Mathematica Basics

Mathematica Basics
Mathematica notebooks for this Intro are kept online at
http://sitelic.colorado.edu/mathematica/graphs
When you are new to Mathematica, look through the Getting Started items from the Welcome
Screen; "Get a quick overview", "Learn with guided examples", "Short video tour".
1. Resources for learning Mathematica:
• Download software to your computer from http://sitelic.colorado.edu/mathematica. Be sure
to use your CU (ucdenver.edu) email address for requesting an activation code
• Watch screencasts online at http://www.wolfram.comlbroadcast/screencasts/.
• Run Mathematica and experiment with it while viewing screencasts
• Choose and explore "Documentation Center" from Mathematica's Help menu.
• Press the Fl key for quick access to documentation. This is very useful!
Visit and explore the "Learning Center" at http://www.wolfram.com/learningcenter/.
• Consulting help at CU; Bruce Fast, sitelic@colorado.edu, 303-­‐492-­‐8995.
2. Syntax, palettes -­‐ how we tell Mathematica what to do
Get acquainted with these syntax rules for Mathematica's "standard
form":
• Capitalization -­‐ all predefined functions/commands are
capitalized
• Square brackets -­‐ f[x] -­‐ enclose function parameters
• Curly brackets -­‐ {a,b,c} -­‐ are used for lists / vectors / matrices
• Parentheses -­‐ (x+1)(x-3) -­‐ are only for algebraic grouping
• Double-­‐square-­‐brackets -­‐ v[[3]] -­‐ are for indexing lists
• Double-­‐equal-­‐signs (==) in equations, e.g., eq= x+y==5
• Semicolons separate/end expressions -­‐ a=3 ; b=4 ; c=a+b
• Postfix notation is an alternative: Pi // N instead of N [Pi]
• Asterisk (*) optional for multiplication: 4Pi==4*Pi, n
Pi==n*Pi
• Dot (.) for matrix multiplication -­‐ A.B or A.v or v.w
All input can be typed from a standard keyboard, but palettes can make some input easier (and
prettier!). Palettes can be selected and opened from the Palettes menu.
3. Overview
A Mathematica notebook can include not only mathematical input and output, but also formatted
text, to make a structured document with headers and sections. Contents can be formatted (fonts,
colors, alignment, etc.) as in any word processor. Choose a Stylesheet (from the Format menu) to set
the overall formatting scheme for the notebook.
To evaluate the expressions in one cell, put the cursor anywhere inside the cell and press Shift-­‐
Return. To evaluate all the cells in a notebook, in order from top to bottom, choose "Evaluate
Notebook" from the Kernel menu.
Just press the Fl key for complete and convenient documentation for all the thousands of
Mathematica commands/functions. This includes tutorials and examples for all topics and
commands. You can copy the examples directly from the documentation and paste them (and edit)
into your own notebook!

4. Lists
Vectors, matrices, sets, arrays of data in an Excel spreadsheet -­‐-­‐ all are lists, and learning to deal with
them appropriately makes everything else easier.
• Create lists either manually (remember the curly brackets), or using Table or Range, or
reading in data from files using Import.
• Add or delete or select portions of a list using the double-­‐bracket notation, e.g.,
m[[5,2 ]], or using specialized commands like Append, Take, Part, Extract, Drop, First,
Last, Rest
• Do arithmetic with lists without using programming loops. For simple calculations just
treat lists as single entities; for more complicated functions use Map. Also be aware of basic
functions specifically defined for use with lists, e.g., Total, Tally, Accumulate, Min, Max,
Differences, Norm, Mean, Median, Commonest. And there are many functions for
matrices.
• Rearrange or restructure lists using Flatten, Transpose, Partition, Reverse, Sort, Riffle
...
• Display lists graphically using ListPlot or (for 3D data) ListPlot3D, ListContourPlot
• Display lists textually using TableForm or MatrixForm.
5. Graphics vs. Graphs, two different kinds of objects
All the commands for graphing functions or data include the word "Plot", e.g., Plot, ListPlot, Plot3D,
ContourPlot, etc. For a complete list, evaluate *Plot*. The command GraphPlot will produce a
Graphic (diagram) of a graph. On the other hand, the commands Graph and TreeGraph produce an
actual Graph object, which one can work with in different ways than just pictures of graphs.
Graph[{1->2, 2->3, 3->4, 2->4, 1->5}] produces a Graph object. By itself it gets
displayed in a standard node/edge diagram, but it can also be used in commands like VertexCount,
EdgeList, and AdjacencyMatrix.
6. Functional Programming
For those with experience with procedural programming languages like Fortran and C, it is worth
"unlearning" the reliance on loops. There is almost always a simpler and more elegant way to
accomplish anything in Mathematica, by learning a functional programming approach.
• Sums and products have their own functions, Sum and Product. Or, use the Palettes menu
to use the traditional mathematical symbols ∑ and Π . No looping involved!
• Defining your own functions is easy and practical using Mathematica's notation, and makes
subsequent programming simpler. Usually it is easiest to use pattern-­‐matching notation,
such as f[x_]:=Sin[x^2]+3, but the same can be accomplished using "pure function"
notationlike f=Function[x,Sin[x^2]+3], or even shorter, f=(Sin[#A2]+3)&
• Applying functions to lists can be done elegantly using Map (/@) and Apply (@@)
7. Saving results
Mathematica notebooks themselves can be read by anyone using the free Mathematica Player (free
download from www.wolfram.com), or any evaluated expressions can be saved to files of any
appropriate format (e.g., XLS, CSV, PDF, PNG, JPEG) using Export. (See $ExportFormats !)
Bruce Fast, January 2012