Krull–Schmidt category: Difference between revisions

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'''Rayleigh's method of dimensional analysis''' is a conceptual tool used in [[physics]], [[chemistry]], and [[engineering]]. This form of [[dimensional analysis]] expresses a [[functional relationship]] of some [[variable (mathematics)|variables]] in the form of an [[exponential equation]]. It was named after [[John Strutt, 3rd Baron Rayleigh|Lord Rayleigh]].
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The method involves the following steps:
 
# Gather all the [[independent variable]]s that are likely to influence the [[dependent variable]].
# If ''R'' is a variable that depends upon independent variables ''R''<sub>1</sub>,&nbsp;''R''<sub>2</sub>,&nbsp;''R''<sub>3</sub>,&nbsp;...,&nbsp;''R''<sub>''n''</sub>, then the [[functional equation]] can be written as ''R'' =&nbsp;''F''(''R''<sub>1</sub>,&nbsp;''R''<sub>2</sub>,&nbsp;''R''<sub>3</sub>,&nbsp;...,&nbsp;''R''<sub>''n''</sub>).
# Write the above equation in the form <math>X = C X_1^a X_2^b X_3^c \cdots X_n^m \,</math> where ''C'' is a [[dimensionless constant]] and ''a'',&nbsp;''b'',&nbsp;''c'',&nbsp;...,&nbsp;''m'' are arbitrary exponents.
# Express each of the quantities in the equation in some [[fundamental unit]]s in which the solution is required.
# By using [[Dimensional analysis#Commensurability|dimensional homogeneity]], obtain a [[set (mathematics)|set]] of [[simultaneous equations]] involving the exponents&nbsp;''a'',&nbsp;''b'',&nbsp;''c'',&nbsp;...,&nbsp;''m''.
# [[Equation solving|Solve]] these equations to obtain the value of exponents&nbsp;''a'',&nbsp;''b'',&nbsp;''c'',&nbsp;...,&nbsp;''m''.
# [[Simultaneous equations#Substitution method|Substitute]] the values of exponents in the main equation, and form the [[non-dimensional]] [[parameter]]s by [[Combining like terms|grouping]] the variables with like exponents.
 
'''Drawback''' – It doesn't provide any information regarding number of dimensionless groups to be obtained as a result of dimension analysis
 
== See also ==
* [[Physical quantity]]
* [[Buckingham pi theorem]]
 
[[Category:Dimensional analysis]]
 
 
{{applied-math-stub}}

Latest revision as of 13:57, 5 June 2014

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