Mathematica

Table of Contents

Bracketing

Parenthesis ()
change the priorities in computing.
Square bracket []
function parameters, e.g. f[x].
Curly brace {}
list, e.g. {a, b, c}.
Double bracket [[]]
abbreviation of Part, e.g. v[[i]].

Constants

Command Value
Pi \(\pi\)
E \(e\)
I \(i = \sqrt{-1}\)
Infinity \(\infty\)

Data objects

Command Meaning
2 + 8 I complex
"text" string

Defining variables, functions, and rules

Command Meaning
x = value assignment
x := value delayed assignment
x = . or Clear[x] remove value assigned to x
f[x_] := expression define a function f(x)
expression /. x->a replace x by a in expression
expression //. x->a replace repeatedly
lhs :> rhs /; test apply rule if test is True

Linear algebra

Command Meaning
{a, b, c} vector, list
{{a, b}, {c, d}} 2 × 2 matrix
n.m matrix multiply
Inverse[m] inverse of matrix
Transpose[m] transpose
Det[m] determinant
MatrixPower[m, n] m^n
MatrixExp[m] e^m
LinearSolve[m, b] solve mx = b
Eigenvalues[m] eigenvalues
Eigenvectors[m] eigenvectors
Eigensystem[m] eigenvalues and eigenvectors

Algebraic calculation

Command Meaning
Solve[lhs==rhs, x] solve algebraic equation for x
DSolve[eqn, y[x], x] solve differential equation
Reduce[eqn, x] reduce equations
Eliminate[eqn, x] eliminate variable x

Calculus

Command Meaning
D[f, x] \(\partial f/\partial x\)
D[f, {x,n} \(\partial^n f/\partial x^n\)
Dt[f] \(df\)
Integrate[f, x] \(\int fdx\)
Integrate[f, {x, a, b}] \(\int_a^b f dx\)
Sum[f, {i, m, n}] \(\sum_{i=m}^n f\)
Product[f, {i, m, n}] \(\prod_{i=m}^n f\)
Limit[f, x->a] \(\lim_{x\to a} f\)

Input and output

Command Meaning
<<file read expressions from file, return last expression.
expression>>file write expression to file.
expression>>>file append expression to file.
!!file display the content of file.
Save["file", x] save the definition of x to file.
!command issue a UNIX command.

Expression in different formats

Command Meaning
FullForm[e] full form
InputForm[e] input
OutputForm[e] out
CForm[e] C codes
FortranForm fortran
MatrixForm[e] matrix
StandardForm[e] math
TeXForm[e] TEX

Programming

Table[expression, {i, max}]
make a list of values of expression with i from 1 to max.
Module[{a, b, c}, expression1; expression2;…]
a procedure with local variables a, b, c return value of last expression.
Do[expression, {i, min, max, di}]
evaluate expression with i run from min to max in steps of di.
While[test, body]
evaluate body repeatedly, so long as test is True.
For[star,test,inc,body]
evaluate start, then repeatedly evaluate body and inc, until test fails.
If[test, then, else]
evaluate then if test is True, and else if it is False.
Which[test1,value1,test2,value2,…]
give the value associated with the first test that is True.
Switch[expression, form1, value1, form2, value2, …]
give the value associated with first form matching expression.
Function[x, body]
specify a pure function.
Nest[f, x, n]
apply the function f nested n times to x.
Apply[f, {a, b, c}]
f(a, b, c).
Map[f, {a, b, c}]
apply f to each elements, {f(a), f(b), f(c)}.