令，详情可以用?local命令查看帮助文档

例子5

plot(plot::Point2d([a, sin(a)], a = 0..2*PI,

Frames = 100, TimeRange = 0..5)):

//5秒100帧

plot(plot::Point2d([a, sin(a)], a = 0..2*PI,

Frames = 200, TimeRange = 0..5)):

//5秒200帧，增加帧可以使动画更光滑，默认是50帧，根据人眼特性，n<25(t1-t0)是合适的帧数

例子6

p1 := plot::Point2d(-1, sin(a), a = 0..PI, Color = RGB::Red,

PointSize = 5*unit::mm,

TimeBegin = 0, TimeEnd = 5):

p2 := plot::Point2d(0, sin(a), a = 0..PI, Color = RGB::Green,

PointSize = 5*unit::mm,

TimeBegin = 6, TimeEnd = 12):

p3 := plot::Point2d(1, sin(a), a = 0..PI, Color = RGB::Blue,

PointSize = 5*unit::mm,

TimeBegin = 9, TimeEnd = 15):

plot(p1, p2, p3, PointSize = 3.0*unit::mm,

YAxisVisible = FALSE):

//多个对象的动画设定，动画的先后顺序通过timebegin和timeend来控制

例子7

p2::VisibleAfterEnd := FALSE:

p3::VisibleBeforeBegin := FALSE:

plot(p1, p2, p3, PointSize = 3.0*unit::mm,

YAxisVisible = FALSE):

//通过Visible命令控制动画前后的可见性

//常见命令有VisibleAfter = t0，VisibleBefore = t1，VisibleFromTo = t0..t1等，详见帮助文档

例子8

p1 := plot::Point2d(-1, sin(a), a = 0..PI, Color = RGB::Red,

PointSize = 5*unit::mm,

TimeBegin = 0, TimeEnd = 5):

p2 := plot::Point2d(0, sin(a), a = 0..PI, Color = RGB::Green,

PointSize = 5*unit::mm,

TimeBegin = 6, TimeEnd = 12):

p3 := plot::Point2d(1, sin(a), a = 0..PI, Color = RGB::Blue,

PointSize = 5*unit::mm,

TimeBegin = 9, TimeEnd = 15):

p2::VisibleAfterEnd := FALSE:

p3::VisibleBeforeBegin := FALSE:

p4 := plot::Point2d(0.5, 0.5, Color = RGB::Black,

PointSize = 5*unit::mm,

VisibleFromTo = 7..13):

plot(p1, p2, p3, p4, PointSize = 3.0*unit::mm,

YAxisVisible = FALSE):

//多个对象，静止和运动交互出现的控制

例子9

for i from 0 to 101 do

t[i] := i/10;

end_for:

for i from 0 to 100 do

x := i/100*PI;

myframe[i] := plot::Group2d(

plot::Point2d([x, sin(x)], Color = RGB::Red),

plot::Point2d([x, cos(x)], Color = RGB::Blue),

VisibleFromTo = t[i]..t[i + 1]);

end_for:

plot(myframe[i] $ i = 0..100, PointSize = 5.0*unit::mm):

//这是所谓的"Frame by frame animations"，通过编程实现动画

例子10

delete i:

plot(plot::Point2d([i/100*PI, sin(i/100*PI)], i = 0..100,

Color = RGB::Red),

plot::Point2d([i/100*PI, cos(i/100*PI)], i = 0..100,

Color = RGB::Blue),

Frames = 101, TimeRange = 0..10,

PointSize = 5.0*unit::mm):

//例9同样可以用一般动画命令来实现

例子11

f := x -> -sqrt(1 - x^2):

plot(// The static rim:

plot::Function2d(f(x), x = -1..1, Color = RGB::Black),

// The incoming rays:

plot::Line2d([x, 2], [x, f(x)], VisibleAfter = 5*x

) $ x in [-1 + i/20 $ i = 1..39],

// The reflected rays leaving to the right:

plot::Line2d([x, f(x)],

[1, f(x) + (1-x)*(f'(x) - 1/f'(x))/2],

Color = RGB::Orange, VisibleAfter = 5*x

) $ x in [-1 + i/20 $ i = 1..19],

// The reflected rays leaving to the left:

plot::Line2d([x, f(x)],

[-1, f(x) - (x+1)*(f'(x) - 1/f'(x))/2],

Color = RGB::Orange, VisibleAfter = 5*x

) $ x in [-1 + i/20 $ i = 21..39],

ViewingBox = [-1..1, -1..1]):

//运用visibleafter命令，将已经画好的图像逐步呈现，以达到动画效果，这是一种常见的简单动画制作方法

应用实例展示

例子12

DIGITS := 2:

// the function:

f := x -> cos(x^2):

// the anti-derivative:

F := x -> numeric::int(f(y), y = 0..x):

// the graph of f(x):

g := plot::Function2d(f(x), x = 0..6, Color = RGB::Blue):

// the graph of F(x):

G := plot::Function2d(F(x), x = 0..6, Color = RGB::Black):

// a point moving along the graph of F(x):

p := plot::Point2d([a, F(a)], a = 0..6, Color = RGB::Black):

// hatched region between the origin and the moving point p:

h := plot::Hatch(g, 0, 0 ..a, a = 0..6, Color = RGB::Red):

// the right border line of the hatched region:

l := plot::Line2d([a, 0], [a, f(a)], a = 0..6,

Color = RGB::Red):

// a dashed vertical line from f to F:

L1 := plot::Line2d([a, f(a)], [a, F(a)], a = 0..6,

Color = RGB::Black, LineStyle = Dashed):

// a dashed horizontal line from the y axis to F:

L2 := plot::Line2d([-0.1, F(a)], [a, F(a)], a = 0..6,

Color = RGB::Black, LineStyle = Dashed):

// the current value of F at the moving point p:

t := plot::Text2d(a -> F(a), [-0.2, F(a)], a = 0..6,

HorizontalAlignment = Right):

plot(g, G, p, h, l, L1, L2, t,

YTicksNumber = None, YTicksAt = [-1, 1]):

delete DIGITS:

//积分动画

例子13

c := a -> 1/2 *(1 - 1/sin(PI/2*a)):

mycolor := (u, v, x, y, z) -> [(u - 0.8)/0.4, 0, (1.2 - u)/0.4]:

rectangle2annulus := plot::Surface(

[c(a) + (u - c(a))*cos(PI*v), (u - c(a))*sin(PI*v), 0],

u = 0.8..1.2, v = -a..a, a = 1/10^10..1,

FillColorFunction = mycolor, Mesh = [3, 40], Frames = 40):

plot(rectangle2annulus, Axes = None,

CameraDirection = [-11, -3, 3]):

//圆盘动画

annulus2moebius := plot::Surface(

[((u - 1)*cos(a*v*PI/2) + 1)*cos(PI*v),

((u - 1)*cos(a*v*PI/2) + 1)*sin(PI*v),

(u - 1)*sin(a*v*PI/2)],

u = 0.8..1.2, v = -1..1, a = 0..1,

FillColorFunction = mycolor, Mesh = [3, 40], Frames = 20):

plot(annulus2moebius, Axes = None,

CameraDirection = [-11, -3, 3]):

//折叠动画

rectangle2annulus::VisibleFromTo := 0..5:

annulus2moebius::VisibleFromTo := 5..7:

plot(rectangle2annulus, annulus2moebius, Axes = None,

CameraDirection = [-11, -3, 3]):

//连接前两个动画

//莫比乌斯带

例子14

ms := 1: m1 := 0.04: m2 := 0.0001:

//设定三体质量

Y := numeric::odesolve2(numeric::ode2vectorfield(

{xs''(t) =

-m1*(xs(t)-x1(t))/sqrt((xs(t)-x1(t))^2 + (ys(t)-y1(t))^2)^3

-m2*(xs(t)-x2(t))/sqrt((xs(t)-x2(t))^2 + (ys(t)-y2(t))^2)^3,

ys''(t) =

-m1*(ys(t)-y1(t))/sqrt((xs(t)-x1(t))^2 + (ys(t)-y1(t))^2)^3

-m2*(ys(t)-y2(t))/sqrt((xs(t)-x2(t))^2 + (ys(t)-y2(t))^2)^3,

x1''(t) =

-ms*(x1(t)-xs(t))/sqrt((x1(t)-xs(t))^2 + (y1(t)-ys(t))^2)^3

-m2*(x1(t)-x2(t))/sqrt((x1(t)-x2(t))^2 + (y1(t)-y2(t))^2)^3,

y1''(t) =

-ms*(y1(t)-ys(t))/sqrt((x1(t)-xs(t))^2 + (y1(t)-ys(t))^2)^3

-m2*(y1(t)-y2(t))/sqrt((x1(t)-x2(t))^2 + (y1(t)-y2(t))^2)^3,

x2''(t) =

-ms*(x2(t)-xs(t))/sqrt((x2(t)-xs(t))^2 + (y2(t)-ys(t))^2)^3

-m1*(x2(t)-x1(t))/sqrt((x2(t)-x1(t))^2 + (y2(t)-y1(t))^2)^3,

y2''(t) =

-ms*(y2(t)-ys(t))/sqrt((x2(t)-xs(t))^2 + (y2(t)-ys(t))^2)^3

-m1*(y2(t)-y1(t))/sqrt((x2(t)-x1(t))^2 + (y2(t)-y1(t))^2)^3,

xs(0) = -m1 , x1(0) = ms, x2(0) = 0,

ys(0) = 0.7*m2, y1(0) = 0, y2(0) = -0.7*ms,

xs'(0) = -1.01*m2, x1'(0) = 0, x2'(0) = 1.01*ms,

ys'(0) = -0.9*m1, y1'(0) = 0.9*ms, y2'(0) = 0},

[xs(t), xs'(t), ys(t), ys'(t),

x1(t), x1'(t), y1(t), y1'(t),

x2(t), x2'(t), y2(t), y2'(t)]

)):

//解ODE方程并输入相关公式

dt := 0.05: imax := 516:

plot(// The sun:

plot::Point2d(Y(t)[1], Y(t)[3], Color = RGB::Orange,

VisibleFromTo = t..t + 0.99*dt,

PointSize = 4*unit::mm

) $ t in [i*dt $ i = 0..imax],

// The giant planet:

plot::Point2d(Y(t)[5], Y(t)[7], Color = RGB::Red,

VisibleFromTo = t..t + 0.99*dt,

PointSize = 3*unit::mm

) $ t in [i*dt $ i = 0..imax],

// The orbit of the giant planet:

plot::Line2d([Y(t - dt)[5], Y(t - dt)[7]],

[Y(t)[5], Y(t)[7]], Color = RGB::Red,

VisibleAfter = t

) $ t in [i*dt $ i = 1..imax],

// The small planet:

plot::Point2d(Y(t)[9], Y(t)[11], Color = RGB::Blue,

VisibleFromTo = t..t + 0.99*dt,

PointSize = 2*unit::mm

) $ t in [i*dt $ i = 0..imax],

// The orbit of the small planet:

plot::Line2d([Y(t - dt)[9], Y(t - dt)[11]],

[Y(t)[9], Y(t)[11]], Color = RGB::Blue,

VisibleAfter = t

) $ t in [i*dt $ i = 1..imax]

):

//运用"frame by frame"制定三体运动动画

Mupad学习记录3

这节我基于Mupad,对螺旋线作图及其动画做一个探究

及其过程中学习到的控制命令

在这一过程中，学习到了以下几个功能及其命令：

镜头 plot::Camera

排版 Layout = #

标题 Footer, Header

字体 HeaderFont

字体位置 HeaderAlignment

变光滑 AdaptiveMesh

划分 UMesh

多图合并 plot::Scene3d

坐标数据 TicksNumber = None (plot settings)

坐标标示 TicksAt

颜色调整 FillColorFunction, Color=#

原比例尺 Scaling=Constrained

点列 plot::PointList3d

扫面 plot::Sweep

组图 plot::Group3d

点法式平面 plot::Plane

三点式平面 plot::Polygon3d

空间圆 plot::Circle3d

切平面 plot::Parallelogram3d

摆线 WheelRadius

坐标变换 Transformations

这些都是在帮助文档中可以找到详细用法，需要相关命令时查阅帮助文档就可以了

Mupad学习记录4

动画遇到的问题及其解决办法

以下是我在实践中遇到的种种问题，经过反复调试才得到解决，我把这些经验写在这里

1，布局问题

用Layout命令对全是动画或全是静图的Scened2d/3d是奏效的，但是当两者一起，plot命令总是把静图优先放置，导致Horizontal命令失效，这时需要用Layout = Relative命令，自行编制动画位置，方可解决

Mupad学习记录5（关于符号）

Mupad notebook提供了良好的编辑界面，对于想基于该界面，制作报告课件等的用户，了解如何编辑符号公式等就显得很必要。

这一节的帮助文档，详见“Symbols and Special Characters”

Symbol::accentAsterisk 带星号

Symbol::accentTilde 带上弯线

Symbol::accentHat 带上角标

Symbol::accentRightArrow 带向量箭头

Symbol::accentDot 带点

Symbol::accentDoubleDot 带双点

Symbol::accentTripleDot 带三点

Symbol::accentOverBar 带上划线

Symbol::accentUnderBar 带下划线

Symbol::new 调取通用字符（参见Typeset Symbols）

Symbol::subScript 下标

Symbol::subSuperScript 上下标

Symbol::superScript 上标

output::mathText 输出含数学公式的语句

output::ordinal 输出数字序数

output::roman 输出罗马序数

output::tableForm 输出表列

output::tree 输出树状图

修改结果，不缩写化简 Pref::abbreviateOutput(FALSE):

关闭优化字符显示 PRETTYPRINT := FALSE

常见错误分析：

Error: The argument is invalid. [xmlprint::beginElement]

Evaluating: plot

定义的自变量有问题,且只要出现 invalid 一般都说明是拼写出现错误，特别是大小写

Error: The argument 'Umesh = 10' is unexpected. [plot::Curve3d::new]

一般是拼写有误

Error: Unexpected 'identifier'. [line 6, col 1]

缺少冒号

The attribute 'VMesh' in the 'Surface' object must be a positive integer number. [plot]

循环语句i的初始值或末值超出范围，或初值为0，或末值有误

画图结果多出对象，检查最后的plot（中是否多打字母）