Mathematica for Physicits

Useful knowledge on Mathematica®
for Physicists
Miroslav Mihaylov
UIC Physics Feb-17-2016

The notebook inrerface and the
kernel
◼ The user enters input in the notebook
◼ The input is sent to the Math Kernel via Shift+Enter or Numpad Enter
◼ The kernel performs the computation ans send the result back to the frontend for display
Frontend
Integrate[Exp[-x^2],x]
π
2
Erf[x]
Math Kernel

◼ MathLink the protocol for comunication of the frontend with the Kernel and other
2 PresnetationSlides-Feb-Apr-2016.nb

Lists {a1, a2, ..., an}
◼ Generating lists with Table[]
Table expr ,  i , i min , i max ,
Table expr ,  i , i min , i max , di , Table expr ,  i ,  i 1 , i 2 , … 
Tablei, i, 10, 20, 2
{10, 12, 14, 16, 18, 20}
Tablei, 2×i, i, 10, 20, 2
{{10, 20}, {12, 24}, {14, 28}, {16, 32}, {18, 36}, {20, 40}}
Table10i+j, i, 4, j, 3
{{11, 12, 13}, {21, 22, 23}, {31, 32, 33}, {41, 42, 43}}
◼ Generating lists with Range[]
Range i max , Range i min , i max , Range i min , i max , di 
Range[5]
{1, 2, 3, 4, 5}
Range2, 5, 0.5
{2., 2.5, 3., 3.5, 4., 4.5, 5.}
◼ Table with Random
TableRandom [], 3
{0.929755, 0.0990263, 0.960832}
RandomInteger 10, 4, 3
{{5, 1, 9}, {4, 3, 10}, {6, 9, 3}, {6, 3, 2}}
PresnetationSlides-Feb-Apr-2016.nb 3

Lists Representations
◼ Usefull for easy inspection of the elemetns of the list
Table10i+j, i, 4, j, 3 // TableForm
11 12 13
21 22 23
31 32 33
41 42 43
0, -ⅈ, ⅈ, 0 // MatrixForm

0 -ⅈ
ⅈ 0

MatrixForm [{a1, a2}]

a1
a2


Programing in Mathematica
Say we have the list bellow and we would like to reverse the elements of each item in the list
lis= {α, 1}, β, 2, γ, 3;
With procedural programing we would write
tempLis = lis;
Fori = 1, i ≤ Lengthlis, i++,

tempLis i, 1, tempLis i, 2 = lisi, 2, lisi, 1

;
tempLis
{{1, α}, {2, β}, {3, γ}}
Slightly better approach
Table lisi, 2, lisi, 1,
i, 1, Lengthlis

{{1, α}, {2, β}, {3, γ}}

The Mathematica Way
For each element of the list we want to reverse
MapReverse, lis
{{1, α}, {2, β}, {3, γ}}
Same as above written in an alternative syntax
Reverse/@lis
{{1, α}, {2, β}, {3, γ}}

Map[] /@ and Apply[] @@
◼ Map applies a function to each element in a list
Mapf, a, b, c
{f[a], f[b], f[c]}
◼ Apply changes the structure of the expression
Applyf, a, b, c
f[a, b, c]
ApplyPlus, a, b, c, d
a+b+c+d
ApplyPlus, a, b, c, {α, β, γ}
{a+α, b+β, c+γ}

Slots # with functions & and user-
defined functions
◼ #&
Unset[x];
(* Function3+#[x] *)
3+# &[x]
3+x
#^2+#^4 &[x]
x2
+x4
#1^2+#2^4 &[x, y]
x2
+y4
(* Map#^2+#^4&,{x,y,z} *)
#^2+#^4 &/@{x, y, z}
x2
+x4
, y2
+y4
, z2
+z4

◼ User-defined functions
f[x_] := Sin[πx]
f0.5
1.

Operations On Lists
varLis= Table10i+j, i, 4, j, 3
{{11, 12, 13}, {21, 22, 23}, {31, 32, 33}, {41, 42, 43}}
SelectvarLis, #2 < 22 &
{{11, 12, 13}}
GathervarLis, Abs#1[[1]]-#2[[1]] ⩵ 20 &
{{{11, 12, 13}, {31, 32, 33}}, {{21, 22, 23}, {41, 42, 43}}}
#[[1]], #3 &/@varLis
{{11, 13}, {21, 23}, {31, 33}, {41, 43}}
No loops!

There is a Function for that,...
◼ If you wanted to bo something in Matheatica
chances are that there exists a built in function called
DoSomething[]
Solve expr , vars , FindRoot lhs == rhs ,  x , x0 , FindPeaks data , σ ,
EstimatedBackground  data , σ  , Gather list , test , Select list , crit ,
GroupBy elem 1 , elem 2 , … , spec , red , Split list , test ,
ImageConvolve  image , ker , ListConvolve ker , list , k , ImageData  image , type ,
Convolve f , g , x , y , FourierTransform  expr , t , ω , Fold f , x , list ,
NDSolve eqns , u ,  x , xmin , xmax , NIntegrate f ,  x , xmin , xmax ,
Evaluate expr , FourierSeries expr ,  t1 , t2 , … ,  n1 , n2 , … , ListPlot y1 , y2 , … ,
ListLinePlot y1 , y2 , … , ListLogPlot y1 , y2 , … , Plot f ,  x , xmin , xmax ,
TimeSeriesModelFit data , mspec , NonlinearModelFit data ,  form , cons ,  β1 , … ,  x1 , … 
◼ Your core logic should not take more than a
single line.

Math Input Shortcuts
Switching between input form and Standard Form for selected cell in the notebook menu select
Cell-> Convert to-> StandardForm or the shortcut Shift+Ctr+N
Input form Standard Form Keyboard Shortcut
x^2 xn
x Ctr-[6] n
Subscript[x, j] xj x Ctr+_ j
Sqrt[x] x x Ctr+[2] x
x/n x
n
x Ctr+/ x

Shortcuts Continued
Navigating throight different placeholders of the exoressins with the Tab
To exit the Ctrl + Space
Input form Standard Form Keyboard Shortcut
Integrate[, {, , }] ∫

ⅆ Esc-dintt-Esc
Sum[x, {, , }] ∑=
  Esc-sumt-Esc
Product[, {, , }] ∏=
  Esc-prodt-Esc
You can access the input palettes from the notebook menu Palletes->Basic Math Assistant

Special Characters Shortcuts
Standard
Form
Keyboard
Shortcut
Notes
π Esc-pp-Esc The Pi 3.1415..
ⅇ Esc-ee-Esc The natural exp
ⅈ Esc-ii-Esc The imaginaryunit
∞ Esc-inf-Esc Infinitysymbol
ϵ Esc-e-Esc Greek epsilon
η Esc-et-Esc Greek etha
Similarly for the upper case greek letters Esc-G-Esc wouth output Γ Δ Ω

Speed Math Typing Demo

Importing data from XLS files
SetDirectoryNotebookDirectory[];
data = Import "table3InLab1.xls"[[1]];
data // TableForm
L1 Delta L1 V1 L2 Delta L2 V2 V
589.041 0.046 23.4279 589.635 0.043 21.8779 22.6529
589.051 0.056 28.5205 589.654 0.062 31.5439 30.0322
589.05 0.055 28.0112 589.644 0.052 26.4566 27.2339
589.03 0.035 17.8259 589.629 0.037 18.8254 18.3257
589.001 0.006 3.05602 589.603 0.011 5.59699 4.3265
588.983 -0.012 -6.11223 589.58 -0.012 -6.10604 -6.10914
588.964 -0.031 -15.7904 589.56 -0.032 -16.2833 -16.0369
588.964 -0.031 -15.7904 589.562 -0.03 -15.2656 -15.528
588.976 -0.019 -9.67781 589.579 -0.013 -6.61489 -8.14635
589.005 0.01 5.09334 589.604 0.012 6.10579 5.59956
589.033 0.038 19.3538 589.63 0.038 19.3342 19.344
dataV = Tablej-1, dataj+1[[7]], j, 1, Lengthdata-1;

Nonlinear Model Fit
F[Vcm_ , Vo_, ϕ_, T_] := Vcm +Vo×Sinϕ+
2π
T
t;
nlm = NonlinearModelFitdataV, F[Vcm , Vo, ϕ, T], {Vcm , Vo, ϕ, T}, t;
pl = PlotNormal nlm , t, 0, 11;
lp = ListPlotdataV, ImageSize → 600, Frame → True,
Axes → False, FrameTicksStyle → DirectiveThick, 20,
PlotStyle → DirectiveRed, PointSize.02;
Showlp, pl
nlm "BestFitParameters "
0 2 4 6 8 10
-10
0
10
20
30
{Vcm → 6.59807, Vo → 23.1787, ϕ → 2.35962, T → 1.1067}

Set initial conditions for the fit
nlm = NonlinearModelFitdataV, F[Vcm , Vo, ϕ, T], Vcm , Vo, ϕ, T, 10, t;
pl = PlotNormal nlm , t, 0, 11;
lp = ListPlotdataV, ImageSize → 600, Frame → True,
Axes → False, FrameTicksStyle → DirectiveThick, 20,
PlotStyle → DirectiveRed, PointSize.02;
Showlp, pl
Totalnlm "FitResiduals"
0 2 4 6 8 10
-10
0
10
20
30
{Vcm → 6.59807, Vo → 23.1787, ϕ → 0.781968, T → 10.372}
-9.9476×10-14

Determining the goodness of fit
nlm "ANOVATable"
DF SS MS
Model 4 3513.73 878.433
Error 7 5.11795 0.731136
Uncorrected Total 11 3518.85
Corrected Total 10 2912.13
nlm "FitResiduals"^2 // Total
5.11795
ListPlotnlm "FitResiduals", Filling→ Axis
2 4 6 8 10
-0.5
0.5
1.0
1.5
5.91404

Fitting a subset of parameters
◼ Somtimes we might need to impose restrinctions on
some of the parameters in our model
F[Vcm , Vo, ϕ, T] /. Vo → 23.05, ϕ → 0.81
Vcm +23.05Sin0.81+
2πt
T

◼ Fitting only the subset of parameters while keeping
the other ones fixed
nlm = NonlinearModelFitdataV,
F[Vcm , Vo, ϕ, T] /. Vo → 23.05, ϕ → 0.81, Vcm , T, 10, t;
{Vcm → 6.68274, T → 10.4396}
◼ Getting our final model
Normal nlm  /. Vo → 23.05, ϕ → 0.81
6.68274+23.05Sin[0.81+0.601862t]

Dynamic Interactivity, Panels,
Controllers and Something else

Manipulate[]
Manipulate
PlotSin[x(1+ax)], x, 0, 2π
, a, 0, 2
a
1 2 3 4 5 6
-1.0
-0.5
0.5
1.0
ManipulateSolute,
Solute, "H2O", "DCE", "Hexane", "Toluene", "Dodecane"

Solute H2O DCE Hexane Toluene Dodecane
Hexane

Manipulate with different
ControlType
ManipulateSolute,
Solute, "H2O", "DCE", "Hexane", "Toluene", "Dodecane", ControlType -> PopupMenu

Solute Toluene
Toluene
Manipulatey, y, 0, 1, ControlType → Slider2D
y
0
◼ Some of the many possible values for ControlType
Animator  u ,  umin , umax , Checkbox x ,
InputField x , RadioButtonBar x ,  val1 , val2 , … 

Working with dynamically updated
content
◼ Dynamic[expr ]
Controlx, 0, 0, {1, 1}
Dynamic [x]
{0., 0.}
Dynamic GraphicsLocatorDynamic [x], PlotRange → 1, Dynamic [x]
 , {0., 0.}
◼ DynamicModule[{x = x0 , y = y0 , … }, expr ]
Maintains a local instance of x,y,.. and preserves their values across notebook sessions

DynamicModule x = 0, 0, GraphicsLocatorDynamic [x], PlotRange → 1, Dynamic [x]
 , {0.0111111, 0.244444}

Manipulate vs DynamicModule
Manipulate
a×b,
a, 6, "Label for A", FieldSize→ 6,
b, 7, "Label for A", FieldSize→ 6

Label for A 6
Label for A 7
42
DynamicModule a = 6, b = 7,
PanelColumn 
"Label A", InputFieldDynamic [a], Number , ContinuousAction→ True,
"Label B", InputFieldDynamic b, Number ,
"A×B", InputFieldDynamic a×b



Label A
6
Label B
7
A×B
42

Example Standalone Panel
Manipulate
ModuleElDenCalc,
ElDenCalcρo_, epmo_ , molwto_  :=
ρoepmo 6.0221415×1023
molwto 1024
;
IfSolute ⩵ "DCE", ρ = 1.24889, epm = 50, molwt = 98.9592;
IfSolute ⩵ "H2O", ρ = 0.99754, epm = 10, molwt = 18.0152;
IfSolute ⩵ "Toluene", ρ = 0.839, epm = 50, molwt = 92.1402;
IfSolute ⩵ "Hexane", ρ = 0.6548, epm = 50, molwt = 86.1754;
StyleElDenCalcρ, epm , molwt , Red, Bold, FontSize→ 22
,
Solute, "H2O", "DCE", "Toluene", "Hexane", ControlType -> PopupMenu,
ρ, .997, "ρmass ", FieldSize→ 6,
epm , 10, "e-
/mol ", FieldSize→ 6, molwt , 18.0152, "mw ", FieldSize→ 6
, FrameLabel →
"", "", Tooltip"Electron Density ", Style"
ρepm NA
mw 1024
", Blue, FontSize→ 30
Electron Density
Solute DCE
ρmass 1.24889
e-
/mol 50
mw 98.9592
0.380005

Another Example Aplication
Manipulate
Module{},
(*Display *)
Column 
Row"Sign Label:", sign, " checkbox is:", cx,
Row"Slider Valie:", SliderVar,
Row" min :", minVar , " max :", maxVar 

, (*END Module*)
(*BEGIN Manipulate Controls *)
sign, "+1", "-1",
SliderVar, 2, "Slider Label", 0, 5
, cx, 1, "Check Label", 0, 1,
minVar , 0.017, maxVar , 0.03

sign +1 -1
Slider Label
Check Label
minVar 0.017
maxVar 0.03
Sign Label:+1 checkbox is:1
Slider Valie:2
min :0.017 max :0.03

A complete example
Fit
C 10
A 15
ϕ 0.5
T 4
0 T->4 T chk->False
Model usemodel
Fit Res res
0 2 4 6 8 10
-10
0
10
20
30

All work no play makes Jack a dull
boy
◼ Game Controller integration
ControllerInformation []
Controller Device 1: Logitech Logitech Dual Action
Raw Product Name "Logitech Logitech Dual Action "
Device Type Linux Joystick Device
Raw Controller Type Joystick
Mathematica Controls 36 controls
X $Failed
Y $Failed
Z $Failed
X1 $Failed
Y1 $Failed
Z1 $Failed
X2 $Failed
Y2 $Failed
X3 $Failed
Y3 $Failed
X4 $Failed
Y4 $Failed
X5 $Failed
Y5 $Failed
X6 $Failed
B1 $Failed
B2 $Failed
B3 $Failed
B4 $Failed
B5 $Failed
B6 $Failed
B7 $Failed
B8 $Failed
B9 $Failed
B10 $Failed
B11 $Failed
B12 $Failed
BLB $Failed
BRB $Failed
JB $Failed
JB1 $Failed
JB2 $Failed
Select Button $Failed
Start Button $Failed
TLB $Failed
TRB $Failed
Show Dynamic Values
Raw Controls 18

Dynamic ControllerState"XY", "B1"

Computing without the frontend
◼ Create an ASCII file scriptname.m
#!/usr/local/bin/MathematicaScript -script
<<myPackageWmyFunctions`;
a=22;b=34;
For[i=0, i<30, i++,
For[j=0, j<30, j++,
For[k=0, k<30, k++,
(*Do the work here and print some results*)
parA=j*a;parB=k*c
SomeResult=MyFunc[parA,parB]
Print[l,” “, i,” “, j,” “, k,SomeResult];
]
]
]
◼ From the command line run
$math -run <scriptname.m

Calling Exteranl programs
◼ Opening system program
Run"notepad"
Run"calc"
◼ Executing command line script and outputing the
result to a file
Run"awk -f AWK4math /myawkscript .awk "<>filename
" > tmplocal /fileprefix_"<>fillevar<>".dat"
◼ Open the specified targeth path with the default
program
SystemOpen "phys.uic.edu"
SystemOpen "filename .docx"

Interacting with MySQL database
◼ Connect to the database
Needs"DatabaseLink`";
JDBCDrivers"MySQL(Connector/J)";
conn = OpenSQLConnectionJDBC"MySQL(Connector/J)", "localhost/test_db",
"Username " → "root", "Password" → ""
◼ Select from the database
SQLSelectconn, "table_name ", "MaxRows" → 4, "ShowColumnHeadings " → True // TableForm
◼ Inserting
SQLInsertconn, "table_name ", "col_a", "col_b", 3, 10.5

And many more applictions
THANK YOU

Mathematica for Physicits

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