Maple

用戶能够直接使用傳統數學符號进行輸入，也可以定制个性化的界面。对于数值计算有额外的支持，能够扩展到任意精度，同时亦支持符號演算及可视化。符號演算的例子参见下文。Maple内建有一种动态的命令行风格的编程语言，该语言支持具有作用域的变量。同时亦有其他語言的接口（C、FORTRAN、Java、Matlab和Visual Basic）。还具有与Excel进行交互的接口。

Maple由一个很小的由C语言编写的内核提供Maple语言。许多功能由各种来源的函数库提供。许多数值计算由NAG数值计算库, ATLAS库, GNU多精度库提供。大部分库由Maple语言编写，并且可查看源代码。

Maple中不同的功能需要不同格式的数值数据。符号表达式在内存中以有向无环图的形式存储。标准界面和计算界面由Java语言编写。经典界面由C语言编写。

evalf[100](2^1/12)

1.059463094359295264561825294946341700779204317494185628559208431458761646063255722383768376863945569

f:=x^2-63*x+99=0;

solve(f,x);

${\displaystyle {\frac {63}{2}}+{\frac {3}{2}}*{\sqrt {(}}397)}$, ${\displaystyle {\frac {63}{2}}-{\frac {3}{2}}*{\sqrt {(}}397)}$

f := x^7+3*x = 7;

solve(f,x);

RootOf(${\displaystyle Z^{7}+3Z-7}$, index = 1),

RootOf(${\displaystyle Z^{7}+3Z-7}$, index = 2),

RootOf(${\displaystyle Z^{7}+3Z-7}$, index = 3),

RootOf(${\displaystyle Z^{7}+3Z-7}$, index = 4),

RootOf(${\displaystyle Z^{7}+3Z-7}$, index = 5),

RootOf(${\displaystyle Z^{7}+3Z-7}$, index = 5),

RootOf(${\displaystyle Z^{7}+3Z-7}$, index =7),

evalf(%);

• (1.1922047171828134),
• (0.8658388666792263) + (0.9230818802764879) I,
• (0.2099602786426775) + (1.3442579297631496) I,
• (1.2519809466279554) + (0.6424819505558892) I,
• (1.2519809466279554) - (0.6424819505558892) I,
• (0.2099602786426775) - (1.3442579297631496) I,
• (0.8658388666792263) - (0.9230818802764879) I

f := sin(x)^3+5*cosh(x) = 0;

${\displaystyle sin^{3}(x)+5cosh(x)=0}$

> solve(f, x);

RootOf(${\displaystyle sin^{3}(Z)-arccosh({\frac {-1}{5}}sin(Z)))}$

> evalf(%);

0.2873691672 - 1.111497506 I

根据${\displaystyle x-y>6}$，寻找${\displaystyle (x+y)^{5}=9}$的所有实数解。

solve({x-y > 6, (x+y)^5 = 9}, [x, y])[];

答案： ${\displaystyle [x=3^{2/5}-y,\quad y<{\frac {1}{2}}3^{2/5}-3]}$

代数方程组

> p1 := x*y*z-x*y^2-z-x-y; p2 := x*z-x^2-z-y+x; p3 := z^2-x^2-y^2;

> sys := {p1, p2, p3};

> var := {x, y, z};

> solve(sys, var);

{x = 0, y = y, z = -y}, {x = 3, y = 4, z = 5}, {x = 1, y = 0, z = -1}

三角方程组

> f1 := cos(x)+sin(3*y)+tan(5*z) = 0;

> f2 := cos(3*z)+tan(3*y^2)-sin(2*z^3) = 33;

> f3 := tan(4*x+y)-sin(5*y-4*z) = 2*x;

> sys1 := {f1, f2, f3};

> var1 := {x, y, z};

{x, y, z}

> fsolve(sys1, var1);

{x = -10.77771790, y = -2.397849343, z = -7.382158103}

计算矩阵的行列式。

M:= Matrix([[1,2,3]], [a,b,c], [[x,y,z]]);  # 矩阵样例

${\displaystyle {\begin{bmatrix}1&2&3\\a&b&c\\x&y&z\end{bmatrix}}}$

with(LinearAlgebra)

m:=Determinant(M);

答案：${\displaystyle bz-cy+3ay-2az+2xc-3xb}$

朗斯基行列式

with(VectorCalculus);

w:=Wronskian([1,x,x^3+x-1],x)

Matrix(3, 3, {(1, 1) = 1, (1, 2) = x, (1, 3) = x^3+x-1, (2, 1) = 0, (2, 2) = 1, (2, 3) = 3*x^2+1, (3, 1) = 0, (3, 2) = 0, (3, 3) = 6*x})

d:=Determinant(w);

6x

雅可比矩阵

J := Jacobian([r*sin(t)), r^2*cosh(t)], [r, t]);

m:=Matrix(2, 2, {(1, 1) = cos(t), (1, 2) = -r*sin(t), (2, 1) = sinh(t), (2, 2) = r*cosh(t)})

d:=Determinant(m);

sin(t)*r^2*sinh(t)-2r^2cos(t)cosh(t)

海森矩阵

f := x^3+y*cos(x)+t*tan(y))

with(VectorCalculus);

h:=hessian(f,[x,y,t]);

${\displaystyle {\begin{bmatrix}6*x-y*cos(x)&-sin(x)&0\\-sin(x)&2*t*tan(y)*(1+tan(y)^{2})&1+tan(y)^{2}\\0&1+tan(y)^{2}&0\end{bmatrix}}}$

求${\displaystyle \int \cos \left({\frac {x}{a}}\right)dx}$.

int(cos(x/a), x);

答案：${\displaystyle a\sin \left({\frac {x}{a}}\right)}$

求${\displaystyle \int \sin \left({\frac {x}{a}}\right)dx}$.

int(sin(x/a), x);

答案：${\displaystyle -a\cos \left({\frac {x}{a}}\right)}$

注意：Maple在积分时不提供常数项C，必须自行补上。

定积分

> int(cos(x/a), x = 1 .. 5);

16 a sin（1/a)* cos^4(1/a) - 12 a sin^2(1/a)

计算以下线性常微分方程的一个精确解${\displaystyle {\frac {d^{2}y}{dx^{2}}}(x)-3y(x)=x}$初始条件为${\displaystyle y(0)=0,\quad \left.{\frac {dy}{dx}}\right|_{x=0}=2}$

dsolve( {diff(y(x),x,x) - 3*y(x) = x, y(0)=0, D(y)(0)=2}, y(x) );

答案：${\displaystyle y(x)={\frac {7}{18}}{\sqrt {3}}e^{{\sqrt {3}}x}-{\frac {7}{18}}{\sqrt {3}}e^{-{\sqrt {3}}x}-{\frac {1}{3}}x}$

dsolve(diff(y(x), x, x) = x^2*y(x))

解：

${\displaystyle y(x)=C_{1}{\sqrt {(}}x)}$BesselI(${\displaystyle 1 \over 4}$,${\displaystyle 1 \over 2}$${\displaystyle x^{2}}$)

+${\displaystyle C_{2}{\sqrt {(}}x)}$BesselK(${\displaystyle 1 \over 4}$,${\displaystyle 1 \over 2}$${\displaystyle x^{2}}$)

series(tanh(x),x=0,15)

${\displaystyle x-{\frac {1}{3}}\,x^{3}+{\frac {2}{15}}\,x^{5}-{\frac {17}{315}}\,x^{7}}$

${\displaystyle +{\frac {62}{2835}}\,x^{9}-{\frac {1382}{155925}}\,x^{11}+{\frac {21844}{6081075}}\,x^{13}+{\mathcal {O}}(x^{15})}$

f:=int(exp^cosh(x),x)
series(f,x=0,15);

${\displaystyle ex+{\frac {1}{6}}ex^{3}+{\frac {1}{30}}ex^{5}+{\frac {31}{5040}}ex^{7}+{\frac {379}{362880}}ex^{9}}$

${\displaystyle +{\frac {149}{907200}}ex^{11}+{\frac {150349}{6227020800}}ex^{13}+{\frac {4373461}{1307674368000}}ex^{15}+{\mathcal {O}}(x^{17})}$

with(inttrans);

拉普拉斯变换

> f := (1+A*t+B*t^2)*exp(c*t);

${\displaystyle (1+A*t+B*t^{2})*e^{c*t}}$

> laplace(f, t, s);

${\displaystyle {\frac {1}{s-c}}+{\frac {A}{(s-c)^{2}}}+{\frac {2B}{(s-c)^{3}}}}$

反拉普拉斯变换

invlaplace(1/(s-a),s,x)

${\displaystyle e^{ax}}$

z := y(t);

y(t)

f := diff(z, t, t)+a*(diff(z, t)) = b*z;

${\displaystyle {\frac {d^{2}}{dt^{2}}}y(t)+a{\frac {d}{dt}}y(t)=by(t)}$

with(inttrans);

g := laplace(f, t, s);

s^2*laplace(y(t), t, s) - D(y)(0) - s y(0)

+ a s^2 laplace(y(t), t, s) - a y(0) = b laplace(y(t), t, s)

invlaplace(g, s, t);

${\displaystyle {\frac {d^{2}}{dt^{2}}}y(t)+a{\frac {d}{dt}}y(t)=by(t)}$

with(inttrans);

fourier(sin(x),x,w)

${\displaystyle \Pi }$*(Dirac(w-1)+Dirac(w+1))

绘制函数${\displaystyle y=x\cdot \sin x}$，${\displaystyle x\in (-10,10)}$

plot(x*sin(x),x=-10..10);

绘制函数${\displaystyle x^{2}+y^{2}}$，${\displaystyle x}$和${\displaystyle y}$的范围为 -1到1

plot3d(x^2+y^2,x=-1..1,y=-1..1);

維基教科書中的相關電子：绘制函数动画图

二维动画

${\displaystyle f:=2*k^{2}/cosh(k*(x-4*k^{2}*t))^{2}}$

with(plots);

animate(subs(k = .5, f), x = -30 .. 30, t = -10 .. 10, numpoints = 200, frames = 50, color = red, thickness = 3);

钟形孤立子

三维函数动画

三维动画

with(plots)

animate3d(cos(t*x)*sin(3*t*y), x = -Pi .. Pi, y = -Pi .. Pi, t = 1 .. 2)

求解偏微分方程组

${\displaystyle {\frac {\partial }{\partial x}}v\left(x,t\right)=-u\left(x,t\right)v\left(x,t\right)}$

${\displaystyle {\frac {\partial }{\partial t}}v\left(x,t\right)=-v\left(x,t\right){\frac {\partial }{\partial x}}u\left(x,t\right)+v\left(x,t\right)\left(u\left(x,t\right)\right)^{2}}$

${\displaystyle {\frac {\partial }{\partial t}}u\left(x,t\right)+2\,u\left(x,t\right){\frac {\partial }{\partial x}}u\left(x,t\right)-{\frac {\partial ^{2}}{\partial {x}^{2}}}u\left(x,t\right)=0}$

条件为${\displaystyle v(x,t)\neq 0}$.

eqn1:= diff(v(x, t), x) = -u(x,t)*v(x,t):
eqn2:= diff(v(x, t), t) = -v(x,t)*(diff(u(x,t), x))+v(x,t)*u(x,t)^2:
eqn3:= diff(u(x,t), t)+2*u(x,t)*(diff(u(x,t), x))-(diff(diff(u(x,t), x), x)) = 0:
pdsolve({eqn1,eqn2,eqn3,v(x,t)<>0},[u,v]): op(%);

答案： ${\displaystyle v\left(x,t\right)={e^{{\sqrt {{\it {\_c}}_{1}}}x}}{\it {\_C3}}\,{e^{{\it {\_c}}_{1}t}}{\it {\_C1}}+{\frac {{\it {\_C3}}\,{e^{{\it {\_c}}_{1}t}}{\it {\_C2}}}{e^{{\sqrt {{\it {\_c}}_{1}}}x}}},\ \ u\left(x,t\right)=-{\frac {{\sqrt {{\it {\_c}}_{1}}}\left({\it {\_C1}}\,\left({e^{{\sqrt {{\it {\_c}}_{1}}}x}}\right)^{2}-{\it {\_C2}}\right)}{{\it {\_C1}}\,\left({e^{{\sqrt {{\it {\_c}}_{1}}}x}}\right)^{2}+{\it {\_C2}}}}}$

寻找函数${\displaystyle f}$满足积分方程 ${\displaystyle f(x)-3\int _{-1}^{1}(xy+x^{2}y^{2})f(y)dy=h(x)}$.

eqn:= f(x)-3*Integrate((x*y+x^2*y^2)*f(y), y=-1..1) = h(x):
intsolve(eqn,f(x));

答案：${\displaystyle f\left(x\right)=\int _{-1}^{1}\!\left(-15\,{x}^{2}{y}^{2}-3\,xy\right)h\left(y\right){dy}+h\left(x\right)}$