# Basics of Mathematica

## Syntax

Welcome to Mathematica!

First things are first: you evaluate input by selecting a cell and pressing “Shift-Enter”. You assign values to a variable using =; the names of a variables cannot start with a number.

x = 1
x1234 = 2
2 x


You cannot change the value of predefined variables, like Pi or E.
If Mathematica currently holds a value for a variable, it shows up in black; if it is undefined, then it shows up in blue.

Mathematica tries to be exact unless you specify otherwise; if you want the decimal approximation of a number, use N[number]

{E, e, Pi, applesauce}
N[{E, e, Pi, applesauce}]


You can remove the value of a variable using Clear[x]

Clear[x, x1234]


Square brackets are used for function arguments; Parentheses are used for order of operations; Curly brackets denote lists.

func = Sin[x]
poly = (x + y)^2
list = {w, x, y, z}
list2 = {a, b, c, d}


The built-in Help in Mathematica is amazing. If you don’t know what something does or how it’s used, either press F1 or use ? with the function name:

?LegendreP


You can find more information by using ?? before the function:

?? Plot


* is a wildcard; Plot*, for example, will return every function which starts with the characters “Plot”. You can use two of them at once as well:

?*Bessel*


## List Manipulations

As we stated, curly brackets denote lists.

list = {w, x, y, z}
list2 = {a, b, c, d}


Two square brackets take only the specified element of the list:

list[]


Lists in Mathematica are much like vectors; nested lists (i.e. lists of lists) represent matrices.

Multiplication by a constant is pretty straightforward:

2 list


You can take the dot product of two lists:

list.list2


It’s sometimes useful to remove a piece of the list with Drop or select a specific part of a list with Take:

Drop[list, 1]
Drop[list, -1]
Take[list, 2]


To make two lists into one (Union also removes duplicated elements):

list3 = Union[list, list2]


To split a long list into a series of sublists:

Partition[list3, 2]


You can generate lists by hand using curly brackets, but Table does it automatically by evaluating an expression which runs over the index:

listInt = Table[i, {i, 1, 12}]
listSq = Table[i^2, {i, 1, 12}]


You can make two lists into a list of ordered pairs using Transpose:

Transpose[{listInt, listSq}]


## Replacement Rules vs Functions

list


You can temporarily set the value of a variable using the following commands:

list /. x -> 2
list /. {w -> 1, y -> 3}


(Semicolons suppress the output of a command, by the way.)

User-defined functions can take any number of arguments, but you have to specify them initially.

func2[x_] = Sin[x];
func3[a_,b_,c_] = a + b^2 + c^4;


(Note the necessary underscores!)

Evaluaion of functions is easy! Mathematica N[func2]

N[func3[2,4,10] 

If you don’t define the argument of a function, then you can still work with it using replacement rules:

func = Sin[x];

N[func /. x -> 1]


## Basic Plotting

Visualization in Mathematica is an intricate subject; we will take it up in detail later. For now, only the basics.

Continuous functions can be plotted using Plot:

Plot[Sin[x]^2, {x, 0, 2 Pi}, ImageSize -> Large]


For lists, you use ListPlot:

ListPlot[pairs, ImageSize -> Large]


## Integrals and Derivatives

Usually, functions in Mathematica are fully spelled out. Not so for derivatives! The function is just a capital D.

D[Sin[w t], t]
D[BesselJ[n, x], x]


Mathematica can do both definite and indefinite integrals:

psi[x_] = A Sin[3 x];
Integrate[psi[x]^2, {x, -a, a}]

int = Integrate[(Sin[t^2]^4 - 10 Tan[2 t])/Csc[t], t]
N[int /. t -> 0]


As we all know, some integrals don’t have exact solutions (or they are really tough to find). You can do a numerical integration using NIntegrate.

NIntegrate[x^2/(E^x - 1), {x, 0, Infinity}]


## A Bunch of Shortcuts

• To make a cell into a text cell, use Alt-7
• (There are a bunch more formatting options with Alt-number)
• To make Greek letters, type Esc-letter-Esc (e.g. Esc-o-Esc gives [Omega])
• Superscripts: Ctrl-6 (e.g. x^2)
• Subscripts: Ctrl - (Control-dash, e.g. Subscript[x, 2])
• To put comments into input cells, use (* ... *)
• (You can also highlight the comment and type Alt-/ )
• To abort the evaluation of a cell, use Alt-. (use when things get stuck!)