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This is my Mathematica cheat sheet. I keep in it useful things I learn about Mathematica and keep forgetting, and some things I see on the net. If something here is from the net, I try to make sure I put a reference or credit to where I saw it, else it will be something I wrote myself, in that case, all blames to me.

This was written using Latex and converted to HTML using tex4ht and to pdf using pdflatex.

1 Mathematica links?

Few are here

1. Archive of Mathematica weekly newsletters
2. Mathematica stackexchange forum
3. The large links page for Mathematica at stackexchnage
4. Language reference
5. What is new in V 10
6. Summary of New Features in 10
7. wolfram cloud
8. How to solve basic engineering and mathematics problems using Mathematica and Matlab

2 Mathematica old signal processing package

Here is a link to information about old Mathematica signal processing package. Currently as of Mathematica version 9, it contains number of build-in DSP functions.

3 Reading Mathematica commands source code?

Say you want to read source code of LaplaceTransform. First load it with some call then apply ?? on it

4 How do pure functions work?

I made this simple diagram to help me understand pure functions.

5 Mathematica directories after installation

I made this diagram to show the installation tree structure.

6 How to write a package?

Small note here

7 How to use context and packages?

8 How to load an m file or a package?

Append to the Path the folder name where the package is located in. In this example, assuming there is a package control.m located in folder C:\data then type the following to load the package

9 Finding names of functions in a package

10 Finding a package that has specific function

use Context.

11 Finding all contexts that belong to a package?

Use Contexts["packageName*"]

12 How to remove packages?

to do

13 Finding partial fraction expansion

?Apart

14 Some keyboard shortcuts

15 Tracing a function

Trace[Integrate[x, {x, 1, 2}], TraceInternal -> True]

16 Removing Big O notation from Series expansion

17 How to plot circle?

18 Solve implicit differentiation of equations

Suppose we are given \(z=x e^{-y}, x=\cosh (t), y=\cos (s)\) and need to find \(\frac{dz}{ds}\)

Another example: \(u=x^2 y^3 z, x=\sin (s+t), y=\cos (s+t), z=e^{s t}\) and we need to find \(\frac{du}{ds}\) and \(\frac{du}{dt}\)

19 Drawing roots of complex equation?

20 DSolveIntegrals package

Where did I get this from?

”Mathematica can handle partial differential equations via the DSolveIntegrals package. These arise in chemical contexts in the 1D wave equation, 3D wave equation, 3D diffusion equation, Time-dependent and Time independent Schrödinger equation. Hermite showed that the quintic equation could be solved by elliptic functions”

21 How to simplify with conditions?

For example, to integrate this below, for n positive integer we do

22 Smart replacement everywhere?

23 Plotting real and imaginary parts

24 Plotting mapping of complex numbers?

25 Saving a plot as eps

26 Extracting DSolve solutions

one way

27 Mathematica not evaluate its arguments?

28 Combining more than plot

One way is to use Show

29 Speed of functional and procedural

By Bill Rowe http://forums.wolfram.com/mathgroup/archive/2004/Apr/msg00357.html

30 Using subscript variables in function definitions

Use notation package

31 Using zero as index

See using_zero_index_in_Mathematica

32 Extracting the LHS and RHS of equation

33 Making animated GIF of a manipulate

One way is to use Vitaliy Kaurov ManToGif. Another way is to run the manipulate and do screen capture using program such as LICEcap

34 How to do convolution?

see ?ListConvolve

35 How to use Piecewise function?

see ?Piecewise

36 Making labeles for frame plot

from the net. Using Times font family is the idea.

37 Doing some matrix operations

Rememebr: Position and Cases return result that can be used by Extract directly. But can’t be used by Part directly.

37.1 How to extract first column in matrix

37.2 How to extract first 3 rows in the first column?

37.3 How to find some matrix rows based on some condition on value in say the first column??

Find rows which has elements in first column less than 3 in the following

Reference: how-to-extract-rows-from-matrix-based-on-value-in-first-entry

The solution using pattern below (by WRech) is interesting since the same pattern can be used by Cases and Position.

solution by me

by Simon

By Asim

By WReach

By WReach

37.4 How to generate a diagonal matrix?

Random values on the diagonal

Ones on the diagonal

37.5 How to generate upper diagonal matrix?

one way, using SparseArray

Or using Table. But notice that in SparseArray, the ’zeros’ are already the default case, so using SparseArray is simpler.

37.6 How to find the trace of a matrix?

see ?Tr[a]

37.7 How to find product of elements on the Trace?

37.8 How to check if a Matrix is diagonal matrix?

by Jon MacLoone

By Paul Abbott

37.9 How to find locations of all zeros (or any other value) in a matrix?

Find location of zeros in this matrix

one way

Another way

37.10 How to find locations of elements subject to some test?

find all elements between 4 and 8

37.11 How to insert an element in specific position?

Using Part to inser 99 in position (1,1)

37.12 How to insert a row into a matrix?

To insert this row in the second row in matrix above

or just use ’2’, it will also work

37.13 How to insert a column into a matrix?

To insert this column in the second column position in above matrix

one way

another way

Another way by Leonid Shifrin how-to-insert-a-column-into-a-matrix-the-correct-mathematica-way

Another by Leonid Shifrin

37.14 How to build a large matrix from blocks of smaller matrices?

Given

and we want to make matrix { {a,a},{a,a} }

37.15 How to apply a function to each element in a 2D matrix?

Given

and we want to apply the this function to it

Then using Map

38 How to find if an expression implies another?

39 Displaying matrices in MatrixForm

One way

Another otpion is to use TraditionalForm. Can change default form from the menu, so this way no need to change $PrePrint

40 Making 3D axes normal instead of boxed

Use Boxed -> False

41 How to use ListPlot on set of \(x,y\) data?

One way to use Transpose

42 How to do autocorrelation

In Matlab it is

43 How to make a spring?

From the net, lost reference

See also how-to-draw-a-spring

44 How to list files in a directory?

45 Where is init.m and how to use it?

Possible locations for init.m files include the following:

1. $BaseDirectory/Kernel kernel initialization code for all users
2. $UserBaseDirectory/Kernel kernel initialization code for the currently logged-in user
3. $BaseDirectory/FrontEnd front end initialization code for all users
4. $UserBaseDirectory/FrontEnd front end initialization code for the currently logged-in user
5. I have my init.m in the following folder

   C:\Documents and Settings\All Users\Application Data\Mathematica\Kernel\init.m

46 Making plot range a slider

47 On Mathematica accuracy and precision

47.1 From class notes: UC Davis, Dr Rocke

some notes below

Precision means the variability between estimates Accuracy means the amount of deviation between the estimate and the ”true value”

The condition number is the ratio of the output error to the input error. if the condition number is about 10k, then one loses about k digits of accuracy.

The main sources of inaccuracy (= error) is truncation error and round-off error.

From the above dart diagram, then we can say this: a value is accurate if it is near the bull-eye. But if is away from the bull-eye, but it is always away from the bull-eye and in the same place, then it is precise. So something can be precise but not accurate. So precise has to do with repeated values. i.e. one can’t say a value is precise, but must talk about an experiment being precise, it is produced same result each time (or very close results each time).

So, it is best of course to be both accurate and precise. So what does it mean to be accurate but not precise? using the above dart diagram, it means values generated from the experiment are always close to the pull eye, but not in the same locations.

47.2 by Andrzej Kozlowski

From http://forums.wolfram.com/mathgroup/archive/2010/Jan/msg00917.html

48 Guidelines when writing Mathematica demonstration

see also http://demonstrations.wolfram.com/guidelines.html

I made small copy here so I do not have to keep looking for this all the time.

To put label on plot, example

To typeset math for display on the demo use this type of coding

To add invisible space use ESC is ESC

49 Making condition as pattern

By Bob Hanlon from math group:

50 Reading Mathematica example data and location

on windows, V 8, example data is located in

and it can be read like this

51 Stopping 3D plot from changing size

Use SphericalRegion->True

52 Selecting elements from a list that satisfy a condition

This question was posted on the net. Given

select from it elements \(x,y\) which satsify \(x+y>9\)

some answers

Adriano Pascoletti answer

Bill Row answer

Murray Eisenberg answer

53 Selecting and replace elements from matrix

Given a matrix, say which has Indeterminate and we want to change all these entries in the matrix by zero.

another example, given a matrix of values, replace those values which are less than \(0.5\) by NULL

54 What are the AppearanceElements names?

see full-documentation-for-appearanceelements

55 How to make listplot

one way

56 Getting points from plot once it is plotted

and

57 Notes on dynamics

Useful notes taken from different places from Mathematica documentation. And some added by me.

1. (added 7/5/14) There is a race condition between when updating a variable in the second argument of dynamics, which is also updated in the Manipulate expression. The problem is easy to explain
   Manipulate[.... n=n+1;...., Manipulator[Dynamic[f,{f=#;n=0}&....]

   Now, what happens, sometimes, is that when moving the slider, and setting \(n=0\), that this update to \(n\) can be lost, since it depends on the timing of when this happens. The Manipulate expression could be in the middle on updating \(n\) itself. This is classical lost update problem in multi-threading. The way to avoid this, is to follow this rule of thumb: When using second argument of dynamics in a Manipulate control, do not update a shared variable which can also be updated inside the Manipulate expression, as the update can be lost. The write to the shared variable/updating should only happen either inside the Manipulate expression or in the second argument of dynamics. But not in both places

2. Dynamic is wrapped around the whole expression, so evaluation of the Table command is delayed until the output is displayed in the notebook. Any time the value of x is changed, the Table command will be reevaluated.
3. Remember that Dynamic has the effect of delaying evaluation until the expression reaches the front end
4. Because it has the attribute HoldFirst, Dynamic does not evaluate its first argument. This is fundamental to the workings of Dynamic, but it can lead to a somewhat unexpected behavior
5. Ordinary variables in Mathematica are owned by the kernel. Their values reside in the kernel, and when you ask Mathematica to display the value in the front end, a transaction is initiated with the kernel to retrieve the value.
6. Variables declared with DynamicModule, on the other hand, are owned by the front end. Their values reside in the front end, and when the front end needs a value, it can be retrieved locally with very little overhead.
7. The most important is the fact that values of all DynamicModule variables are saved in the file when the notebook is saved.
8. By default, dynamic outputs triggered by changes in variable values are updated no faster than twenty times per second (this rate can be changed with the SystemOption  "DynamicUpdateInterval").
9. Dynamic outputs are only updated when they are visible on screen.
10. Remember to add synchorization->False to all dynamics, else will time out. When using Refresh also.
11. Never make a refresh[] tracks on 2 of my own symbols (not control variables). Use tick, only. Causes major synchronization problems with I update 2 variables inside a refresh, and have the refresh tracks both. Only make one track local variable, such as ticks
12. Ok, Found out that finishDynamic[] can causes annoying refresh on the UI whenever I move sliders. So removed it.
13. Remember to use :> and not -> for TrackedSymbols

57.1 Notes from John Fultz collected on dynamics

By John Fultz on math group, jan 24/2012

”Generally, you should construct controls so that they’re not inside Dynamics that will trigger while you’re interacting with those controls, since this can create instability”

By John Fultz on math group, feb 3/2012

”CDF files which you expect to deploy cannot rely on Shift+Enter evaluations to prime the pump. You need to make sure that all of the code dependencies are in the dynamic evaluations somewhere. Some possible ways of doing this, all of which have been discussed at various points on MathGroup, include:

* Using the Initialization option of Dynamic, DynamicModule, or Manipulate

* Using the SaveDefinitions option of Manipulate

* Adding code to the NotebookDynamicExpression option of the notebook (if it’s initialization code, then wrapped in Refresh[#,None]& to prevent it from evaluating more than once per session).

* Not such a good idea for function definitions, but if you simply have code that needs to run before Dynamic code runs, nesting a DynamicWrapper around it might be appropriate, too.”

Notes from WRI tech support

see how-to-define-constants-for-use-with-with-in-one-place-and-then-apply-them-lat

Some useful posts and links on dynamics why-wont-this-work-dynamic-in-a-select

When DynamicModule is first evaluated, initial assignments for local variables are made during the evaluation. Any setting for the Initialization option is evaluated only when the output of DynamicModule is displayed.

see how-to-make-dynamicmodule-work-without-an-extra-enter

Here is a trick to allow one to control one slider based on another

58 Making struct/record and array of structs

There is no build-in struct or record in Mathematica. But this is what I do. Since in M a matrix can include non-numeric data, I use a list for a record, and then use a matrix to make an array of records (or array of structs). I just need to make a constant to indicate the field name in the record, to make it easier to reference. Here is an example

Ok, not very fancy, but easy to setup and data works with all M other functions, since it is just a list of lists.

Some more links on the subject

1. struct-data-type-in-mathematica
2. setting-up-a-struct-in-mathematica-safely
3. using-a-struct-inside-manipulate-to-help-manage-control-variables-how-to- initia
4. struct-equivalent-in-mathematica

59 Applying a function using 2 arguments from a list

Or

60 Using Sow and Reap

61 Making comments use monospaced fonts

Thanks to Alexey Popkov for this

62 How to do long division of 2 polynomials?

This came about when I was trying to convert 1/(1-x^2/2) to normal form, i.e. tell Mathematica to change the above to 1+x^2/2 But doing Simplify[1/(1-x^2/2)] or Expand does not work. So the only solution I found is to use Series command, as follows

63 Common patterns for function parameters

Use these in parameter ”declaration” of functions to make more robust. From the help

64 How to add rational polynomials?

65 How to use options in functions?

This below is also a useful post by David Park on the net on options usage in packages

msg00335.html

66 How to replace patterns in expressions

66.1 example 1

by Andrzej Kozlowski on math group, July 2010:

Suppose in the expression 2/3 I + x/y I you wish to replace all fractions (that is 2/3 and x/y) by r and I by d. Without worrying about evaluation you can do this as follows:

If you allow the expression to evaluate the patterns will no longer match. For example, with only one Unevaluated you will get

66.2 example 2

question: I want to replace y for x everywhere except in Exp[x].

Answer by Bob Hanlon on the net. messageID=7120881&tstart=0

67 Finding which folders are on trusted path

Thanks to Mike for these commands, see http: //stackoverflow.com/questions/8583521/why-do-i-get-security-warning-message-this-file-contains-potentially-unsafe-dyn

Now to find if your current notebook is on the trusted path type NotebookDirectory[] and see if the output shows up in the trusted path of not. To add a folder to trusted path go to ”Preferences ¿ Advanced ¿ Open Options Inspector”. Then under Global Preferences search for trusted

68 Difference between Block, With and Module

Block and Module have values, the last expression evaluated is their value, we can see this by making a Grid (or just printing). But module leaked symbols have $ signs

Modules and Blocks both execute if they are in the path of code, without calling them. Block:

Module:

69 Generating T.O.C. to sections in a notebook

These are the steps I use to make TOC which is just HTML links to internal tags in the notebook, where these cell tags are sections. This way, when I exprt to HTML, I end up with TOC which is hyperlinks to internal locations within the web page.

1. create the section as normal. As in right-click the mouse, and select INSERT new cell, and select Section. Now in the new cell, write the section title.
2. Copy, using the mouse the title of the Section you just wrote so that the title is in the buffer. Now go to Cell->Cell tags->Add/remove and in the little window, paste the title of the section there and click Add.
3. Go to the top of the document itself, where the TOC is located, and also PASTE the name of the section there. It will be plain text now.
4. Now, using the mouse again, select the text you just pasted, and do right-click and select MAKE hyperlink. This will bring up a menu like this

   

5. Select the option Current notebook from above, and this will bring up a list of all cell tags below. Scroll down looking for the same title there and click on it. This will make the TOC entry HTML link.
6. Now do SAVE AS HTML, and the notebook will be saved as HTML and the TOC will be links to the sections

70 Extracting values in a list of the form x-¿value

one way

another way

71 Aligning inside Framed environment

One way us to use Item

72 Aligning individual row of a grids

One way us to use Item

73 Printing numerical value not in scientific notation?

Use NumberForm

74 Clearing notebook cache from any old symbols

Sometimes I get the case that the notebook retain old definitions and symbols even after I deleted them from the notebook. This happened when I was using a Demonstration stylesheet and had an separate initilization cell, and had added SaveDefinitions->True in the Manipulate cell.

To make sure the notebook clears any old symbols, enter this command in the notebook once

In addition, I change the preferences like this:

75 Difference between RotationMatrix and RotationTransform

The call for each is as follows

76 How to change the head of a list?

A list has a Head at its zero index position

By changing the head we use Apply. For example, to add the numbers of the above lst, we need to change the Head from List to Plus. There is a command in Mathematica to change the Head, called Apply

We could have used the zero index trick, but it is better to use Apply:

If we have a list of lists, like this

And we wanted to change the head of each list in it, for example, we want the product of each list shown, then we need to change the head of each list to Times. To do that, the follwing short but might be strange looking command

Another way to do the above is to Map the Apply function

or, little shorter version of the above:

77 Displaying polynomial from higher to lower order

Mathematica default display of polynomial is reverse the traditional form:

use Traditional Form with ParameterVariables to make it appear as in text books

78 How to understand symbol shadwing?

See this article by David Wagner http://www.mathematica-journal.com/issue/v6i2/columns/wagner/wagner62.pdf

And why-are-some-function-names-red

79 How to do OO in Mathematica?

Some note here

80 How to sort a list of numbers?

Sort[] sorts numbers from small to large by default. By supplying a function, one can change this as needed

81 Copy/paste code to Stackexchange or email messages?

82 Making automatic subscripted matrix notation

83 Finding Names of build-in function context and options

To find say all names in NDSolve and options used by that name if any do (this example is for NDSolve)

then call it with

getList["NDSolve"]

It will produce large table. Here is part of it

getList["FindMinimum"]

84 Making escape key add around elements

if one types in 1,2,3,4 is there is a way to select these and have {} automatically put around them to make a list {1,2,3,4} using escape key shortcut?

Answer by Chris Degnen who wrote this

put it in the init file to load at start-up. See wrap-text-selection-in-brackets-in-mathematica

85 Searching for substring inside a larger string

I needed to do this when I was parsing some output. The problem is like this: given a string say "foo[] boo[] more goo[] more" and wanted to look for pattern like this "__["

In other words, a letter or more that end with "[", but needed to find the first one. Hence in the above, I wanted to find "foo".

2 ways to do this:

and

86 Dynamically change the layout of Manipulate

87 What are most common commands?

87.1 Mape or /@

Takes function and applies it to each element in a list

87.2 Thread

Use when function needs to be called with arguments taken from more than one list, else use Map if argument come from one list

87.3 MapThread

In this case gives the same answer as using Thread

This is only when the lists are one level. For 2 levels we have to use MapThread. This shows the difference

88 common signature definitions

see tutorial/PatternsOverview this below from tutorial/PuttingConstraintsOnPatterns

See also what-is-the-recommended-way-to-check-that-a-list-is-a-list-of-numbers-in-argumen

88.1 some signatures collection

88.1.1 integer

88.1.2 integer strictly positive

88.1.3 integer strictly negative

88.1.4 integer zero or positive

88.1.5 integer zero or negative

88.1.6 integer in some range

88.1.7 Real

88.1.8 Real strictly positive

88.1.9 Real strictly negative

88.1.10 Real zero or positive

88.1.11 Real zero or negative

88.1.12 Real in some range

88.1.13 Boolean

88.1.14 any numerical parameter

88.1.15 checks for Head Real, Integer, Ratioal and Complex

88.1.16 general complex number

88.1.17 list of any dimension, ragged lists, 1D vectors, 2D, any content

88.1.18 1D list (i.e. vector)

88.1.19 Numeric 1D list

88.1.20 Numeric 1D list

88.1.21 2D matix of numbers

88.1.22 2D matrix numeric but contains no complex numbers

88.1.23 2D matrix of strings

89 How to check for Head of expression?

use MatchQ

90 How to make different Grids

91 Common Patterns

From help

91.1 string matching in list

See select-and-blank

91.2 how to find if one symbolic term starts with minus sign or not?

See given-a-symbolic-expression-how-to-find-if-starts-with-a-minus-or-not

92 Compare Manipulate to DynamicModule

See convert_manipulate_to_dynamicModule

93 Replacing dependent variables and its derivatives in an equation

Suppose we have \(u''(t)+u'(t)+u(t)=3 \cos (2 t)\) and we wanted to find a particula solution by replacing \(u\) in the differential equation by some guess for a particular solution. Then do

94 things to remember

Watch out for adding the extra third argument to trigger as show below (which is 1 now). This seems to cause a problem. Was using it in Manipulate and when I added, sometimes the trigger stops firing on its own. When I remove it, it never does stop.

95 Copying outout cells to another notebook

see how-to-select-and-delete-all-output-cells

96 Mathematica equivalent command to Matlab blkdiag

To make a matrix, which contains on its diagonal matrices, Matlab uses the command blkdiag. In Mathematica use the following

97 floating points stuff

Below is from ”accuracy and stability of numerical algorithms”, by Highma, page 36

98 Mathematica functions that does the same thing