Compare Prefactors for Nondimensionalized Formulas

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Remove physical dimensions from formulas on the left or right hand sides and compare the remaining prefactors

Get the dimensionless equation for Coulomb's Law. Note that 4π is a prefactor, a coefficient that precedes a variable or component of a formula:

In[1]:=

NondimensionalizationTransform[First@FormulaData["CoulombsLaw"],{

},{f,q1,q2,ρ}]//.a_?NumericQb_Style[a,Blue]b

Out[1]=

f(4π)

q1q2

Get a list of all available formulas:

In[2]:=

entities=FormulaData[];

Obtain the equation for all formulas:

In[3]:=

formulas=FormulaData/@entities;

Select formulas that have the form of a variable equal to a combination of

Quantity

and

QuantityVariable

objects:

In[4]:=

formulas=Cases[Transpose[{entities,formulas}],{n_,f:Equal[_QuantityVariable,_]}];formulas=Cases[formulas,{n_,f_?(FreeQ[#,Plus]&)}{f,FormulaData[n,"QuantityVariables"]}];

Calculate the dimensionless forms of the equations:

In[5]:=

sol=Quiet[Check[NondimensionalizationTransform[#〚1〛,#〚2,;;-2〛,Table[C[i],{i,Length[#〚2〛]-1}]],Nothing]]&/@formulas;Length[sol]

Out[5]=

271

Remove the new variable and any placeholders for introduced quantities as well as any factors of exactly 1:

In[6]:=

prefactors=DeleteCases[sol〚All,1〛/.{C[1]1,K[_]1},1];

Visualize the distribution of smaller prefactor values:

In[7]:=

Histogram[Normal/@Quiet[Select[prefactors,#<10^2&]],{1}]

Out[7]=

Examine the 10 most frequent prefactors:

In[8]:=

Grid[Prepend[(Flatten[{First[#],N[First[#]],Last[#]}]&/@Reverse@SortBy[Tally[prefactors],Last])〚;;10〛,{"prefactor","numerical value","frequency"}],FrameAll]

Out[8]=

prefactor	numerical value	frequency
2	2.	15
1 π	0.31831	14
1 2π	0.159155	10
2π	6.28319	7
4 π	1.27324	6
Sec[1]	1.85082	4
25000000000000/207861565453831 	1.1278	4
1 2	0.5	4
3 4π	0.238732	3
1 4π	0.0795775	3

Display all prefactors, sized by frequency:

In[9]:=

WordCloud[prefactors]

Out[9]=

Related Symbols

NondimensionalizationTransform
FormulaData
QuantityVariable
SpectralLineData
PlanckRadiationLaw
PsychrometricPropertyData
ThermodynamicData

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Contributed by: Wolfram Staff