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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

LaplaceTransform[x,x,t]; 
ClearAttributes[LaplaceTransform,ReadProtected] 
??LaplaceTransform
 

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?

{"JLink`", "GetFEKernelInit`", "ResourceLocator`", 
"PacletManager`", "QuantityUnits`", "WebServices`", 
 "System`", "Global`"}

{"PacletManager`", "QuantityUnits`", "WebServices`", 
"System`", "Global`"}

 
{"System`ComplexExpand`AbsExpr", 
"System`ComplexExpand`ArgExpr", 
"System`ComplexExpand`ConjugateExpr", 
"System`ComplexExpand`ReImExpr", 
"System`ComplexExpand`ReImFail", 
"System`ComplexExpand`SignExpr"} 

Contexts[] gives a list of all contexts. 
Contexts["string"] gives a list of the contexts 
                   which match the string.

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

AppendTo[$Path, "C:\\data"] 
<< control.m
 

9 Finding names of functions in a package

{"JLink`", "GetFEKernelInit`", "ResourceLocator`", "PacletManager`", 
"QuantityUnits`", "WebServices`", "System`", "Global`"}

{"JLink`AddPeriodical", "JLink`AddToClassPath",...

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

Normal[Series[f[x], {x, 0, 3}]]
 

f[0]+x (f^')[0]+1/2 x^2 (f^'')[0]+1/6 x^3 (f^(3))[0]

17 How to plot circle?

ParametricPlot[{Sin[u], Cos[u]}, {u, 0, 2 Pi}, AspectRatio -> 1]
 

18 Solve implicit differentiation of equations

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

x[t_] := Cosh[t] 
y[s_] := Cos[s] 
z[x_, y_] := x[t] Exp[-y[s]] 
D[z[x, y], s]
 

(Cosh[t]*Sin[s])/E^Cos[s]

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 \(\frac {du}{dt}\)

x[s_, t_] := Sin[s + t]; 
y[s_, t_] := Cos[s + t]; 
z[s_, t_] := Exp[s t]; 
u[s_, t_] := x[s, t]^2 y[s, t]^3 z[s, t]; 
Clear[s, t]; 
D[u[s, t], s]
 

2*E^(s*t)*Cos[s + t]^4*Sin[s + t] + 
  E^(s*t)*t*Cos[s + t]^3*Sin[s + t]^2 - 
  3*E^(s*t)*Cos[s + t]^2*Sin[s + t]^3

19 Drawing roots of complex equation?

rootsPlot[poly_, z_] := ListPlot[{Re[z], Im[z]} 
  /. NSolve[poly == 0, z], 
  PlotStyle -> Directive[PointSize[0.015]]]; 
 
rootsPlot[z^7 - 1, z]
 

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

Integrate[Sin[n x]^2, {x, 0, Pi}]
 

Assuming[Element[n, Integers] && n > 0, 
   Integrate[Sin[n x]^2, {x, 0, Pi}]]
 

22 Smart replacement everywhere?

eq = f == x^3 + 6/x^3; 
Reduce[{eq, x^3 == z}]
 

(x == z^(1/3) || x == (-(-1)^(1/3))*z^(1/3) || 
  x == (-1)^(2/3)*z^(1/3)) && 
   z != 0 && f == (6 + z^2)/z

23 Plotting real and imaginary parts

f[t_] := BesselK[3, I t] 
Plot[{Re[f[t]], Im[f[t]]}, {t, 0.01, 20}]
 

24 Plotting mapping of complex numbers?

f[z_] := z^2 
ParametricPlot[Through[{Re, Im}[f[x + I y]]], 
  {x, -2, 2}, {y, -1, 1}, PlotStyle -> Red]
 

25 Saving a plot as eps

Plot3D[Sin[x y], {x, -3, 3}, {y, -3, 3}] 
Export["plot.eps", %]
 

26 Extracting DSolve solutions

one way

Clear[y, x] 
DSolve[{y'[x]^2 == 1 - y[x]^2, y[0] == 0}, y[x], x]
 

{{y[x] -> -Sin[x]}, {y[x] -> Sin[x]}}

27 Mathematica not evaluate its arguments?

In[60]:= s = HoldForm[1 + 2] 
Out[60]= HoldForm[1 + 2] 
In[61]:= ReleaseHold[s] 
Out[61]= 3
 

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

In[1]:= 
Timing[For[sum = 0; n = 1,  n < 100001, n++, sum += n]] 
 
Out[1]= 
{1.2999999999999998*Second,  Null} 
 
In[2]:= 
Timing[Plus @@ Range[10000]] 
 
Out[2]= 
{0.009999999999999787*Second, 50005000}
 

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

In[62]:= eq = x^2 + Sin[4*a] == 3 - Derivative[1][y][t] 
Out[62]= x^2 + Sin[4*a] == 3 - Derivative[1][y][t] 
 
In[63]:= lhs = eq /. (lhs_) == (rhs_) -> lhs 
Out[63]= x^2 + Sin[4*a] 
 
In[64]:= rhs = eq /. (lhs_) == (rhs_) -> rhs 
Out[64]= 3 - Derivative[1][y][t]
 

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.

data = Table[{x, Random[Real, {0, x}]}, {x, 0, 10}]; 
ListPlot[data, Frame -> True, 
   PlotStyle -> {FontFamily -> "Times"}, 
   PlotLabel -> "64Cycles in FIFO", 
   FrameLabel -> {"S/N dB", "RMS error mm"}]
 

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

a = Table[RandomInteger[100], {4}, {4}]
 

Out[69]= {{55, 63, 78, 45}, 
          {13, 45, 67, 1}, 
          {94, 32, 48, 90}, 
          {31, 75, 43, 60}}

a[[All,1]]

Out[70]= {55, 
          13, 
          94, 
          31}

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

a[[1 ;; 3,1]]

Out[71]= {55, 13, 94}

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

a = {{1, 2, 3}, {4, 5, 8}, {7, 8, 9}}

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

pos = Position[a[[All,1]], _?(#1 <= 4 & )] 
Out[73]= {{1}, {2}} 
 
Extract[a, pos] 
Out[74]= {{1, 2, 3}, {4, 5, 8}}

by Simon

pos = Position[a, _List?(First[#1] <= 4 & ), {1}] 
Out[75]= {{1}, {2}} 
 
Extract[a, pos] 
Out[76]= {{1, 2, 3}, {4, 5, 8}}

By Asim

Pick[a, a[[All,1]], _?(#1 <= 4 & )] 
Out[77]= {{1, 2, 3}, {4, 5, 8}}

By WReach

Cases[a,{n_,__}/;n<=4,{}] 
Out[78]= {{1,2,3},{4,5,8}}

By WReach

pos=Position[a,{n_,__}/;n<=4,{}] 
Extract[a,pos] 
 
Out[79]= {{1},{2}} 
Out[80]= {{1,2,3},{4,5,8}}

37.4 How to generate a diagonal matrix?

Random values on the diagonal

DiagonalMatrix[Table[Random[], {3}]]

Ones on the diagonal

DiagonalMatrix[Table[1, {3}]]

37.5 How to generate upper diagonal matrix?

one way, using SparseArray

Normal[SparseArray[{{i_, i_} :> 2*i, {i_, j_} :> i + j /; i < j}, {5, 5}]] 
Out[81]= {{2, 3, 4, 5, 6}, 
          {0, 4, 5, 6, 7}, 
          {0, 0, 6, 7, 8}, 
          {0, 0, 0, 8, 9}, 
          {0, 0, 0, 0, 10}}
 

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

Table[If[i == j, 2*i, If[i < j, i + j, 0]], {i, 5}, {j, 5}] 
Out[82]= ... same as above
 

37.6 How to find the trace of a matrix?

see ?Tr[a]

37.7 How to find product of elements on the Trace?

a = {{1, 2, 3}, 
     {4, 5, 8}, 
     {7, 8, 9}} 
Tr[a, Times] 
 
Out[84]= 45
 

37.8 How to check if a Matrix is diagonal matrix?

by Jon MacLoone

DiagonalQ[m_List] /; ArrayDepth[m] === 2 && Equal @@ Dimensions[m] := 
   And @@ Flatten[MapIndexed[#1 === 0 || Equal @@ #2 & , m, {2}]]; 
 
DiagonalQ[m_] := Return[False]; 
a = {{1, 2}, {2, 4}} 
b = {{1, 0}, {0, 2}} 
 
DiagonalQ[a] 
Out[89]= False 
 
DiagonalQ[b] 
Out[90]= True
 

By Paul Abbott

Clear[DiagonalQ]; 
DiagonalQ[(m_)?MatrixQ] /; SameQ @@ Dimensions[m] := m === DiagonalMatrix[Tr[m, List]] 
DiagonalQ[a] 
 
Out[93]= False 
 
DiagonalQ[b] 
Out[94]= True
 

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

Find location of zeros in this matrix

a = {{1, 2, 3}, 
     {4, 0, 8}, 
     {7, 8, 0}}

one way

Position[a, 0] 
Out[96]= {{2, 2}, {3, 3}}

Another way

Position[a, _?(#1 == 0 & )] 
Out[97]= {{2, 2}, {3, 3}}

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

find all elements between 4 and 8

a = {{1, 2, 3}, 
     {4, 0, 8}, 
     {7, 8, 0}} 
 
Position[a, _?(#1 >= 4 && #1 <= 8 & )] 
Out[99]= {{2, 1}, {2, 3}, {3, 1}, {3, 2}} 
 
Extract[a, %] 
Out[100]= {4, 8, 7, 8}

37.11 How to insert an element in specific position?

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

a = {{1, 2, 3}, 
     {4, 0, 8}, 
     {7, 8, 0}} 
a[[1,1]] = 99; 
a 
 
Out[103]= {{99, 2, 3}, 
           {4, 0, 8}, 
           {7, 8, 0}} 

37.12 How to insert a row into a matrix?

a = {{1, 2, 3}, 
     {4, 0, 8}, 
     {7, 8, 0}}

To insert this row in the second row in matrix above

row = {97, 98, 99}; 
newa = Insert[a, row, {2}] 
 
Out[106]= {{1, 2, 3}, 
           {97, 98, 99}, 
           {4, 0, 8}, 
           {7, 8, 0}}

or just use ’2’, it will also work

newa = Insert[a, row, 2] 
 
Out[107]= {{1, 2, 3}, 
           {97, 98, 99}, 
           {4, 0, 8}, 
           {7, 8, 0}}

37.13 How to insert a column into a matrix?

a = {{1, 2, 3}, 
     {4, 0, 8}, 
     {7, 8, 0}}

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

column = {97, 98, 99};

one way

newa = Transpose[Insert[Transpose[a], column, 2]] 
Out[110]= {{1, 97, 2, 3}, 
           {4, 98, 0, 8}, 
           {7, 99, 8, 0}}

another way

Normal[SparseArray[{{i_, j_} :> column[[i]] /; j == 2, 
       {i_, j_} :> a[[i,j]] /; j == 1, {i_, j_} :> a[[i,j - 1]] /; j > 1}, 
   {3, 4}]] 
 
Out[111]= {{1, 97, 2, 3}, 
           {4, 98, 0, 8}, 
           {7, 99, 8, 0}}

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

MapThread[Insert, {a, column, Table[2, {Length[column]}]}] 
 
Out[112]= {{1, 97, 2, 3}, 
           {4, 98, 0, 8}, 
           {7, 99, 8, 0}}

Another by Leonid Shifrin

ArrayFlatten[{{a[[All,1 ;; 1]], Transpose[{column}], a[[All,2 ;; All]]}}] 
 
Out[113]= {{1, 97, 2, 3}, 
           {4, 98, 0, 8}, 
           {7, 99, 8, 0}} 

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

Given

a = {{1, 2, 3}, 
     {4, 0, 8}, 
     {7, 8, 0}}

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

b = ArrayFlatten[ {{a, a}, {a, a}}] 
 
Out[118] {{1, 2, 3, 1, 2, 3}, 
          {4, 0, 8, 4, 0, 8}, 
          {7, 8, 0, 7, 8, 0}, 
          {1, 2, 3, 1, 2, 3}, 
          {4, 0, 8, 4, 0, 8}, 
          {7, 8, 0, 7, 8, 0}}]

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

Given

a = {{1, 2, 3}, 
     {4, 0, 8}, 
     {7, 8, 0}}

and we want to apply the this function to it

f[x_] := x + 2*Sin[x]

Then using Map

r = Map[f[#1] & , a, {2}] 
 
Out[123]= {{1 + 2*Sin[1], 2 + 2*Sin[2], 3 + 2*Sin[3]}, 
           {4 + 2*Sin[4], 0, 8 + 2*Sin[8]}, 
           {7 + 2*Sin[7], 8 + 2*Sin[8], 0}}

38 How to find if an expression implies another?

Remove["Global`*"] 
Refine[Sin[x]^2 + Cos[x]^2 == q, q == 1] 
 
Out[125]= True

39 Displaying matrices in MatrixForm

One way

$PrePrint = If[MatrixQ[#], MatrixForm[#], #] &; 
m = {{1, 2}, {3, 4}}

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

Plot3D[2 x + 7 y, {x, -4, 4}, {y, -4, 4}, 
 Boxed -> False, 
 AxesEdge -> {{-1, -1}, {-1, -1}, {-1, -1}}, 
 AxesLabel -> {x, y, z}]

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

One way to use Transpose

x = Table[i, {i, 0, 2 Pi, Pi/10}]; 
y = Sin[x]; 
data = Transpose[{x, y}]; 
ListPlot[data, Joined -> True]

42 How to do autocorrelation

v = {1, 2, 3}; 
ListCorrelate[v, v, {-1, 1}, 0] 
 
Out[139]= {3, 8, 14, 8, 3}

In Matlab it is

v=[1,2,3] 
xcorr(v,v) 
 
ans = 
     3     8    14     8     3

43 How to make a spring?

From the net, lost reference

ParametricPlot3D[{Sin[u], Cos[u], 0.2 u}, {u, 0, 10*2 Pi}, 
  PlotStyle -> {Tube[0.1]}, 
  ViewPoint -> {3.38378, 0, 0}, 
 ViewVertical -> {0, 1, 0}, 
 ViewAngle -> Automatic, 
 Axes -> False, Boxed -> False, 
 ImageSize -> 200]

See also how-to-draw-a-spring

44 How to list files in a directory?

SetDirectory[$BaseDirectory]; 
FileNames["*"]

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

maxy = Pi; 
Row[{VerticalSlider[Dynamic[maxy], {-2 Pi, 2 Pi}, 
  Appearance -> "LeftArrow"], 
  Dynamic@Plot[Sin[x], {x, -Pi, Pi}, 
  PlotRange -> {{-Pi, Pi}, {-maxy, maxy}}, 
  ImageSize -> 200]}]

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

The definition of precision in Mathematica is this. Suppose x is a 
number known up to an error of epsilon, that is it can be viewed as 
lying in the interval (x-epsilon/2,x+epsilon/2). Then its precision is 
 
-Log[10,epsilon/x]. Its accuracy is -Log[10,epsilon]. The two are 
related by the equation: 
Precision[x] - Accuracy[x] == RealExponent[x] 
 
The interpretation in terms of digits is only approximate. Both accuracy 
and precision can be negative - this depends on the scale of the number 
i.e. RealExponent. A number will have negative accuracy if its absolute 
error is large. It is easy to produce such numbers by cancellation 
 
With[{x = N[10^100, 50] - N[10^100, 50]}, 
     Accuracy[x]] 
 
-50.301 
 
On the other hand, since 
 
$MinPrecision 
 
0 
 
You won't normally in Mathematica see numbers with negative Precision. 
Precision is the main concept, Accuracy is only used because Precision 
is singular at 0 (remember - its relative error). 
 
It's all perfectly documented so this tired scape goat is not available 
this time.

48 Guidelines for writing demonstration

In math italicize single Roman letters that are variables or 
functions  (example, x,y,f(x),t 
2. Exception to above is capital letters for points like P and Q in 
geometry. 
3. Do not italicize Greek letters (example, alpha, gamma, beta, etc..), 
   and units like sec or rad, or  punctuation like ( ). 
4. Styling the control labels is optional. 
5. Do not use strings with <> for such formatting. Use Row[{ăăă }] 
6. put () around units in plot labels. As (sec) or (hz) 
7. do not italicize function names longer than one letter. So Style["exp",Italic] 
   should just be "exp" 
8. The t in delta(t) should be italic--but not the delta, Greek letter are not 
   Italian letters is how I remember that. 
9. Log should be log. 
10. Is j^2= -1? Better say so in the caption for non EE.

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

Text@Style[TraditionalForm[HoldForm[Sin[x] + Cos[y]], 12]]

To add invisible space use ESC is ESC

49 Making condition as pattern

By Bob Hanlon from math group:

Clear[x, $PrePrint] 
expr = {E^x, x, x^2, Log[x]}; 
Position[expr, _?( !PolynomialQ[#1, x] & ), 1] 
 
Out[146]= {{1}, {4}}

50 Reading Mathematica example data and location

on windows, V 8, example data is located in

C:\Program Files\Wolfram Research\Mathematica\8.0\Documentation\English\System\ExampleData

and it can be read like this

str = OpenRead["ExampleData/USConstitution.txt"] 
 
Out[147]= InputStream[ExampleData/USConstitution.txt, 127]

51 Stopping 3D plot from changing size

Use SphericalRegion->True

ListPlot3D[Table[RandomReal[], {5}, {5}], AxesLabel -> {"x", "y", "z"}, 
  ImageSize -> {200, 200}, ImagePadding -> 20, 
  SphericalRegion -> True];

52 Selecting elements from a list that satisfy a condition

This question was posted on the net. Given

b:=Table[{x,y},{x,1,6},{y,1,6}]

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

some answers

(me) 
Select[Flatten[b, 1], #1[[1]] + #1[[2]] > 9 & ] 
 
Out[152]= {{4, 6}, 
           {5, 5}, 
           {5, 6}, 
           {6, 4}, 
           {6, 5}, 
           {6, 6}}

Adriano Pascoletti answer

Cases[b, {x_Integer, y_Integer} /; x + y > 9, {2}]

Bill Row answer

Cases[Flatten[b, 1], _?(Total[#1] > 9 & )]

Murray Eisenberg answer

Select[Flatten[b, 1], First[#1] + Last[#1] > 9 & ]

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.

mat = {{-1., -1., Indeterminate, -1., -1.}, 
       {-1., -1., Indeterminate, -1., -1.}, 
       {Indeterminate, Indeterminate, Indeterminate, Indeterminate, Indeterminate}, 
       {-1., -1., Indeterminate, -1., -1.}, 
       {-1., -1., Indeterminate, -1., -1.}} 
 
p = Position[mat, Indeterminate] 
mat = ReplacePart[mat, p -> 0] 
 
Out[169]= {{-1., -1., 0, -1., -1.}, 
           {-1., -1., 0, -1., -1.}, 
           {0, 0, 0, 0, 0}, 
           {-1., -1., 0, -1., -1.}, 
           {-1., -1., 0, -1., -1.}}

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

n = 5; 
a = Table[RandomReal[], {n}, {n}]; 
p = Position[a, x_ /; x < 0.5]; 
a = ReplacePart[a, p -> Null] 
 
Out[173]= {{Null, Null, Null, 0.6781657418995635, 0.7290662037036753}, 
           {Null, 0.7084980071179792, Null, Null, 0.5811489862295911}, 
           {Null, Null, 0.8467863882617719, Null, 0.8891915946646993}, 
           {0.8173279058333203, 0.7272894246356278, Null, Null, 0.8665880423275274}, 
           {Null, Null, 0.662026816962838, 0.5982839657423036, 0.6603967280952212}}

54 What are the AppearanceElements names?

see full-documentation-for-appearanceelements

list = { 
        "AutorunPlayButton", 
        "BookmarksButton", 
        "BookmarksPlayButton" 
        "ContentResizeArea", 
        "DirectionButton", 
        "FasterSlowerButtons", 
        "HideControlsButton", 
        "InteractiveTradingChartMenu", 
        "InteractiveTradingChartSnapshotButton", 
        "InteractiveTradingChartResetButton", 
        "InputField", 
        "InlineInputField", 
        "ManipulatePlayButton", 
        "ManipulateMenu", 
        "PlayPauseButton", 
        "ProgressSlider", 
        "ResetButton", 
        "SnapshotButton", 
        "StepLeftButton", 
        "StepRightButton", 
        "UpdateButton", 
        None};

55 How to make listplot

one way

p = Plot[Sin[x], {x, 0, Pi}, MaxRecursion -> 0, 
         PlotPoints -> 10]; 
data = Cases[Normal[p], x_Line :> First[x], Infinity]; 
Show[p, ListPlot[data, PlotStyle -> Red]]

56 Getting points from plot once it is plotted

p = ContourPlot3D[x^2 + y^3 - z^3 == 0, {x, -2, 2}, {y, -2, 2}, {z, -2, 2}, 
    PlotPoints -> Automatic] 
data = (InputForm@p)[[1, 1, 1]]

and

data = Reap[DensityPlot[Sin[x*y], {x, 0, 2 Pi}, {y, 0, 2 Pi}, 
     EvaluationMonitor :> Sow[{x, y, Sin[x*y]}]]][[2, 1]]; 
 
ListPlot3D[data]

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

Module variables should *never* appear inside Dynamics or 
Manipulates internal to that Module. 
 
To be clear with some examples (all using Dynamic, but they could equally well 
use Manipulate, which is implemented using Dynamic)... 
 
(* OK *) Dynamic[Module[{a}, a]] 
(* OK *) Module[{a}, 
                (* expression involving a*); 
                Dynamic[(* expression *not* involving a *)] 
(* BAD *) Module[{a}, Dynamic[a]]

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

This is the support explanation of why this error came showed up: 
 
The issue is specifically with the section: 
Evaluate@env[{{age, 100, "age"}, 10, 200, 1}] 
 
Manipulate doesn't really evaluate until it gets to the Initialization 
option, but it will check its input for correct form. Mathematica 
reads the main body of the Manipulate before running the 
Initialization option. This is can be verified by using a Print statement: 
 
Initialization -> (Print["Test"]; 
  makeCustomEnvironmentAlt = 
   Function[Null, Function[code, With @@ Hold[{##}, code], HoldAll], 
    HoldAll]; 
  env = makeCustomEnvironmentAlt[$age = 1, $salary = 2]; 
  Print["Test"]) 
 
Test does not print. 
 
Getting around this will be probably not be clean. 
.... 
Having  the code for the controller for age depend on evaluation of 
some function which must be initialized does not appear to be possible 
with simply Manipulate. 

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

Manipulate[{a, b}, 
 Grid[{ 
 
   {"a", Manipulator[Dynamic[a, {(a = #) &, (a = #; If[b > a, b = a]) &}], 
          {1, 10, 1}], Dynamic[a]}, 
 
   {"b", Manipulator[Dynamic[b, {(b = #) &, (b = #; If[b > a, b = a]) &}], 
          {1, 10, 1}], Dynamic[b]}} 
  ], 
 
 {{a, 5}, None}, 
 {{b, 3}, None} 
 
 ]

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

id = 1;  (*first field name*) 
pop = 2; (*second field name*) 
name = 3; (*third field name*) 
 
(*now build the array of record, each row is a record*) 
m = {{1, 3000, "London"}, 
     {1, 6000, "New York"}, 
     {3, 7300, "Moscow"}}; 
 
(*now can iterate over the records*) 
 
Do[ 
 Print@m[[i, id]]; 
 m[[i, pop]] += 1, 
 {i, Length[m]} 
]

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

Remove[a, b, c, d, e, f] 
Apply[#1 + 3*#2 & , {{a, b}, {c, d}, {e, f}}, 1] 
 
Out[183]= {a + 3*b, c + 3*d, e + 3*f}

Or

Apply[#1 + 3*#2 & , {{a, b}, {c, d}, {e, f}}, {1}] 
 
Out[184]= {a + 3*b, c + 3*d, e + 3*f}

60 Using Sow and Reap

ListAnimate[Flatten[Reap[Do[Sow@Plot[Sin[x - c], {x, 0, 4 Pi}, 
      Ticks -> None], {c, 0, 2 Pi - Pi/10, Pi/10}]]]]

61 Making comments use monospaced fonts

Thanks to Alexey Popkov for this

SetOptions[EvaluationNotebook[], 
 AutoStyleOptions -> {"CommentStyle" -> {FontWeight -> Plain, 
     FontColor -> GrayLevel[0.6], ShowAutoStyles -> False, 
     ShowSyntaxStyles -> False, AutoNumberFormatting -> False, 
     FontFamily -> "Consolas"}}]

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

Normal[Series[1/r, {x, 0, 2}]] 
Out[189]= 1 + x^2/2

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?

Clear[s]; 
f1 = 2/(s + 3); 
f2 = 7/(s^2 + 2.5*s + 7); 
Simplify[Together[f1 + f2]] 
 
Out[193]= (35. + 12.*s + 2.*s^2)/(21. + 14.5*s + 5.5*s^2 + s^3)

65 How to use options in functions?

Options[myFun] = {form -> "linear"}; 
myFun[x_, OptionsPattern[]] := Module[{}, Print["x=", x, "form=", OptionValue[form]]; ] 
myFun[3, form -> "quadratic"]

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:

Unevaluated[Unevaluated[(2/3)*I + (x/y)*I] /. HoldPattern[(x_)/(y_)] -> r] 
       /. HoldPattern[I] -> d 
 
Out[200]= -d + d*(1 - x^2/2)

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

Unevaluated[(2/3)*I + (x/y)*I] /. HoldPattern[(x_)/(y_)] -> 
       r /. HoldPattern[I] -> d 
 
Out[201]= -I + I*(1 - x^2/2)

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

Remove["Global`*"] 
expr = a*x + b*x^2 - c*Exp[x]; 
expr /. {Exp[x] -> z, x -> y} /. z -> Exp[x] 
 
Out[211]= (-c)*E^x + a*y + b*y^2

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

CurrentValue[$FrontEnd, {"NotebookSecurityOptions", "TrustedPath"}] 
 
Out[212]= {FrontEnd`FileName[{$InstallationDirectory}], 
          FrontEnd`FileName[{$BaseDirectory}], 
          FrontEnd`FileName[{$UserBaseDirectory}]} 
 
CurrentValue[$FrontEnd, {"NotebookSecurityOptions", "UntrustedPath"}] 
 
Out[213]= {FrontEnd`FileName[{FrontEnd`$DesktopDirectory}], 
           FrontEnd`FileName[{FrontEnd`$DownloadsDirectory}], 
  FrontEnd`FileName[{FrontEnd`$LocalApplicationDataDirectory}], 
  FrontEnd`FileName[{FrontEnd`$RemoteApplicationDataDirectory}], 
  FrontEnd`FileName[{FrontEnd`$ProgramFilesDirectory}], 
  FrontEnd`FileName[{FrontEnd`$ProgramFilesX86Directory}], 
  FrontEnd`FileName[{$TemporaryPrefix}]}

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

Remove["Global`*"] 
Grid[{{Module[{x}, x]}}, Frame -> All]

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

Remove["Global`*"] 
x = 4; 
Block[{}, If[x == 4, x = 3]]; 
x 
 
Out[217]= 3

Module:

Remove["Global`*"] 
x = 4; 
Module[{}, If[x == 4, x = 3]]; 
x 
 
Out[221]= 3

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

lst = {{x -> 4, y -> 7}, {x -> 2, y -> 5}, {x -> -1, y -> 10}}

one way

({#1[[1,2]], #1[[2,2]]} & ) /@ lst 
 
Out[223]= {{4, 7}, {2, 5}, {-1, 10}}

another way

Cases[lst, {_ -> n_, _ -> m_} :> {n, m}] 
 
Out[224]= {{4, 7}, {2, 5}, {-1, 10}}

71 Aligning inside Framed environment

One way us to use Item

mat = Table[Random[], {3}, {3}]; 
Framed[ 
  Item[Grid[mat, Frame -> All, Alignment -> Center], 
  Alignment -> {Center, Top}], ImageSize -> {300, 200}]

72 Aligning individual row of a grids

One way us to use Item

Grid[{{"row1,row1"}, {"row2"}, {"row3"}}, 
    Frame -> All] 
 
Grid[{{"row1,row1"}, 
  {Item["row2", Alignment -> Left]}, 
  {"row3"}}, Frame -> All]

73 Printing numerical value not in scientific notation?

Use NumberForm

NumberForm[1./10^6, ExponentFunction -> (Null & )] 
 
Out[227]  0.000001

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

SetOptions[EvaluationNotebook[], 
  PrivateNotebookOptions -> {"FileContents" -> {"NotebookData"}, 
   "FileOutlineCache" -> False}]

In addition, I change the preferences like this:

75 RotationMatrix and RotationTransform

The call for each is as follows

p1 = {1, 1}; 
p2 = RotationMatrix[30 Degree].p1; 
o = {0, 0}; 
Graphics[{ {Red, Arrow[{o, p1}]}, 
  Arrow[{o, p2}]}, Axes -> True, ImageSize -> 100]

p1 = {1, 1}; 
r = RotationTransform[30 Degree]; 
o = {0, 0}; 
Graphics[{ {Red, Arrow[{o, p1}]}, Arrow[{o, r[p1]}]}, 
   Axes -> True, ImageSize -> 100]

76 How to change the head of a list?

A list has a Head at its zero index position

lst = {1, 2, 3}; 
Head[lst] 
 
Out[242]= List 
 
lst[[0]] 
Out[243]= List

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

Plus @@ lst 
 
Out[244]= 6

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

lst[[0]] = Plus 
Out[245]= Plus 
 
lst 
Out[246]= 6

If we have a list of lists, like this

lst = {{1, 2, 3}, {4, 5, 6}, {7, 8, 9}} 
Out[247]= {{1, 2, 3}, {4, 5, 6}, {7, 8, 9}}

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

Apply[Times, lst, {1}] 
Out[248]= {6, 120, 504}

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

(Times @@ #1 & ) /@ lst 
Out[249]= {6, 120, 504}

or, little shorter version of the above:

(Times @@ #1 & ) /@ lst 
Out[250]= {6, 120, 504}

77 Displaying polynomial from higher to lower order

Mathematica default display of polynomial is reverse the traditional form:

poly = x^3 + a*x + b*x^2 + c + d 
Out[251]= 27 + 3*a + 9*b + c + d

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

TraditionalForm[poly, ParameterVariables :> {a, b, c, d}]

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

lst = {1, 2, 5, 3, 7}; 
Sort[lst] 
Out[255]= {1, 2, 3, 5, 7} 
 
Sort[lst, #1 < #2 & ] 
Out[256]= {1, 2, 3, 5, 7} 
 
Sort[lst, #1 > #2 & ] 
Out[257]= {7, 5, 3, 2, 1}

81 Copy/paste code to Stackexchange?

82 Making automatic subscripted matrix notation

Format[t_a] := Subscripted[t] 
t = Table[a[i, j], {i, 2}, {j, 3}]

83 Finding Names of buildin contexts and options

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

getList[name_String] := Module[{options, idx}, 
   options = Names[name <> "`*"]; 
   options = ToExpression /@ options; 
   options = {#, Options[#]} & /@ options; 
   idx = Range[Length[options]]; 
   options = {#[[1]], TableForm[#[[2]]]} & /@ options; 
   options = Insert[options[[#]], #, 1] & /@ idx; 
   options = Insert[options, {"#", "Option", "Options to this option"}, 1]; 
   Grid[options, Frame -> All, Alignment -> Left, FrameStyle -> Directive[Thickness[.005], Gray]] 
   ];

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

FrontEndExecute[FrontEnd`AddMenuCommands["DuplicatePreviousOutput", 
   {Delimiter, MenuItem["Make List", FrontEnd`KernelExecute[nb = SelectedNotebook[]; 
sel = NotebookRead[nb]; 
       NotebookWrite[nb, Cell[BoxData[RowBox[{"{", sel, "}"}]]]]], 
     MenuKey["u", Modifiers -> {"Control"}], 
     MenuEvaluator -> Automatic]}]]

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:

s = "foo[] boo[] more goo[] more"; 
StringCases[s, RegularExpression["^\\w*\\["]] 
Out[265]= {foo[}

and

StringCases[s, Shortest[StartOfString~~__~~"["], Overlaps -> False] 
Out[266]= {foo[}

86 Dynamically change the layout of Manipulate

Manipulate[x, 
 
 {x, {True, False}}, 
 Grid[{ 
   {Dynamic@If[x, 
      Control[{y, {True, False}}], 
      Control[{z, 0, 1, .1}] 
      ] 
    } 
   } 
  ] 
 ] 

87 What are most common commands?

87.1 Mape or /@

Takes function and applies it to each element in a list

f /@ {a, b, c} 
Out[267]= {f[a], f[b], f[c]} 
 
(1 + g[#1] & ) /@ {a, b, c} 
Out[268]= {1 + g[a], 1 + g[b], 1 + g[c]} 
 
f /@ {a, b, c} 
Out[269]= {f[a], f[b], f[c]}

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

Thread[f[{a, b, c}]] 
Out[270]= {f[a], f[b], f[c]} 
 
f /@ {a, b, c} 
Out[271]= {f[a], f[b], f[c]} 
 
Thread[f[{a, b, c}, {1, 2, 3}]] 
Out[272]= {f[a, 1], f[b, 2], f[c, 3]}

87.3 MapThread

MapThread[f, {{a, b, c}, {1, 2, 3}}] 
Out[273]= {f[a, 1], f[b, 2], f[c, 3]}

In this case gives the same answer as using Thread

Thread[f[{a1, a2, a3}, {b1, b2, b3}]] 
Out[274]= {f[a1, b1], f[a2, b2], f[a3, b3]}

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

MapThread[f, {{{a, b}, {c, d}}, {{1, 2}, {3, 4}}}] 
Out[275]= {f[{a, b}, {1, 2}], f[{c, d}, {3, 4}]} 
 
Thread[f[{{{a, b}, {c, d}}, {{1, 2}, {3, 4}}}]] 
Out[276]= {f[{{a, b}, {c, d}}], f[{{1, 2}, {3, 4}}]}

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

foo[(x_)?(Element[#1, Integers] & )] := x 
foo[x_Integer] := x 
foo[x_Integer] := x

88.1.2 integer strictly positive

foo[(x_)?(IntegerQ[#1] && #1 > 0 & )] := x 
foo[x_Integer /; x > 0] := x 
foo[(x_Integer)?Positive] := x 
foo[x_Integer /; x > 0] := x

88.1.3 integer strictly negative

foo[(x_)?(IntegerQ[#1] && #1 < 0 & )] := x 
foo[x_Integer /; x < 0] := x 
foo[(x_Integer)?Negative] := x 
foo[x_Integer /; x < 0] := x

88.1.4 integer zero or positive

foo[(x_)?(IntegerQ[#1] && #1 >= 0 & )] := x 
foo[x_Integer /; x >= 0] := x 
foo[(x_Integer)?NonNegative] := x 
foo[x_Integer /; x >= 0] := x

88.1.5 integer zero or negative

foo[(x_)?(IntegerQ[#1] && #1 <= 0 & )] := x 
foo[x_Integer /; x <= 0] := x 
foo[(x_Integer)?NonPositive] := x 
foo[x_Integer /; x <= 0] := x

88.1.6 integer in some range

foo[x_Integer /; x > 3 && x < 7] := x

88.1.7 Real

foo[x_?(Element[#, Reals] &)] := x 
foo[x_Real] := x

88.1.8 Real strictly positive

foo[x_Real /; x > $MachineEpsilon] := x 
foo[x_Real /; x > $MachineEpsilon] := x 
foo[x_Real /; Positive[x]] := x 
foo[x_ (Element[#, Reals] && Positive[#] &)] := x

88.1.9 Real strictly negative

foo[x_Real /; x < $MachineEpsilon] := x 
foo[x_Real /; x < $MachineEpsilon] := x 
foo[x_Real /; Negative[x]] := x 
foo[x_?(Element[#, Reals] && Negative[#] &)] := x

88.1.10 Real zero or positive

foo[x_Real /; x >= $MachineEpsilon] := x 
foo[x_Real /; x >= $MachineEpsilon] := x 
foo[x_Real /; Positive[x] || x == 0] := x 
foo[x_ (Element[#, Reals] && (Positive[#] || # == 0) &)] := x

88.1.11 Real zero or negative

foo[x_Real /; x <= $MachineEpsilon] := x 
foo[x_Real /; x <= $MachineEpsilon] := x 
foo[x_Real /; Negative[x] || x == 0] := x 
foo[x_ (Element[#, Reals] && (Negative[#] || # == 0) &)] := x

88.1.12 Real in some range

foo[x_ (Element[#, Reals] && ((# - 3) > $MachineEpsilon && (7 - #) > $MachineEpsilon) &)] := x 
foo[x : _Real /; (x - 3) > $MachineEpsilon && (7 - x) > $MachineEpsilon] := x

88.1.13 Boolean

foo[x_?(Element[#, Booleans] &)] := x

88.1.14 any numerical parameter

foo[x_?(Element[#, Reals] &)] := x 
foo[x_?(NumericQ[#] &)] := x 
foo[x : _?NumericQ] := x

88.1.15 checks for Head Real, Integer, Ratioal and Complex

foo[x_?(NumberQ[#] &)] := x

88.1.16 general complex number

foo[x_Complex] := x 
foo[x_?(Not@FreeQ[#, _Complex] &)] := x

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

foo[x_List] := x

88.1.18 1D list (i.e. vector)

foo[x_?(VectorQ[#] &)] := x

88.1.19 Numeric 1D list

foo[x_?(VectorQ[#, NumericQ] &)] := x

88.1.20 Numeric 1D list

foo[x_?(VectorQ[#, NumericQ] &)] := x 
foo[x : {_?NumericQ ..}] := x 
foo[x : {__?NumericQ }] := x 
foo[x_?(VectorQ[#, IntegerQ] &)] := x

88.1.21 2D matix of numbers

foo[x_?(MatrixQ[#, NumericQ] &)] := x 
foo[x : {{_?NumericQ ..}}] := x 
foo[x : {{__?NumericQ }}] := x

88.1.22 2D matrix numeric but contains no complex numbers

foo[x_?(MatrixQ[#, NumericQ] && FreeQ[#, _Complex] &)] := x

88.1.23 2D matrix of strings

foo[x_?(MatrixQ[#, StringQ] &)] := x

89 How to check for Head of expression?

use MatchQ

MatchQ[1/3, _Rational] 
Out[277]= True 
 
MatchQ[3, _Integer] 
Out[278]= True 
 
Or for the above can do 
 
IntegerQ[3] 
Out[279]= True

90 How to make different Grids

Grid[{ 
  {Item[a, Alignment -> Center], b}, 
  {SpanFromAbove, c}}, Frame -> All] 
 
Grid[{ 
  {Item[a, Alignment -> Center], Item[Column[{b, c}]]}}, Frame -> All] 
 
Grid[{ 
  {Item[a, Alignment -> Center], Item[Column[{b, c}, Frame -> All]]}}, Frame -> All] 
 
Grid[{ 
  {a, Item[b, Alignment -> Center]}, 
  {c, SpanFromAbove}}, Frame -> All] 
 
Grid[{ 
  {Item[a, Alignment -> Center], Item[b, Alignment -> Center], c}, 
  {SpanFromAbove, SpanFromAbove, d}, 
  {SpanFromAbove, e, f}}, Frame -> All] 

91 Common Patterns

From help

91.1 string matching in list

See select-and-blank

test = {{"String1", "a"}, {"String2", "b"}, {"String3", "a"}, {"String4", "a"}}; 
Cases[test, {_String, "a"}] 
Out[281]= {{String1, a}, {String3, a}, {String4, a}} 
 
Select[test, MatchQ[#1, {_String, "a"}] & ] 
Out[282]= {{String1, a}, {String3, a}, {String4, a}}

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

Clear[x] 
p = (_.)*_?Negative; 
MatchQ[-3*x^2, p] 
Out[285]= True 
 
MatchQ[3*x^2, p] 
Out[286]= False 
 
expr = -3*x^2; 
(expr /. Thread[Variables[expr] -> 1]) < 0 
Out[288]= True 
 
expr = 3*x^2; 
(expr /. Thread[Variables[expr] -> 1]) < 0 
Out[290]= False

92 Compare Manipulate to DynamicModule

See convert_manipulate_to_dynamicModule

93 Replacing \(y,y',y''\) 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

ode = Derivative[2][u][t] + Derivative[1][u][t] + u[t] == 3*Cos[2*t]; 
ode /. u -> (c1*Cos[#1] + c2*Sin[#1] & ) 
 
Out[2]= c2*Cos[t] - c1*Sin[t] == 3*Cos[2*t]

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.

Trigger[Dynamic[t0, {t0 = #} &], {0, 10000, 1}, ....] 
ToString[#] & /@ a| is same as ToString /@ a

95 Copying outout cells to another notebook

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

96 Mathematica command like Matlab blkdiag

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

a = {{1, 2, 3}, {4, 5, 6}} 
b = {{7, 8}, {9, 10}} 
SparseArray[Band[{1, 1}] -> {a, b}]

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

ClearAll["Global`*"] 
lst = {{a, b}, {c, d}}; 
MapThread[f, lst] 
Out[14]= {f[a, c], f[b, d]} 
 
Thread[f[Sequence @@ lst]] 
Out[15]= {f[a, c], f[b, d]}

99 Complex rules to help simplifications

ClearAll[x, y, z, g, foo]; 
p1 = Conjugate[x_]*Conjugate[y_] :> Conjugate[x*y]; 
p2 = (x_)*Conjugate[x_] :> Abs[x]^2; 
p3 = Abs[(x_)*(y_)]^(n_.) :> Abs[x]^n*Abs[y]^n; 
p4 = (x_)*Conjugate[y_] + (y_)*Conjugate[x_] :> 2*(Re[x]*Re[y] + Im[x]*Im[y]); 
p5 = (x_) + Conjugate[x_] :> 2*Re[x]; 
allRules = {p1, p2, p3, p4, p5};

test it

expr = {{foo = x*Conjugate[y] + y*Conjugate[x]; foo, foo //. allRules}, 
 {foo = x*Conjugate[y] + y*Conjugate[x] + z*Conjugate[g] + g*Conjugate[z]; foo, foo //. allRules}, 
 {foo = x*Conjugate[x]; foo, foo //. allRules}, {foo = x*y*Conjugate[x*y]; foo, foo //. allRules}, 
 {foo = x*y*z*Conjugate[x*y*z]; foo, foo //. allRules}, {foo = x + Conjugate[x]; foo, foo //. allRules}, 
 {foo = x*y + Conjugate[x*y], foo; foo //. allRules}, {foo = x*y*z + Conjugate[x*y*z]; foo, foo //. allRules}, 
 {foo = x*y + Conjugate[x]*Conjugate[y]; foo, foo //. allRules}, 
 {foo = x*y*z*g + Conjugate[x]*Conjugate[y]*Conjugate[z]*Conjugate[g]; foo, foo //. allRules}}; 
Grid[expr, Frame -> All, Spacings -> {0.5, 1}, Alignment -> Left]

100 How to find names of named characters?

See mathematica/guide/ListingOfNamedCharacters.html

101 How to understand views for 3D Graphics?

From extract-values-for-viewmatrix-from-a-graphics3d/3538

by Yu-Sung Chang (Wolfram Research)

102 On imagepadding, imageMargins etc.....

1. ref/ImagePadding.html
2. tutorial/GridsRowsAndColumns.html
3. ref/PlotRangePadding.html

103 How to thread functions over equations?

eq = E^(0.002/t) + E^(0.032/t) == 2*E^(0.03/t) 
Thread[Log[eq], Equal] 
 
Out[26]= Log[E^(0.002/t) + E^(0.032/t)] == Log[2*E^(0.03/t)]

104 Usages of Manipulate

I wrote these for an answer here http://community.wolfram.com/groups/-/m/t/153862?p_p_auth=GLyok3xN

The manipulate expression is anything between the start of Manipulate and the first ","

Manipulate[Plot[Sin[x (1 + a x)], {x, 0, 6}], {a, 0, 2}]

It has the form

Manipulate[expression, controlVariables, Initialization :> ()]

104.1 case 1

foo[] := Plot[Sin[x (1 + a x)], {x, 0, 6}] 
Manipulate[Evaluate@foo[], {a, 0, 2}]

104.2 case 2

move foo[] in the above example to inside Manipulate, in the initialization section, add literal symbol a so Manipulate will track it

Manipulate[a; 
 foo[], {a, 0, 2}, 
Initialization :> 
 (foo[] := Plot[Sin[x (1 + a x)], {x, 0, 6}]) 
]

Notice the trick above. a was added so that it appears in the Manipulate expression. Otherwise, it will not be tracked. You’ll get an initial plot, but nothing will happen when moving the slider

104.3 case 3

move foo[] to global context, but have to tell Manipulate that LocalizeVariable is false

foo[] := Plot[Sin[x (1 + a x)], {x, 0, 6}]; 
Manipulate[a; 
 foo[], {a, 0, 2}, LocalizeVariables -> False]

But notice, we still need to put a somewhere in the expression for it to tracked. Not enough just to say LocalizeVariables -> False

104.4 case 4

It is not enough to use TrackedSymbols :>a, if the symbol itself do not show up in the expression. Hence this does not work

foo[] := Plot[Sin[x (1 + a x)], {x, 0, 6}]; 
Manipulate[foo[], {a, 0, 2}, LocalizeVariables -> False, TrackedSymbols :> a]

Notice there is no a in the expression. Manipulate will not track a even though we told it to !

104.5 case 5

Same as case 4, even if we put the function inside the Initialization section, it will still not track a . One will get an initial plot, but that is all.

Manipulate[foo[], {a, 0, 2}, 
  TrackedSymbols :> a, 
  Initialization :> 
   (foo[] := Plot[Sin[x (1 + a x)], {x, 0, 6}];)]

104.6 case 6

Putting the function definition of foo[] itself inside Manipulate expression, now Manipulate sees a there and will automatically track it. No need to do anything more:

Manipulate[Module[{}, foo[] := Plot[Sin[x (1 + a x)], {x, 0, 6}]; 
  foo[]], {a, 0, 2}]

Or simply

Manipulate[Plot[Sin[x (1 + a x)], {x, 0, 6}], {a, 0, 2}]

104.7 case 7

This is the method I use myself. Put all the functions inside the initialization section, but pass the dependent variables by argument call.

 
Manipulate[foo[a], {a, 0, 2}, 
  Initialization :> ( 
    foo[a_] := Module[{x}, Plot[Sin[x (1 + a x)], {x, 0, 6}]] 
)]

I like this, becuase it achieves both the goal of having all the slider symbols inside the Manipulate expression, hence Manipulate will track them, and at the same time, it avoid the function be in global context, and it is the modular way, since one can see which parameter each function depends on by looking at the signature of the function.

104.8 case 8

Similar to case 7, but the function itself is now in the global context. This is a fine solution as well, if this function needs to be called from somewhere else as well other than from the Manipulate. Otherwise, it is best not to have in the global context and use case 7.

foo[a_] := Module[{x}, Plot[Sin[x (1 + a x)], {x, 0, 6}]]; 
Manipulate[foo[a], {a, 0, 2}]

105 Correct way to define function for Integrate use

Watch out when defining a function such as this:

f[x_] := Integrate[x - t, {t, 0, x}]

The problem is this:

In[45]:= f[t] 
Out[45]= 0

This is becuase the replacement of "x" by "t" changed the integrand to zero.

The correct way is to always use Module symbols for everything inside the function, like this

f[x_] := Module[{t}, Integrate[x - t, {t, 0, x}]]

Now it gives the correct answer regardless of the symbol used as argument

In[46]:= f[t] 
Out[46]= t^2/2 
 
In[47]:= f[x] 
Out[47]= x^2/2

106 How to find list of all distributions

StringCases[#, ___ ~~ "Distribution" ~~ ___ :> #] & /@ Names["System`*"]; 
DeleteCases[%, {}]

Gives

{{"ArcSinDistribution"}, {"BarabasiAlbertGraphDistribution"}, {"BatesDistribution"}, {"BeckmannDistribution"}, \ 
{"BenfordDistribution"}, {"BeniniDistribution"}, {"BenktanderGibratDistribution"}, {"BenktanderWeibullDistribution"}, \ 
{"BernoulliDistribution"}, {"BernoulliGraphDistribution"}, {"BetaBinomialDistribution"}, {"BetaDistribution"}, \ 
{"BetaNegativeBinomialDistribution"}, {"BetaPrimeDistribution"}, {"BinomialDistribution"}, {"BinormalDistribution"}, \ 
{"BirnbaumSaundersDistribution"}, {"BorelTannerDistribution"}, {"CauchyDistribution"}, {"CensoredDistribution"}, \ 
{"ChiDistribution"}, {"ChiSquareDistribution"}, {"CompoundPoissonDistribution"}, {"CopulaDistribution"}, {"CoxianDistribution"}, \ 
{"DagumDistribution"}, {"DataDistribution"}, {"DavisDistribution"}, {"DegreeGraphDistribution"}, {"DirichletDistribution"}, \ 
{"DiscreteUniformDistribution"}, {"DistributionChart"}, {"DistributionDomain"}, {"DistributionFitTest"}, \ 
{"DistributionParameterAssumptions"}, {"DistributionParameterQ"}, {"EmpiricalDistribution"}, {"ErlangDistribution"}, \ 
{"EstimatedDistribution"}, {"ExpGammaDistribution"}, {"ExponentialDistribution"}, {"ExponentialPowerDistribution"}, \ 
{"ExtremeValueDistribution"}, {"FailureDistribution"}, {"FindDistributionParameters"}, {"FirstPassageTimeDistribution"}, \ 
{"FisherHypergeometricDistribution"}, {"FisherZDistribution"}, {"FRatioDistribution"}, {"FrechetDistribution"}, \ 
{"GammaDistribution"}, {"GeometricDistribution"}, {"GompertzMakehamDistribution"}, {"GraphPropertyDistribution"}, \ 
{"GumbelDistribution"}, {"HalfNormalDistribution"}, {"HistogramDistribution"}, {"HotellingTSquareDistribution"}, \ 
{"HoytDistribution"}, {"HyperbolicDistribution"}, {"HyperexponentialDistribution"}, {"HypergeometricDistribution"}, \ 
{"HypoexponentialDistribution"}, {"InverseChiSquareDistribution"}, {"InverseGammaDistribution"}, {"InverseGaussianDistribution"}, \ 
{"JohnsonDistribution"}, {"KDistribution"}, {"KernelMixtureDistribution"}, {"KumaraswamyDistribution"}, {"LandauDistribution"}, \ 
{"LaplaceDistribution"}, {"LevyDistribution"}, {"LindleyDistribution"}, {"LogGammaDistribution"}, {"LogisticDistribution"}, \ 
{"LogLogisticDistribution"}, {"LogMultinormalDistribution"}, {"LogNormalDistribution"}, {"LogSeriesDistribution"}, \ 
{"MarginalDistribution"}, {"MaxStableDistribution"}, {"MaxwellDistribution"}, {"MeixnerDistribution"}, {"MinStableDistribution"}, \ 
{"MixtureDistribution"}, {"MoyalDistribution"}, {"MultinomialDistribution"}, {"MultinormalDistribution"}, \ 
{"MultivariateHypergeometricDistribution"}, {"MultivariatePoissonDistribution"}, {"MultivariateTDistribution"}, \ 
{"NakagamiDistribution"}, {"NegativeBinomialDistribution"}, {"NegativeMultinomialDistribution"}, {"NoncentralBetaDistribution"}, \ 
{"NoncentralChiSquareDistribution"}, {"NoncentralFRatioDistribution"}, {"NoncentralStudentTDistribution"}, {"NormalDistribution"}, \ 
{"OrderDistribution"}, {"ParameterMixtureDistribution"}, {"ParetoDistribution"}, {"PascalDistribution"}, {"PearsonDistribution"}, \ 
{"PERTDistribution"}, {"PoissonConsulDistribution"}, {"PoissonDistribution"}, {"PolyaAeppliDistribution"}, {"PowerDistribution"}, \ 
{"PriceGraphDistribution"}, {"ProbabilityDistribution"}, {"ProductDistribution"}, {"RayleighDistribution"}, \ 
{"ReliabilityDistribution"}, {"RiceDistribution"}, {"SechDistribution"}, {"SinghMaddalaDistribution"}, {"SkellamDistribution"}, \ 
{"SkewNormalDistribution"}, {"SliceDistribution"}, {"SmoothKernelDistribution"}, {"SpatialGraphDistribution"}, \ 
{"SplicedDistribution"}, {"StableDistribution"}, {"StandbyDistribution"}, {"StationaryDistribution"}, {"StudentTDistribution"}, \ 
{"SurvivalDistribution"}, {"SuzukiDistribution"}, {"TransformedDistribution"}, {"TriangularDistribution"}, \ 
{"TruncatedDistribution"}, {"TsallisQExponentialDistribution"}, {"TsallisQGaussianDistribution"}, {"TukeyLambdaDistribution"}, \ 
{"UniformDistribution"}, {"UniformGraphDistribution"}, {"UniformSumDistribution"}, {"VarianceGammaDistribution"}, \ 
{"VoigtDistribution"}, {"VonMisesDistribution"}, {"WakebyDistribution"}, {"WalleniusHypergeometricDistribution"}, \ 
{"WaringYuleDistribution"}, {"WattsStrogatzGraphDistribution"}, {"WeibullDistribution"}, {"WignerSemicircleDistribution"}, \ 
{"ZipfDistribution"}}

107 find number of points used in Plot command?

see how-to-return-the-value-of-automatic-when-it-is-used-in-a-mathematica-function

Thanks to Bob Hanlon for this method:

p1 = Plot[Sin[x], {x, 0, Pi}, PlotPoints -> Automatic] 
Cases[p1, Line[pts_] :> Length[pts], Infinity] 
  (*259*)

Another method due to Simon Woods

Trace[ 
 Plot[Sin[t], {t, 0, 2 Pi}], 
 HoldPattern[PlotPoints | MaxRecursion -> _], 
 TraceInternal -> True] // Flatten // Union 
 
  {MaxRecursion -> 6, MaxRecursion -> Automatic, PlotPoints -> 50, 
     PlotPoints -> Automatic}

108 save matrix to file, and read it again

SetDirectory[NotebookDirectory[]]; 
list = {{3, 4, 5}, {4, 5, 6}}; 
Export["data.txt", list]

To read it later, say after closing and restarting Mathematica again to continue working on the data

SetDirectory[NotebookDirectory[]]; 
list = ToExpression@Import["data.txt", "List"] 
 
    {{3, 4, 5}, {4, 5, 6}}

109 How to solve ODE using power series method?

For an ode, that has just ordinary point at \(x=x_0\), we can find power series solution near \(x_0\) as follows. Assume the ode is

\[ u''(t)+\frac {1}{10} u(t)^3 u'(t)^2+4 u(t)=0 \] with intitial conditions \(u(0)=1,u'(0)=1\), then

findSeriesSolution[t_, nTerms_] := Module[{pt = 0, u, ode, s0, s1, ic, eq, sol}, 
  ic = {u[0] -> 1, u'[0] -> 1}; 
  ode = u''[t] + 4 u[t] + 1/10 u[t]^3 u'[t]^2; 
  s0 = Series[ode, {t, pt, nTerms}]; 
  s0 = s0 /. ic; 
  roots = Solve@LogicalExpand[s0 == 0]; 
  s1 = Series[u[t], {t, pt, nTerms + 2}]; 
  sol = Normal[s1] /. ic /. roots[[1]] 
  ]

and now call it with

seriesSol = findSeriesSolution[t, 6]

It returns \[ \frac {445409479 t^8}{840000000}+\frac {8878343 t^7}{10500000}-\frac {277427 t^6}{600000}-\frac {12569 t^5}{50000}+\frac {1607 t^4}{2000}-\frac {29 t^3}{50}-\frac {41 t^2}{20}+t+1 \]

110 How to delete all input cells?

link

make sure to copy the notebook first, just in case.

Module[{nb}, 
 nb = EvaluationNotebook[]; 
 NotebookFind[EvaluationNotebook[], "Input", All, CellStyle]; 
 NotebookDelete[nb]]

111 Use of ## &[]

reference

To understand what it does, these three do the same thing

Map[If[# == 1, Unevaluated@Sequence[], #] &, {1, 2, 3}]; 
If[# == 1, Unevaluated@Sequence[], #] & /@ {1, 2, 3}; 
If[# == 1, Sequence @@ {}, #] & /@ {1, 2, 3}; 
If[# == 1, ## &[], #] & /@ {1, 2, 3};

All above give {2,3}. So the effect of ## &[] is to remove the entry completely (so we do not end up with a Null or empty slot in there).

112 How to use MapThread to map function on 2 lists?

Given lists a={1,2,3}, b={4,5,6} and we want to do operation from slot 1 from a with slot 1 from b, and so on. For this, we can use ‘MapThread‘. Suppose we want to add each corresponding slot, then

a = {1, 2, 3} 
b = {4, 5, 6} 
MapThread[(#1 + #2) &, {a, b}] 
 
   (*  {5, 7, 9} *)

Of course, in this simple example, doing a+b will work, but this is just an example.

113 How to make shortcut to add [[ and ]] quickly?

see https://mathematica.stackexchange.com/questions/5212/automating-esc-esc-formatting

114 run Mathematica m file as script

Thanks to Rolf Mertig reference for help on this.

Make a file foo.m such as

AppendTo[$Echo, "stdout"] 
SetOptions[ $Output, FormatType -> OutputForm ]; 
Integrate[Sin[x],x]

Now type, from DOS window

"C:\Program Files\Wolfram Research\Mathematica\10.1\math.exe" < foo.m

This will send input and output to screen. To send output to file do

"C:\Program Files\Wolfram Research\Mathematica\10.1\math.exe" < foo.m > log.txt

You can modify the PATH on windows to add the above to environment variable so not to have to type the long command each time

115 How to run CDF in Chrome browser?

Type this in the chrome window

chrome://flags/#enable-npapi

and click enable to enable NPAPI. Chrome now disables NPAPI and so CDF no longer runs inside browser. After doing the above, restart the browser again. Now it should run

116 Installing Mathematica 10.1 on Linux

 
>sudo bash Mathematica_10.1.0_LINUX.sh 
[sudo] password for me: 
Mathematica 10.1.0 for LINUX Installer Archive 
 
Verifying archive integrity. 
Extracting installer. ............. 
 
 
Wolfram Mathematica 10.1 Installer 
 
Copyright (c) 1988-2015 Wolfram Research, Inc. All rights reserved. 
 
WARNING: Wolfram Mathematica is protected by copyright law 
and international treaties. Unauthorized reproduction or 
distribution may result in severe civil and criminalpenalties 
and will be prosecuted to the maximum extent possible under law. 
 
Enter the installation directory, or press ENTER to select 
  /usr/local/Wolfram/Mathematica/10.1: 
 
Now installing... 
 
[****************** 
 
Type the directory path in which the Wolfram Mathematica script(s) 
will be created, or press ENTER to select /usr/local/bin: 
 
 
> 
 
 
Installation complete. 
 
 
>which Mathematica 
/usr/local/bin/Mathematica 
>export DISPLAY=:0 
>Mathematica 

117 get step-by-step solution from Alpha

One method is to just type WolframAlpha["command here"] and then click on the show step by step on top right corner of the result that displays on the notebook, assuming Wolfram Alpha gives an answer.

If the above does not work, try

WolframAlpha["Integrate[x Sin[x],{x,0,Pi}]", {{"Input", 2}, 
  "Content"}, PodStates -> {"Input__Step-by-step solution"}]

Another example

WolframAlpha["solve y''=-y", 
              IncludePods -> "DifferentialEquationSolution", 
              AppearanceElements -> {"Pods"}, 
              TimeConstraint -> {20, Automatic, Automatic, Automatic}, 
              PodStates -> {"DifferentialEquationSolution__Step-by-step solution"}]

or for text output

 WolframAlpha["solve y''=-y", {{"DifferentialEquationSolution", 2}, "Plaintext"}, 
         PodStates -> {"DifferentialEquationSolution__Step-by-step solution"}]

Another example

WolframAlpha["q*c'[x]==k*c[x], c[0]==c0",  PodStates -> {"Step-by-step solution"}]

118 How to nest Map inside Map?

see http://mathematica.stackexchange.com/questions/15480/how-do-i-designate-arguments-in-a-nested-map

From about url by Halirutan:

Map[Function[p2, Map[Function[p1, f[p1, p2]], list1]], list2]

119 TeXForm handling of derivative higher than two

How to make TeXForm handle higher derivatives better.

See http://mathematica.stackexchange.com/questions/134936/texform-handling-of-derivative-higher-than-two

From above URL by Carl Woll

Derivative /: MakeBoxes[Derivative[n_Integer?Positive][h_],TraditionalForm] 
     := SuperscriptBox[MakeBoxes[h,TraditionalForm], 
    StringJoin@ConstantArray["\[Prime]",n] 
] 
 
TeXForm[y'''''[t] + 2 x'''[t] - y'[t] == x[t]]

120 How to classify singular points for ODE?

Given an ode, such as \(x(1-x) y''(x)+(c-(a+b+1)x)y'(x)-a b y(x)=0\) and we want to classify the singular points (finite and infinite).

We write it as \(y''(x)+p(x) y'(x)+q(x)=0\) and then follow standard procedures. In this example, we want to classify points \(0,1,\infty \).

This small function I wrote for a HW will do this. Pass it \(p(x),q(x)\) and list of the points to classify.

checkForSingularity[p_, q_, at_List, x_] := Module[{p0, q0, t, r, r0}, 
  r = First@Last@Reap@Do[ 
       If[at[[n]] === Infinity, 
        p0 = p /. x -> (1/t); 
        p0 = (2 t - p0)/t^2; 
        q0 = (q /. x -> (1/t))/t^4; 
        r0 = checkForSingularity[p0, q0, {0}, t]; 
        Sow[Flatten[{Infinity, Rest[Flatten@r0]}]]; 
        , 
        r0 = {at[[n]], Limit[(x - at[[n]])  p, x -> at[[n]]], 
          Limit[(x - at[[n]])^2 q, x -> at[[n]]]}; 
        Sow@r0 
        ] 
       , {n, 1, Length@at} 
       ]; 
  r 
  ]

To use it on the above example

ClearAll[c, a, b, x]; 
m = checkForSingularity[(c - (a + b + 1) x)/(x (1 - 
        x)), (-a b)/(x (1 - x)), {0, 1, Infinity}, x]; 
Grid[Join[{{"point", "limit x p(x)", "limit x^2 q(x)"}}, m], 
 Frame -> All]

Gives

{{"point", "limit x p(x)", "limit x^2 q(x)"}, 
 {0,            c,                 0}, 
 {1,        1 + a + b - c,         0}, 
 {\[Infinity], 1 - a - b,        a b} 
}

Since all points have finite limits, then all \(0,1,\infty \) are removable singularities.

I did not test this function too much, but it works for the HW I did :)

121 How to replace \(y(x)\) by \(e^{s(x)}\) in an ode?

Given ode \(y^{'''}=x y(x)\) we want to replace \(y(x)\) by \(e^{s(x)}\)

ClearAll[y,x,s] 
eq = y'''[x] == x y[x] 
eq = eq /. y -> Function[{x}, Exp[s[x]]] 
Simplify@Thread[eq/Exp[s[x]], Equal]
 

Gives \(x=s^{'''}(x)+s'(x)^3+3 s'(x) s''(x)\)

122 How to print definitions of some internal functions?

Needs["GeneralUtilities`"]; 
PrintDefinitions@Charting`FindTicks
 

Thanks to https://mathematica.stackexchange.com/questions/132568/extract-ticks-from-plot

The above works if the function is not read protected.

See also this link

123 rewrite ODE as \(y''=\text {RHS}\)

Given y'[x]+3 Sin[x]-3-Cos[x]-2 y[x]==0 how to rewrite it to be y'[x]==3+Cos[x]-3 Sin[x]+2 y[x] i.e. put y'[x] on one side, and the rest on the other side?

ClearAll[f,x,y] 
expr=y'[x]+3 Sin[x]-3-Cos[x]-2 y[x]==0; 
lhs=expr/.lhs_==rhs_:>lhs; 
rhs=expr/.lhs_==rhs_:>rhs; 
rhs=-(lhs/.(y'[x]+any_.):>any)+rhs; 
expr=y'[x]==rhs
 

This makes it easier to make it in form \(y'(x)=f(x,y)\).

124 convert code from stackexchange to 2D math

sometime code is posed at Mathematica stacexchange which is hard to read. To convert it to clear 2D math code, copy into a mathematica cell (in an open notebook) on the computer, then do

CTRL-SHIFT-N

125 How to trace DSolve running?

Use this

Block[{DSolve`print=Print}, 
    DSolve[{ode,ic},y[t],t] 
]
 

126 How to put time out on integrate in DSolve?

see https://mathematica.stackexchange.com/questions/120364/why-cant-dsolve-find-a-solution-for-this-ode/120650#120650 by Michael E2.

ClearAll[withTimedIntegrate]; 
SetAttributes[withTimedIntegrate, HoldFirst]; 
withTimedIntegrate[code_, tc_] := Module[{$in}, 
   Internal`InheritedBlock[{Integrate}, 
    Unprotect[Integrate]; 
    i : Integrate[___] /; ! TrueQ[$in] := 
     Block[{$in = True}, 
      TimeConstrained[i, tc, Inactivate[i, Integrate]] 
      ]; 
    Protect[Integrate]; 
    code 
    ] 
   ]; 
 
withTimedIntegrate[{dsol} = DSolve[ode == 0, y, x], 1]; // AbsoluteTiming 
dsol
 

127 How to replace subexpression with condition?

Given \[ a \sqrt {z}+\sin \left (c \sqrt {r+4}+20\right )+\frac {e^{-c} \sqrt {9 e^{2 c} x^2+8}-x+99}{4 x}+4 \sqrt {h} e^y-99 \] Replace only the subexpression of the form  anything*Exp[anyting]*Sqrt[anything] by the square of the pattern found if any.

expr /. patternName : (Exp[any3_]*any_.)*Power[Any_, Rational[1 | -1, 2]] -> patternName^2
 

This gives \[ a \sqrt {z}+\sin \left (c \sqrt {r+4}+20\right )+\frac {e^{-2 c} \left (9 e^{2 c} x^2+8\right )-x+99}{4 x}+16 h e^{2 y}-99 \]

The code  patternName : above gives a name for the pattern found, this way we can use the name to easily do any transformation needed. In this example, it was just squaring it. But it can be anything else we want.

128 How to set up a function with optional arguments?

Optional arguments are very important and useful. It is easy to make these in Mathematica. lets say we want to make function foo that takes person name and age but the sex is provided as optional with default value as say male. To do this, two things are needed to be defined.

The function foo itself, but with additional argument called opt : OptionsPattern[{foo}] and we need to define an Options[foo] =... which sets up the default values. Here is an example

Clear["Global`*"]; 
 
(*this sets up the DEFAULT values for each optional argument *) 
Options[processPerson] = {"sex" -> "male"}; 
 
(*this is the definition of the function itself*) 
processPerson[name_String, age_?NumericQ, opt : OptionsPattern[{processPerson}]] /;  age > 0 := Module[{sexOfPerson}, 
 
  (* the following line reads the optional value sex->"" if provided *) 
  (* else Mathematica will automatically return the default value set up*) 
 
  sexOfPerson = OptionValue["sex"]; 
 
  Print["Name = ", name]; 
  Print["age = ", age]; 
  Print["sex = ", sexOfPerson] 
  ]
 

Now we are ready to call the function.

processPerson["me",10]
 

Gives

     Name = me 
     age = 10 
     sex = male

We see in the above, that variable sex was automatically set to the default value thanks to the definitions Options[foo]=...

Now we call it with explicitly passing in the optional argument

processPerson["me",10,"sex"->"female"]
 

Gives

    Name = me 
    age = me 
    sex = female

In the above, the option argument sex->"..." must always be the last one passed in. You can not do this for example

processPerson["joe doe","sex"->"female",10]
 

Here is another example

Clear["Global`*"]; 
Options[dsolve] = {"ic" -> {}, "hint" -> "None"}; 
dsolve[ode_, y_[x_], x_, OptionsPattern[]] :=  Module[{ic = OptionValue["ic"], hint = OptionValue["hint"]}, 
   Print["ic=", ic, " hint=", hint] 
   ]; 
 
dsolve[y''[x] + y[x] == 1, y[x], x, "ic" -> {x[0] == 1}] 
dsolve[y''[x] + y[x] == 1, y[x], x] 
dsolve[y''[x] + y[x] == 1, y[x], x, "hint" -> "linear",  "ic" -> {x[10] == 0}]
 

Which prints

ic={x[0]==1} hint="None" 
ic={} hint="None" 
ic={x[10]==0} hint="linear"

To find all options supported by function do as an example Options[LinearSolve] and that will print {Method -> Automatic, Modulus -> 0, ZeroTest -> Automatic}

It is better to use string for the name of the hint as in the above, so not to clash with the same symbol being already defined.

129 checking optional values passed to function

The above showed how to use optional value. This below shows how to check and what to do if an optional value was not what is expected. This is done by explicitly checking, after reading the optional value, that its value is one that is expected. Like this

Clear["Global`*"]; 
Options[processPerson]={"sex"->"male"}; 
 
processPerson::msg="`1`"; 
 
processPerson[name_String,age_?NumericQ,opt:OptionsPattern[{processPerson}]]/;age>0:=Module[{sex}, 
    sex = OptionValue["sex"]; 
 
    If[Not[MemberQ[{"male","female"},sex]], 
       Message[processPerson::msg,"Invalid value for sex found. "<>sex<>" but expected one of {\"male\", \"female\"}"]; 
       Abort[] 
    ]; 
 
    Print["Name = ",name]; 
    Print["age = ",age]; 
    Print["sex = ",sexOfPerson] 
]
 

Now we are ready to call the function. Lets call it with bad value for the optional parameter

processPerson["me",10,"sex"->"car"] 
 
    processPerson::msg: Invalid value for sex found. car but expected one of {"male", "female"} 
    $Aborted
 

130 post Mathematics code with Greek letters

Copy the Mathematica code from the notebook, paste it into http://steampiano.net/msc/ and click convert. Then copy the output and paste that into the post at stackexchange.

131 How to find all variables in expressions?

Clear["Global`*"]; 
expr = y[z] + x + Pi + m/Sin[a] + b[0]; 
DeleteDuplicates@Cases[expr, any_ /; Head[any] === Symbol && Not[MemberQ[Attributes[any], Constant]], Infinity]
 

Gives {x,n,a,x}. Note that the above does not detected indexed variables, such as \(b[0]\). The Attributes check is so not to include \(\pi \) since this is also a symbol. But we do not want to be there.

132 How to find the order of an ODE?

Use

Clear["Global`*"]; 
ode=y''[x]+3*y'[x]+3==0; 
Internal`ProcessEquations`DifferentialOrder[ode,{x},{y}] 
 
(*   {{2}} *)
 

So it is order 2. Notice the input must be lists [ode,{x},{y}], it will hang if you use [ode,x,y]

133 How to find the degree of an ODE?

Degree of an ode is the degree of the highest derivative in the oder. This needs a helper function

(*This function thanks to  Carl Woll, see https://mathematica.stackexchange.com/questions/151850/using-cases-and-when-to-make-input-a-list-or-not*) 
Clear["Global`*"]; 
getPatterns[expr_, pat_] :=   Last@Reap[expr /. a : pat :> Sow[a], _, Sequence @@ #2 &];
 

And now do

getDegreeOfOde[ode_,y_,x_]:=Module[{maxDer,p,d,der}, 
    der    = getPatterns[ode,Power[Derivative[_.][y][x],_.]] ; 
    p      = Flatten[Internal`ProcessEquations`DifferentialOrder[#,{x},{y}]&/@der]; 
    maxDer = Extract[der,Position[p,Max[p]]]; 
    Abs[First[maxDer/.(Derivative[_.][y][x])^(d_.):>d]] 
]
 

Examples of usage

ode=y'''[x]+y''[x]^4+3*y'[x]^8+3*y[x]^8==0; 
getDegreeOfOde[ode,y,x] 
   (*1*) 
 
ode=y''[x]^4+3*y'[x]^8+3*y[x]^8==0; 
getDegreeOfOde[ode,y,x] 
   (*4*) 
 
ode=y''[x]+3*y'[x]^8+3*y[x]^8==0; 
getDegreeOfOde[ode,y,x] 
   (*1*)