Maple 9: A Quick Reference
Prepared by:
Douglas Meade
Department of Mathematics
University of South Carolina
March 2004, updated for Maple 9
(Earlier editions for Maple 8, Maple 7, Maple 6, Maple V, Release 5, and Maple V, Release 4)
Symbols and Abbreviations
Symbol Description Example
:= assignment f := x^2/y^3;
; terminate command; display result int( x^2, x );
: terminate command; hide result int( x^2, x ):
.. specify a range or interval plot( t*exp(-2*t), t=0..3 );
{ } set delimiter (a set is an unordered list) { y, x, y };
[ ] list delimiter (lists are ordered) [ y, x, y ];
% refers to previous result (percent)
Note: Was " until Maple V, Release 5
Int( exp(x^2), x=0..1 ):
% = evalf( % );
" "
(see ?strings)
string delimiter (double quote)
Note: Changed in Maple V, Release 5 (see %)
plot( sin(10*x) + 3*sin(x), x=0..2*Pi,
title="An interesting plot" );
` `(see ?names) name delimiter (back quote) `A name` := `This is a name.`;
||
(see also ?cat)
concatenate string or name
Note: Was . prior to Maple 6
a||3;
a||(1..3);
´ ´(see ?uneval) delayed evaluation (single quote) x := ´x´;
->
(see ?-> and ?proc)
mapping (procedure) definition f := (x,y) -> x^2*sin(x-y);
f(Pi/2,0);
@ composition operator (cos@arcsin)(x);
@@ repeated composition operator (D@@2)(ln);
Mathematical Operations, Functions, and Constants
Symbol Description Example
+, -, *, /, ^ add, subtract, multiply, divide, power 3*x^(-4) + x/Pi;
sin, cos, tan,
cot, sec, csc
trigonometric functions sin( theta-Pi/5 ) - sec( theta^2 );
arcsin, arccos,
arctan, arccot,
arcsec, arccsc
inverse trigonometric functions arctan( 2*x );
exp exponential function exp( 2*x );
ln natural logarithm ln( x*y/2 );
log10 common logarithm (base 10) log10( 1000 );
abs absolute value abs( (-3)^5 );
sqrt square root sqrt( 24 );
! factorial k!;
=, <>, <, <=, >, >= equations and inequalities
Note: E no longer exists; use exp(1)
diff( y(x), x ) + x*y(x) = F(x);
exp(Pi) > Pi^exp(1);
Pi, I π, i (mathematical constants)
Note: Maple is case-sensitive
exp( Pi*I );
infinity infinity (∞) int( x^(-2), x=1..infinity );
NOTES:
• The document is also available on the World Wide Web in either PDF (http://www.math.sc.edu/ meade/maple/mapleref.pdf)
or PostScript (http://www.math.sc.edu/ meade/maple/maple-ref.ps).
• Please send comments, corrections, and suggestions for improvements to meade@math.sc.edu.
Commands
Command Description Example
restart clear all Maple definitions restart:
with load a Maple package with( DEtools ); with( plots ):
help (also ?) display Maple on-line help ?DEplot
limit calculate a limit limit( sin(a*x)/x, x=0 );
diff compute the derivative of an expression diff( a*x*exp(b*x^2)*cos(c*y), x )
int definite or indefinite integration int( sqrt(x), x=0..Pi );
Limit inert (unevaluated) form of limit Limit( sin(a*x)/x, x=0 );
Diff inert (unevaluated) form of diff Diff( a*x*exp(b*x^2)*cos(c*y), x );
Int inert (unevaluated) form of int Int( sqrt(x), x=0..Pi );
value evaluate an inert expression
(typically used with Limit, Diff, or Int)
G := Int( exp(-x^2), x );
value( G );
plot create a 2-dimensional plot plot( u^3, u=0..1, title="cubic" );
plot3d create a 3-dimensional plot plot3d(sin(x)*cos(y),x=0..4*Pi,y=0..Pi);
display combine multiple plot structures into a single
plot or modify optional settings in a plot
(in plots package)
F:=plot( exp(x), x=0..3, style=line );
G:=plot( 1/x, x=0..3, style=point );
plots[display]([F,G], title="2 curves");
solve solve equations or inequalities solve( x^4 - 5*x^2 + 6*x = 2, { x } );
fsolve solve using floating-point arithmetic fsolve( t/10 + t*exp(-2*t) = 1, t );
dsolve solve ordinary differential equations;
see ?dsolve for a list of available options
dsolve( diff(y(x),x)-y(x)=1, y(x) );
odeplot create 2D and 3D plots from solutions obtained
by dsolve (with type=numeric);
see ?odeplot for more options
(in plots package)
S:=diff(x(t),t)=-y(t),diff(y(t),t)=x(t):
IC:=x(0)=1,y(0)=1:
P:=dsolve({S,IC}, {x(t),y(t)}, numeric):
odeplot(P, [[t,x(t)],[t,y(t)]], 0..Pi);
odeplot(P, [x(t),y(t)], 0..Pi);
DEplot create plot associated with an ODE or system of
ODEs; see ?DEplot for more information
(in DEtools package)
ODE := diff( y(x),x ) = 2*x*y(x);
DEplot( ODE, [y(x)], x=-2..2,
y=-1..1, arrows=SMALL );
D differential operator
(often used when specifying derivative
initial conditions for dsolve)
ODE := diff(y(x),x$2) +y(x) = 1;
IC := y(0)=1, D(y)(0)=1;
dsolve( { ODE, IC }, y(x) );
simplify apply simplification rules to an expression simplify( exp( a+ln(b*exp(c)) ) );
factor factor a polynomial factor( (x^3-y^3)/(x^4-y^4) );
convert convert an expression to a different form convert( x^3/(x^2-1), parfrac, x );
collect collect coefficients of like powers collect( (x+1)^3*(x+2)^2, x );
rhs right-hand side of an equation rhs( y = a*x^2 + b );
lhs left-hand side of an equation lhs( y = a*x^2 + b );
numer extract the numerator of an expression numer( (x+1)^3/(x+2)^2 );
denom extract the denominator of an expression denom( (x+1)^3/(x+2)^2 );
subs substitute values into an expression subs( x=r^(1/3), 3*x*ln(x^3) );
eval evaluate an expression with specific values eval( 3*x*ln(x^3), x=r^(1/3));
evalf evaluate using floating-point arithmetic evalf( exp( Pi^2 ) );
evalc evaluate a complex-valued expression
(returns a value in the form a+I*b)
evalc( exp( alpha+I*omega ) );
evalb evaluate a Boolean expression
(returns true or false or FAIL)
evalb( evalf( exp(Pi) > Pi^exp(1) ) );
assign perform assignments
(often used after solve or dsolve)
S:=solve( {x+y=1, 2*x+y=3}, {x,y} );
assign( S ); x; y;
seq create a sequence seq( [0,i], i=-3..3 );
for . . . from . . .
to . . . by . . . in . . .
while . . . do
. . . end do
repetition statement; see ?do for syntax
(Note: od is an acceptable substitute for end do)
tot := 0;
for i from 11 by 2 while i < 100 do
tot := tot + i^2
end do;
if . . . then . . . elif
. . . else . . . end if
conditional statement; see ?if for syntax
(Note: fi is an acceptable substitute for end if)
if type(x,name) then ’f’(x)
else x+1 end if;
assume inform Maple of additional properties of objects assume( t>0 );
about check assumptions on Maple objects about( t );