Number of Dimensionless Groups of a Physical System

In a well posed physical problem described by well formed equations, the solution may be written in terms of dimensionless groups. The number of dimensionless groups. The number of dimensionless groups is less than the number of parameters and variables in the system.
Let the relevant physical quantieis - surface tension, viscosity, thermal conductivity etc - be  
\[P_1, \: P_2, \: ... , \: P_n\]
  and let the fundamental quantities - time, length, temperaure etc - be  
\[m_1, \: m_2, \: ... , \: m_k\]
.
We can write  
\[P_j=m_1^{a_{1j}}m_2^{a_{2j}}...m_k^{a_{kj}} , \: j=1, \:2, \: ..., \: n\]
.
We can write a dimensional matrix  
\[A= \{a_{ij} \}\]
.
Considered as a vector space, if the number of linearly independent vecors is  
\[r\]
, each of the remaining  
\[n-r\]
  vectors can be expressed as a linear combination of the  
\[r\]
  independent vectors, so  
\[ P_j = \sum_{i=1}^r w_{ij} P_i , \: j=r+1, \: r+2, \: n\]
  where the  
\[w_{ij}\]
  are constants.
In terms of the physical quantities this becomes  
\[P_j = \prod_{i=1}^r P_i^{w_{ij}}\]
. There will be  
\[n-r\]
  dimensionless groups. This means that any function of all the  
\[m_i\]
  can be expressed in terms of  
\[n-r\]
  dimensionless  
\[P_j\]
.

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