## Dimensional homogeneity of a sum

Recall that dimensional homogeneity^{(ibid. 2.)} is defined mathematically by the theorem

The function
is an identity in the variables
(

⋮

*ƒ*(*x*_{1},*x*_{2}, …,*x*) is dimensionally homogeneous if, and only if,_{n}*K*⋅

*ƒ*(

*x*

_{1},

*x*

_{2}, …,

*x*) =

_{n}*ƒ*(

*K*

_{1}

*x*

_{1},

*K*

_{2}

*x*

_{2}, …,

*K*

_{n}*x*)

_{n}*x*_{1},*x*_{2}, …,*x*,_{n}*A*,*B*,*C*,*D*,*E*,*F*,*G*) where*y*⋅

*K*=

*y*⋅

*A*⋅

^{a}*B*⋅

^{b}*C*⋅

^{c}*D*⋅

^{d}*E*⋅

^{e}*F*⋅

^{f}*G*

^{g}*x*

_{1}⋅

*K*

_{1}=

*x*

_{1}⋅

*A*

^{a1}⋅

*B*

^{b1}⋅

*C*

^{c1}⋅

*D*

^{d1}⋅

*E*

^{e1}⋅

*F*

^{f1}⋅

*G*

^{g1}

*x*

_{2}⋅

*K*

_{2}=

*x*

_{2}⋅

*A*

^{a2}⋅

*B*

^{b2}⋅

*C*

^{c2}⋅

*D*

^{d2}⋅

*E*

^{e2}⋅

*F*

^{f2}⋅

*G*

^{g2}

⋮

*x*⋅

_{n}*K*=

_{n}*x*⋅

_{n}*A*⋅

^{an}*B*⋅

^{bn}*C*⋅

^{cn}*D*⋅

^{dn}*E*⋅

^{en}*F*⋅

^{fn}*G*.

^{gn}
The (sum) function
all terms in the sum have the same dimensions as the sum

*ƒ*(*x*_{1},*x*_{2}, …,*x*) =_{n}*x*_{1}+*x*_{2}+ … +*x*is dimensionally homogeneous if, and only if, in_{n}*K*⋅ (

*x*

_{1}+

*x*

_{2}+ … +

*x*) =

_{n}*K*

_{1}

*x*

_{1}+

*K*

_{1}

*x*

_{2}+ … +

*K*

_{n}*x*

_{n}*K*=

*K*

_{1}=

*K*

_{2}= … =

*K*.

_{n}
Let *y* = *ƒ* (*x*_{1}, *x*_{2}, …, *x _{n}*) be
such that

*ƒ*is the sum of

*x*'s,

_{i}*ƒ*(

*x*

_{1},

*x*

_{2}, …,

*x*) =

_{n}*x*

_{1}+

*x*

_{2}+ … +

*x*.

_{n}*K*⋅

*ƒ*(

*x*

_{1},

*x*

_{2}, …,

*x*) =

_{n}*ƒ*(

*K*

_{1}

*x*

_{1},

*K*

_{2}

*x*

_{2}, …,

*K*

_{n}*x*)

_{n}*K*⋅ (

*x*

_{1}+

*x*

_{2}+ … +

*x*) =

_{n}*K*

_{1}

*x*

_{1}+

*K*

_{2}

*x*

_{2}+ … +

*K*

_{n}*x*

_{n}*K*

*x*

_{1}+

*K*

*x*

_{2}+ … +

*K*

*x*=

_{n}*K*

_{1}

*x*

_{1}+

*K*

_{2}

*x*

_{2}+ … +

*K*

_{n}*x*.

_{n}*x*'s

_{i}*K*=

*K*

_{1}=

*K*

_{2}+ … =

*K*.∎

_{n}*y*,

*x*

_{1},

*x*

_{2}, …,

*x*their corresponding units are given by

_{n}*K*,

*K*

_{1},

*K*

_{2}, …,

*K*respectively, the identity of all the

_{n}*K*'s means all the terms have the same dimension. This can be shown as follows.

We know that,

*K*=

*A*⋅

^{a}*B*⋅

^{b}*C*⋅

^{c}*D*⋅

^{d}*E*⋅

^{e}*F*⋅

^{f}*G*

^{g}*K*=

_{i}*A*⋅

^{ai}*B*⋅

^{bi}*C*⋅

^{ci}*D*⋅

^{di}*E*⋅

^{ei}*F*⋅

^{fi}*G*for all

^{gi}*i*= 1, 2, …,

*n*.

*A*to

*G*are shared among the

*K*'s, from

*K*=

*K*

_{1}=

*K*

_{2}+ … =

*K*we can see that for all

_{n}*i*= 1, 2, …,

*n*

*a*=

*a*

_{i}*b*=

*b*

_{i}⋮

*g*=

*g*.∎

_{i}*necessary*condition.

## Dimensional homogeneity of a product

The (product) function

=
is dimensionally homogeneous if, and only if, the exponents

⋮

where

*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 dimensional exponents.*necessary condition*to

*K*=

*K*

_{1}

^{k1}

*K*

_{2}

^{k2}…

*K*

_{n}*.*

^{kn}
Because the expression
plays an important part in dimensional analysis this expression is called

*y*=

*x*

_{1}

^{k1}⋅

*x*

_{2}

^{k1}⋅ … ⋅

*x*

_{n}^{kn}**products**.
Let *y* = *ƒ* (*x*_{1}, *x*_{2}, …, *x _{n}*) be
such that

*ƒ*is the product of

*x*'s,

_{i}*ƒ*(

*x*

_{1},

*x*

_{2}, …,

*x*) =

_{n}*x*

_{1}

^{k1}⋅

*x*

_{2}

^{k1}⋅ … ⋅

*x*.

_{n}^{kn}*K*⋅

*ƒ*(

*x*

_{1},

*x*

_{2}, …,

*x*) =

_{n}*ƒ*(

*K*

_{1}

*x*

_{1},

*K*

_{2}

*x*

_{2}, …,

*K*

_{n}*x*)

_{n}*K*⋅ (

*x*

_{1}

^{k1}⋅

*x*

_{2}

^{k1}⋅ … ⋅

*x*) = (

_{n}^{kn}*K*

_{1}

*x*

_{1})

^{k1}⋅ (

*K*

_{2}

*x*

_{2})

^{k2}⋅ … ⋅ (

*K*

_{n}*x*)

_{n}

^{kn}*K*⋅ (

*x*

_{1}

^{k1}⋅

*x*

_{2}

^{k1}⋅ … ⋅

*x*) =

_{n}^{kn}*K*

_{1}

^{k1}

*K*

_{2}

^{k2}…

*K*

_{n}*⋅ (*

^{kn}*x*

_{1}

^{k1}⋅

*x*

_{2}

^{k1}⋅ … ⋅

*x*) .

_{n}^{kn}*K*=

*K*

_{1}

^{k1}

*K*

_{2}

^{k2}…

*K*

_{n}*.*

^{kn}*K*=

*A*⋅

^{a}*B*⋅

^{b}*C*⋅

^{c}*D*⋅

^{d}*E*⋅

^{e}*F*⋅

^{f}*G*

^{g}*K*=

_{i}*A*⋅

^{ai}*B*⋅

^{bi}*C*⋅

^{ci}*D*⋅

^{di}*E*⋅

^{ei}*F*⋅

^{fi}*G*for all

^{gi}*i*= 1, 2, …,

*n*.

*y*=

*x*

_{1}

^{k1}⋅

*x*

_{2}

^{k1}⋅ … ⋅

*x*involves three dimensions. Then,

_{n}^{kn}*K*=

*K*

_{1}

^{k1}

*K*

_{2}

^{k2}…

*K*

_{n}

^{kn}*A*⋅

^{a}*B*⋅

^{b}*C*= (

^{c}*A*

^{a1}⋅

*B*

^{b1}⋅

*C*

^{c1})

^{k1}⋅ (

*A*

^{a2}⋅

*B*

^{b2}⋅

*C*

^{c2})

^{k2}⋅ … ⋅ (

*A*⋅

^{an}*B*⋅

^{bn}*C*)

^{cn}

^{kn}*A*⋅

^{a}*B*⋅

^{b}*C*=

^{c}*A*

^{(a1k1 + a2k2 + … + ankn)}⋅

*B*

^{(b1k1 + b2k2 + … + bnkn)}⋅

*C*

^{(c1k1 + c2k2 + … + cnkn)}.

*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}*c*=

*c*

_{1}

*k*

_{1}+

*c*

_{2}

*k*

_{2}+ … +

*c*

_{n}k_{n}*k*

_{1},

*k*

_{2}, …,

*k*are a solution of the linear equations.∎

_{n}
*Next:*

On Vector Space (p:4) ➽