Mathematica

Objective

Basic Commands
Solving Equations
Differential and Integral
Polynomial Operations
Graphics
Table and Array
Simple programming

Introduction

Mathematica is a powerful package for doing mathematics.
It can runs on Windows, Mac, and Linux. It can be used to:

Perform computations, either numeric or symbolic(most remarkable). Result can be visualized.
Programming tool.
Publishing tool(formatting graphics, test, and equations, and export to PS, TeX, HTML ...)

First 5 minutes with mathematica

Mathematica help is a very useful document.
To see help, type ?+command OR PRESS F1.
Usual usage:

[in#]:INPUT
RUN shift + return
[out#]:OUTPUT

Some convention

RUN	shift + return
* or SPACE	multiplication
^	power
*^	power of 10
%	gives the last result generated
%%	gives the result before last
@@	apply an operation on expression
Pi	the value of Pi:3.14159...
()	parentheses for grouping
[]	square brackets for functions
{}	curly braces for lists
[[]]	double brackets for indexing equivalent to Part[]
N[]	evaluate numerically
Sqrt[]	calculate square root
Sin[]/Cos[]	calculate sine/cosine
Log[]/Exp[]	calculate logarithm/power of natural base

Solving Equations

FindRoot[f, {x,x_0}]
searches for a numerical root of f, starting from the point x_0.
- Example:FindRoot[Exp[x]-3x == 0, {x,3}]

NSolve[expr, vars]
attempts to find numerical approximations to the solutions of the system expr of equations or inequalities for the variables vars.
- Example:NSolve[{x^2 + y^2 == 1, 2 x + 3 y == 4}, {x, y}]]

NDSolve[eqns, y, {x, x_min, x_max}]
finds a numerical solution to the ordinary differential equations eqns for the function y with the independent variable x in the range x_min to x_max.
- Example:NDSolve[{y'[x] == y[x] Cos[x + y[x]], y[0] == 1}, y, {x, 0, 30}]
- Plot[Evaluate[y[x] /. %], {x, 0, 30}, PlotRange -> All]

Differential and Integral

D[f, x]
gives the (partial) derivative df/dx.
D[f,{x,n}]
gives the n-th (partial) derivative of x.
- Example: D[Sin[x]^10, {x,4}]

Integrate[f, x]
gives the indefinite integral.
- Example: Integrate[1/(x^3 + 1),x]

Integrate[f, {x, x_min, x_max}]
gives the definite integral from x_min to x_max.
- Example: Integrate[1/(x^3+1), {x,0,1}]

Polynomial operations

Factor[poly]
factors a polynomial over integers.
- Example: Factor[x^2+3x y+2y^2]

Expand[expr]
expand out products and positive integer powers in expr.
- Example: Expand[%]

Together[expr]
puts terms in a sum over a common denominator, and cancels factors in the result.
- Example: Together[a/b + c/d]

Apart[expr]
rewrites a rational expression as a sum of terms with minimal denominators.
- Example: Apart[1/((1+x)(5+x))]

Solve[expr, vars]
attempts to solve the system of equations expr from variables vars
- Example: Solve[{x (x+y) == c, y-2 == 0}, {x, y}]

Series[f, {x, x_0, n}]
generate a power series of f at point x_0 to an error of order of n
- Example: Series[Exp[x], {x, 0, 10}]

Normal[expr]
convert expr to normal expression, from a variety of special forms
- Example: Normal[%]

Simplify[expr]
tries to find the simplist form of expr through a sequece of transformations.
- Example: D[Integrate[1/(x^3+1),x],x]
- Simplify[%]

Collect[expr, x]
collect together terms involving the same powers of x.
- Example: Collect[(1+a+x)^4, x]

Graphics

Plot[f, {x, x_min, x_max}]
- Example: Plot[Sin[x],{x,0,6 Pi}]

Plot3D[f, {x, x_min, x_max}, {y, y_min, y_max}]
- Example: Plot3D[Sin[x + y^2], {x, -3, 3}, {y, -2, 2}]

ContourPlot[f, {x, x_min, x_max},{y, y_min, y_max}]
- Example: ContourPlot[Cos[x] + Cos[y], {x, 0, 4 Pi}, {y, 0, 4 Pi}]

LogPlot[f, {x, x_min, x_max}]
- Example: LogPlot[x^x,{x,1,5}]

BarChart[{y_1, y_2,...}]
- Example: BarChart[{{1,2,3},{1,3,2},{5,2}}, ChartLabels->{"a","b","c"}]

ParametricPlot[{f_x,f_y}, {u, u_min, u_max}]
- Example:g1= ParametricPlot[{Sin[x],Sin[2x]},{x,0,2 Pi}]

ListPlot[{{x_1, y_1}, {x_2, y_2}, ...}]
- Example:g2= ListPlot[Table[{Sin[n],Sin[2n]}, {n, 50}]]

Show[g_1, g_2, ...]
- Example: Show[g1, g2]

Table and Array

Table[expr,{i, i_min, i_max}, {j, j_min, j_max}]
generate a list with i from i_min to i_max and j from j_min to j_max.
- Example: Table[10 i + j, {i, 3}, {j, 3}]
- Table[z^k,{k,0,10}]
- Table[Sin[i x] + Cos[j y], {i, 4}, {j, 4}]

Array[f,n]
generate a list of length n with elements f[i].
- Example: Array[f, 10]
- Array[Exp[#1] + Log[#2] &, {5, 5}]

element of array: [[]]
- Example: a[[1]], a[[2]], a[[{1,2}]]

First[expr], Last[expr], Length[expr]

MatrixForm[list]
print the list in a matrix form.

Transpose[list], Inverse[m], IdentityMatrix[n], EigenValue[m]

Simple Programming

Remove["Global`*"]

f[x_,y_]:=x^y;

var[x_]:=Module[

{m,n},

n=Length[x];

m=(Plus@@x)/n;

(Plus@@((x-m)^2))/(n-1)

];

f[x0_] := Module[

{x = x0},

While[x > 0, x = Log[x]];

]