Core functionality

Users can enter mathematics in traditional mathematical notation. Custom user interfaces can also be created. There is support for numeric computations, to arbitrary precision, as well as symbolic computation and visualization. Examples of symbolic computations are given below.

Maple incorporates a dynamically typed imperative-style programming language (resembling Pascal), which permits variables of lexical scope.^[3] There are also interfaces to other languages (C, C#, Fortran, Java, MATLAB, and Visual Basic), as well as to Microsoft Excel.

Maple supports MathML 2.0, which is a W3C format for representing and interpreting mathematical expressions, including their display in web pages.^[4] There is also functionality for converting expressions from traditional mathematical notation to markup suitable for the typesetting system LaTeX.

Architecture

Maple is based on a small kernel, written in C, which provides the Maple language. Most functionality is provided by libraries, which come from a variety of sources. Most of the libraries are written in the Maple language; these have viewable source code. Many numerical computations are performed by the NAG Numerical Libraries, ATLAS libraries, or GMP libraries.

Different functionality in Maple requires numerical data in different formats. Symbolic expressions are stored in memory as directed acyclic graphs. The standard interface and calculator interface are written in Java.

Integration

Find

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

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

Output:

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

Determinant

Compute the determinant of a matrix.

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

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

LinearAlgebra:-Determinant(M);

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

Series expansion

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

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

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

Solve equations numerically

The following code numerically calculates the roots of a high-order polynomial:

 f := x^53-88*x^5-3*x-5 = 0

 fsolve(f)

 -1.097486315, -.5226535640, 1.099074017

The same command can also solve systems of equations:

 f := (cos(x+y))^2 + exp(x)*y+cot(x-y)+cosh(z+x) = 0:

 g := x^5 - 8*y = 2:

 h := x+3*y-77*z=55;
                    
 fsolve( {f,g,h} );

 {x = -2.080507182, y = -5.122547821, z = -0.9408850733}

Plotting of function of single variable

Plot ${\displaystyle x\sin(x)}$ with ${\displaystyle x}$ ranging from -10 to 10:

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

Plotting of function of two variables

Plot ${\displaystyle x^{2}+y^{2}}$ with ${\displaystyle x}$ and ${\displaystyle y}$ ranging from -1 to 1:

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

Animation of functions

• Animation of function of two variables

${\displaystyle f:={\frac {2k^{2}}{\cosh ^{2}\left(xk-4k^{3}t\right)}}}$

plots:-animate(subs(k = 0.5, f), x=-30..30, t=-10..10, numpoints=200, frames=50, color=red, thickness=3);

2D bell solution

• Animation of functions of three variables

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

3D animation of function

• Fly-through animation of 3-D plots.^[28]

 M := Matrix([[400,400,200], [100,100,-400], [1,1,1]], datatype=float[8]):
 plot3d(1, x=0..2*Pi, y=0..Pi, axes=none, coords=spherical, viewpoint=[path=M]);

Maple plot3D fly-through

Laplace transform

• Laplace transform

f := (1+A*t+B*t^2)*exp(c*t);

${\displaystyle \left(1+A\,t+B\,t^{2}\right)e^{ct}}$

 inttrans:-laplace(f, t, s);

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

• inverse Laplace transform

inttrans:-invlaplace(1/(s-a), s, x);

${\displaystyle e^{ax}}$

Fourier transform

• Fourier transform

 inttrans:-fourier(sin(x), x, w)

${\displaystyle \mathrm {I} \pi \,(\mathrm {Dirac} (w+1)-\mathrm {Dirac} (w-1))}$

Integral equations

Find functions ${\displaystyle f}$ that satisfy the integral equation

${\displaystyle f(x)-3\int _{-1}^{1}(xy+x^{2}y^{2})f(y)dy=h(x)}$.

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

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