## Dimensionless products and complete set of dimensionless products

Recall that in dimensional analysis the **product**^{(ibid. 3.)} is given by the expression

*y*=

*x*

_{1}

^{k1}⋅

*x*

_{2}

^{k1}⋅ … ⋅

*x*

_{n}^{kn}**dimensionless product**. In dimensional analysis the symbol

*π*is used as a convention to denote dimensionless products. It does not equate to the number ≈ 3.14159.

To be more precise, *π* is used to symbolize the complete set of dimensionless products *π _{i}*
of a function.

*π*

_{1}=

*x*

_{1}

^{k1}

^{'}⋅

*x*

_{2}

^{k1}

^{'}⋅ … ⋅

*x*

_{n}^{kn}^{'}

*π*

_{2}=

*x*

_{1}

^{k1}

^{''}⋅

*x*

_{2}

^{k1}

^{''}⋅ … ⋅

*x*

_{n}^{kn}^{''}

⋮

*π*=

_{p}*x*

_{1}

^{k1p}⋅

*x*

_{2}

^{k1p}⋅ … ⋅

*x*

_{n}^{knp}*π*= {

*π*

_{1},

*π*

_{2}, …,

*π*}

_{p}*y*=

*ƒ*(

*x*

_{1},

*x*

_{2}, …,

*x*)

_{n}=

*x*

_{1}

^{k1}⋅

*x*

_{2}

^{k1}⋅ … ⋅

*x*

_{n}^{kn}*k*

_{1},

*k*

_{2}, …,

*k*are a solution of the linear equations

_{n}*a*=

*a*

_{1}

*k*

_{1}+

*a*

_{2}

*k*

_{2}+ … +

*a*

_{n}k_{n}*b*=

*b*

_{1}

*k*

_{1}+

*b*

_{2}

*k*

_{2}+ … +

*b*

_{n}k_{n}⋮

*g*=

*g*

_{1}

*k*

_{1}+

*g*

_{2}

*k*

_{2}+ … +

*g*

_{n}k_{n}*a*,

*b*, …,

*g*are the the dimensional exponents.

*π*=

_{i}*x*

_{1}

^{k1i}⋅

*x*

_{2}

^{k1i}⋅ … ⋅

*x*is dimensionless if, and only if, the exponents

_{n}^{kni}*k*of the independent variables

_{j}^{i}*x*are the solution of

_{j}*a*

_{1}

*k*

_{1}

^{i}+

*a*

_{2}

*k*

_{2}

^{i}+ … +

*a*

_{n}k_{n}^{i}0 =

*b*

_{1}

*k*

_{1}

^{i}+

*b*

_{2}

*k*

_{2}

^{i}+ … +

*b*

_{n}k_{n}^{i}⋮

0 =

*g*

_{1}

*k*

_{1}

^{i}+

*g*

_{2}

*k*

_{2}

^{i}+ … +

*g*

_{n}k_{n}^{i}*j*= 1 to

*n*the total number of independent variables, and the coefficients

*a*,

_{j}*b*, …,

_{j}*g*are the dimensional exponents making up rows in the dimensional matrix of the

_{j}*x*'s.

**complete**if

*x*'s can be displayed in a matrix

x_{1} | x_{2} | … | x_{n} | |
---|---|---|---|---|

π_{1} | k_{1}' | k_{2}' |
… | k_{n}^{'} |

π_{2} | k_{1}'' | k_{2}'' |
… | k''_{n} |

⋮ | ⋮ | ⋮ | ⋮ | ⋮ |

π_{p} | k_{1}^{p} | k_{2}^{p} |
… | k_{n}^{p} |

## Independent dimensionless products

*π*

_{1},

*π*

_{2}, …,

*π*are said to be independent, if other than

_{p}*h*

_{1}=

*h*

_{2}= … =

*h*= 0 there are no constants

_{p}*h*

_{1},

*h*

_{2}, …,

*h*such that

_{p}*π*

_{1}

^{h1}⋅

*π*

_{2}

^{h2}⋅ … ⋅

*π*≡ 1

_{p}^{hp}*linear dependence*. This may be defined from the perspective of linear algebra. But before that we define the

*linear combination*of rows in a matrix as

*at least one row*that is a linear combination of other rows.

In the matrix of exponents, this means that there exists constants

*h*

_{1},

*h*

_{2}, …,

*h*where

_{p}*not all*are zero such that, for a given column the sum of the products of a row element by its corresponding constant is zero, and this is so for all columns. Symbolically this means,

*h*

_{1}

*k*' +

_{i}*h*

_{2}

*k*'' + … +

_{i}*h*= 0, for

_{p}k_{i}^{p}*i*= 1, 2, …,

*n*.

*π*

_{1},

*π*

_{2}, …,

*π*be independent is that the

_{p}*rows in the matrix of exponents be linearly independent*.

**Hypothesis:**Dimensionless products are independent. That is,

*π*

_{1}

^{h1}⋅

*π*

_{2}

^{h2}⋅ … ⋅

*π*≡ 1 iff,

_{p}^{hp}*h*= 0, for

_{i}*i*= 1, 2, …,

*p*.

We know that the matrix of exponents is composed from

*π*

_{1}=

*x*

_{1}

^{k1}

^{'}⋅

*x*

_{2}

^{k1}

^{'}⋅ … ⋅

*x*

_{n}^{kn}^{'}

*π*

_{2}=

*x*

_{1}

^{k1}

^{''}⋅

*x*

_{2}

^{k1}

^{''}⋅ … ⋅

*x*

_{n}^{kn}^{''}

⋮

*π*=

_{p}*x*

_{1}

^{k1p}⋅

*x*

_{2}

^{k1p}⋅ … ⋅

*x*.

_{n}^{knp}*π*

_{1}⋅

*π*

_{2}⋅ … ⋅

*π*=

_{p}*x*

_{1}

^{(k1}

^{'}+

^{k1}

^{''}

^{+ … +}

^{k1p)}⋅

*x*

_{2}

^{(k2}

^{'}+

^{k2}

^{''}

^{+ … +}

^{k2p)}⋅ … ⋅

*x*

_{n}^{(kn}

^{'}+

^{kn}

^{''}

^{+ … +}

^{knp)}.

Definition of linear dependency tell us that there must exist constants
*h*_{1}, *h*_{2}, …, *h _{p}* where

*not all*are zero such that,

*h*

_{1}

*k*' +

_{i}*h*

_{2}

*k*'' + … +

_{i}*h*= 0, for

_{p}k_{i}^{p}*i*= 1, 2, …,

*n*.

*π*

_{1},

*π*

_{2}, …,

*π*by its corresponding constant

_{p}*h*

_{1},

*h*

_{2}, …,

*h*the above equation becomes,

_{p}*π*

_{1}

^{h1}⋅

*π*

_{2}

^{h2}⋅ … ⋅

*π*=

_{p}^{hp}*x*

_{1}

^{(h1k1}

^{'}+

^{h1k1}

^{''}

^{+ … +}

^{h1k1p)}⋅

*x*

_{2}

^{(h2k2}

^{'}+

^{h2k2}

^{''}

^{+ … +}

^{h2k2p)}⋅ … ⋅

*x*

_{n}^{(hpkn}

^{'}+

^{hpkn}

^{''}

^{+ … +}

^{hpknp)}

*π*

_{1}

^{h1}⋅

*π*

_{2}

^{h2}⋅ … ⋅

*π*=

_{p}^{hp}*x*

_{1}

^{0}⋅

*x*

_{2}

^{0}⋅ … ⋅

*x*

_{n}^{0}.

*π*

_{1}

^{h1}⋅

*π*

_{2}

^{h2}⋅ … ⋅

*π*= 1.

_{p}^{hp}Therefore, the matrix of exponents must be linearly independent for the dimensionless products to be linearly independent.∎

**Hypothesis:**Rows in matrix of exponents are linearly independent. That is,

*h*

_{1}

*k*' +

_{i}*h*

_{2}

*k*'' + … +

_{i}*h*= 0 iff,

_{p}k_{i}^{p}*h*= 0, for

_{i}*i*= 1, 2, …,

*n*.

*h*

_{1},

*h*

_{2}, …,

*h*where

_{p}*not all*are zero such that,

*π*

_{1}

^{h1}⋅

*π*

_{2}

^{h2}⋅ … ⋅

*π*≡ 1.

_{p}^{hp}
Substituting the expressions for *π*'s shown belown

*π*

_{1}=

*x*

_{1}

^{k1}

^{'}⋅

*x*

_{2}

^{k1}

^{'}⋅ … ⋅

*x*

_{n}^{kn}^{'}

*π*

_{2}=

*x*

_{1}

^{k1}

^{''}⋅

*x*

_{2}

^{k1}

^{''}⋅ … ⋅

*x*

_{n}^{kn}^{''}

⋮

*π*=

_{p}*x*

_{1}

^{k1p}⋅

*x*

_{2}

^{k1p}⋅ … ⋅

*x*.

_{n}^{knp}*π*

_{1}

^{h1}⋅

*π*

_{2}

^{h2}⋅ … ⋅

*π*=

_{p}^{hp}*x*

_{1}

^{(h1k1}

^{'}+

^{h1k1}

^{''}

^{+ … +}

^{h1k1p)}⋅

*x*

_{2}

^{(h2k2}

^{'}+

^{h2k2}

^{''}

^{+ … +}

^{h2k2p)}⋅ … ⋅

*x*

_{n}^{(hpkn}

^{'}+

^{hpkn}

^{''}

^{+ … +}

^{hpknp)}.

*π*

_{1}

^{h1}⋅

*π*

_{2}

^{h2}⋅ … ⋅

*π*≡ 1 we have,

_{p}^{hp}*x*

_{1}

^{(h1k1}

^{'}+

^{h1k1}

^{''}

^{+ … +}

^{h1k1p)}⋅

*x*

_{2}

^{(h2k2}

^{'}+

^{h2k2}

^{''}

^{+ … +}

^{h2k2p)}⋅ … ⋅

*x*

_{n}^{(hpkn}

^{'}+

^{hpkn}

^{''}

^{+ … +}

^{hpknp)}= 1

_{i = 1 to p, j = 1 to n}(

*h*' +

_{i}k_{j}*h*'' + … +

_{i}k_{j}*h*) = 0.

_{i}k_{i}^{p}*h*

_{1}

*k*' +

_{i}*h*

_{2}

*k*'' + … +

_{i}*h*= 0, ∀

_{p}k_{i}^{p}*i*.

*h*

_{1},

*h*

_{2}, …,

*h*are zero. Therefore, the observed result is contrary to the hypothesis.

_{p}Therefore, the condition that the matrix of exponents must be linearly independent is sufficient for the dimensionless products to be linearly independent.∎

## Computation of dimensionless products

*a*

_{1}

*k*

_{1}+

*a*

_{2}

*k*

_{2}+ … +

*a*= 0

_{n}k_{n}*b*

_{1}

*k*

_{1}+

*b*

_{2}

*k*

_{2}+ … +

*b*= 0

_{n}k_{n}⋮

*g*

_{1}

*k*

_{1}+

*g*

_{2}

*k*

_{2}+ … +

*g*= 0

_{n}k_{n}*k*

_{1},

*k*

_{2}, …,

*k*of a

_{n}*complete set of dimensionless products*of the independent variables

*x*

_{1},

*x*

_{2}, …,

*x*.

_{n}Conversely, the exponents

*k*

_{1},

*k*

_{2}, …,

*k*of a

_{n}*complete set of dimensionless products*of the independent variables

*x*

_{1},

*x*

_{2}, …,

*x*are a fundamental system of solutions of

_{n}*a*

_{1}

*k*

_{1}+

*a*

_{2}

*k*

_{2}+ … +

*a*= 0

_{n}k_{n}*b*

_{1}

*k*

_{1}+

*b*

_{2}

*k*

_{2}+ … +

*b*= 0

_{n}k_{n}⋮

*g*

_{1}

*k*

_{1}+

*g*

_{2}

*k*

_{2}+ … +

*g*= 0

_{n}k_{n}*x*

_{1},

*x*

_{2}, …,

*x*is

_{n}*n*−

*r*, in which

*r*is the rank of the dimensional matrix of the variables.

We know from definition of a dimensionless product that the product has exponents
*k*_{1}, *k*_{2}, …, *k _{n}*
that are a solution of

*a*

_{1}

*k*

_{1}+

*a*

_{2}

*k*

_{2}+ … +

*a*= 0

_{n}k_{n}*b*

_{1}

*k*

_{1}+

*b*

_{2}

*k*

_{2}+ … +

*b*= 0

_{n}k_{n}⋮

*g*

_{1}

*k*

_{1}+

*g*

_{2}

*k*

_{2}+ … +

*g*= 0

_{n}k_{n}*k*

_{1}^{'},

*k*

_{i}^{''}, …,

*k*

_{i}^{(n − r)}, for

*i*= 1, 2, …,

*n*

*r*is the rank of the dimensional matrix of the coefficients

*π*

_{1},

*π*

_{2}, …,

*π*be independent it that the

_{p}*rows in the matrix of exponents be linearly independent*.

*k*

_{1}^{'},

*k*

_{i}^{''}, …,

*k*

_{i}^{(n − r)}are linearly independent, they should yield independent dimensionless products.

Finally, because the above system of solutions possess no more than (*n* − *r*)
linearly independent solutions, it is impossible to have more than (*n* − *r*)
independent dimensionless products.