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My Mathematica cheat sheet
October 10, 2018 Compiled on October 10, 2018 at 3:12pm
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
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Archive of Mathematica weekly newsletters
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Mathematica stackexchange forum
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The large links page for Mathematica at stackexchnage
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Language reference
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What is new in V 10
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Summary of New Features in 10
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wolfram cloud
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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?
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How to find what contexts are loaded?
?$Packages
”gives a list of the contexts corresponding to
all packages which have been loaded in your
current Mathematica session.” |
{"JLink`", "GetFEKernelInit`", "ResourceLocator`",
"PacletManager`", "QuantityUnits`", "WebServices`",
"System`", "Global`"}
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How to find what packages are loaded?
$ContextPath
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{"PacletManager`", "QuantityUnits`", "WebServices`",
"System`", "Global`"}
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Finding packages in specific context
Names["System`ComplexExpand`*"]
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{"System`ComplexExpand`AbsExpr","System`ComplexExpand`ArgExpr",
"System`ComplexExpand`ConjugateExpr",
"System`ComplexExpand`ReImExpr",
"System`ComplexExpand`ReImFail",
"System`ComplexExpand`SignExpr"}
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How to find what contexts are loaded?
?Contexts
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Contexts[] gives a list of all contexts.
Contexts["string"] gives a list of the contexts
which match the string.
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8 How to load an m file or a package?
Append to the Path the folder name where the package is located in. In this example, assuming
there is a package control.m located in folder C:\data then type the following to load the
package
9 Finding names of functions in a package
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$Packages |
{"JLink`", "GetFEKernelInit`", "ResourceLocator`", "PacletManager`",
"QuantityUnits`", "WebServices`", "System`", "Global`"}
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Names["JLink`*"] |
{"JLink`AddPeriodical", "JLink`AddToClassPath",...
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10 Finding a package that has specific function
use Context.
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Context[Integrate] | "System`" |
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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
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To insert I |
esc ii esc |
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To enter \(\pi \) |
esc p esc |
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To enter E | esc ee esc |
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ctrl-6 wil make exponent | |
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ctrl^ will make superscript |
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15 Tracing a function
Trace[Integrate[x, {x, 1, 2}], TraceInternal -> True]
16 Removing Big O notation from Series expansion
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f[0]+x (f^')[0]+1/2 x^2 (f^'')[0]+1/6 x^3 (f^(3))[0]
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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 \(\frac{dz}{ds}\)
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(Cosh[t]*Sin[s])/E^Cos[s]
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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 \(\frac{du}{dt}\)
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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
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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
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Pi/2 - Sin[2*n*Pi]/(4*n) |
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| Pi/2 |
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22 Smart replacement everywhere?
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(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
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23 Plotting real and imaginary parts
24 Plotting mapping of complex numbers?
25 Saving a plot as eps
26 Extracting DSolve solutions
one way
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{{y[x] -> -Sin[x]}, {y[x] -> Sin[x]}}
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y[x] /. % |
{-Sin[x],Sin[x]} |
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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
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Out[69]= {{55, 63, 78, 45}, {13, 45, 67, 1},
{94, 32, 48, 90},
{31, 75, 43, 60}}
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Out[70]= {55, 13, 94, 31}
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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
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
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
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;aOut[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
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
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}]
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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]
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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]
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See also how-to-draw-a-spring
44 How to list files in a directory?
SetDirectory[$BaseDirectory];FileNames["*"]
45 Where is init.m and how to use it?
Possible locations for init.m files include the following:
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$BaseDirectory/Kernel kernel initialization code for all users
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$UserBaseDirectory/Kernel kernel initialization code for the currently logged-in user
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$BaseDirectory/FrontEnd front end initialization code for all users
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$UserBaseDirectory/FrontEnd front end initialization code for the currently logged-in
user
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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]}]
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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 when writing Mathematica demonstration
In math italicize single Roman letters that are variables orfunctions (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[{ }]
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]]
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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.
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(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
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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.
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Remember that Dynamic has the effect of delaying evaluation until the expression reaches the
front end
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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
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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.
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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.
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The most important is the fact that values of all DynamicModule variables are saved in the file
when the notebook is saved.
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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").
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Dynamic outputs are only updated when they are visible on screen.
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Remember to add synchorization->False to all dynamics, else will time out. When using
Refresh also.
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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
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Ok, Found out that finishDynamic[] can causes annoying refresh on the UI whenever I move
sliders. So removed it.
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Remember to use :> and not -> for TrackedSymbols
57.1 Notes from John Fultz collected on dynamics
Module variables should *never* appear inside Dynamics orManipulates 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
-
struct-data-type-in-mathematica
-
setting-up-a-struct-in-mathematica-safely
-
using-a-struct-inside-manipulate-to-help-manage-control-variables-how-to-
initia
-
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]];xOut[217]= 3
Module:
Remove["Global`*"]x = 4;Module[{}, If[x == 4, x = 3]];xOut[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.
-
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.
-
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.
-
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.
-
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
-
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.
-
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]]} & ) /@ lstOut[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 Difference between 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]= Listlst[[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
We could have used the zero index trick, but it is better to use Apply:
lst[[0]] = PlusOut[245]= PluslstOut[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 & ) /@ lstOut[249]= {6, 120, 504}
or, little shorter version of the above:
(Times @@ #1 & ) /@ lstOut[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 + dOut[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 or email messages?
82 Making automatic subscripted matrix notation
Format[t_a] := Subscripted[t]
t = Table[a[i, j], {i, 2}, {j, 3}]
|
|
MatrixForm[t] |
|
|
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)
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] & )] := xfoo[x_Integer] := xfoo[x_Integer] := x
88.1.2 integer strictly positive
foo[(x_)?(IntegerQ[#1] && #1 > 0 & )] := xfoo[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 & )] := xfoo[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 & )] := xfoo[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 & )] := xfoo[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] &)] := xfoo[x_Real] := x
88.1.8 Real strictly positive
foo[x_Real /; x > $MachineEpsilon] := xfoo[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] := xfoo[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] := xfoo[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] := xfoo[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] &)] := xfoo[x_?(NumericQ[#] &)] := xfoo[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] := xfoo[x_?(Not@FreeQ[#, _Complex] &)] := x
88.1.17 list of any dimension, ragged lists, 1D vectors, 2D, any content
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] &)] := xfoo[x : {_?NumericQ ..}] := x
foo[x : {__?NumericQ }] := x
foo[x_?(VectorQ[#, IntegerQ] &)] := x
88.1.21 2D matix of numbers
foo[x_?(MatrixQ[#, NumericQ] &)] := xfoo[x : {{_?NumericQ ..}}] := xfoo[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]= TrueMatchQ[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]= TrueMatchQ[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 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
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 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
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.....
-
ref/ImagePadding.html
-
tutorial/GridsRowsAndColumns.html
-
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:
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^2/2In[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 How to 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 How to save matrix of data to file, and read it again later?
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 How to run Mathematica m file as script on windows?
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? This plugin is not supported issues
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 How to use Mathematica to get step-by-step solution from Alpha
WolframAlpha["Integrate[x Sin[x],{x,0,Pi}]", {{"Input", 2},
"Content"}, PodStates -> {"Input__Step-by-step solution"}]
also try
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"}]
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 \(e^{s(x)}\)
Gives \(x=s^{'''}(x)+s'(x)^3+3 s'(x) s''(x)\)
122 How to print definitions of some internal functions?
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 How to rewrite ODE so that derivative on one side and rest on another side?
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?
This makes it easier to make it in form \(y'(x)=f(x,y)\) so analyze the ode.
124 How to convert code from Mathematica stacexchange to 2D math in notebook?
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 opne notebook) on the computer, then
do
CTRL-SHIFT-N
125 How to trace DSolve running?
Use this
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.
127 How to post Mathematics code with Greek letters to stackexchange?
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.