## Practical definition

An equation of the form *z* = *p* + *q* + *r* + …
is dimensionally homogeneous if, and only if, all the *terms* have the same dimension.

Why is the above practical definition unsatisfactory from a mathematical viewpoint?

Although most think of dimensional homogeneity as the one described in the practical definition
mathematicians find the definition unsatisfactory. This is because the concept of *terms*
does not enter the definition of a function.

*y*is a function of

*x*if, to each value of

*x*, there corresponds a value of

*y*.

In other words, the function *y* can be
defined by a graph.

This is not necessarily the case for the concept of terms. For the terms,
*p*, *q*, *r*, and *z*
the definition does not state the relationship between them. This means that the value for a term
cannot be determined from the remaining terms. Therefore,
there is not enough information in the definition to state that the terms may be defined with a
graph.

## Mathematical definition

*n*-variables such that some operator

*ƒ*is applied to independent variables

*x*

_{1},

*x*

_{2}, …,

*x*to yield dependent variable

_{n}*y*.

We denote this consideration as

*y*=

*ƒ*(

*x*

_{1},

*x*

_{2}, …,

*x*)

_{n}
Furthermore, if the original units are subject to change, the variables
*x*_{1}, *x*_{2}, …, *x _{n}*, and

*y*will be transformed into

*x̄*

_{1},

*x̄*

_{2}, …,

*x̄*, and

_{n}*ȳ*.

The change is expressed as

*ȳ*=

*ƒ*(

*x̄*

_{1},

*x̄*

_{2}, …,

*x̄*)

_{n}^{(ibid. 1.)}for each variable is

*ȳ*=

*y*⋅

*A*⋅

^{a}*B*⋅

^{b}*C*⋅

^{c}*D*⋅

^{d}*E*⋅

^{e}*F*⋅

^{f}*G*=

^{g}*y*⋅

*K*

*x̄*

_{1}=

*x*

_{1}⋅

*A*

^{a1}⋅

*B*

^{b1}⋅

*C*

^{c1}⋅

*D*

^{d1}⋅

*E*

^{e1}⋅

*F*

^{f1}⋅

*G*

^{g1}=

*x*

_{1}⋅

*K*

_{1}

*x̄*

_{2}=

*x*

_{2}⋅

*A*

^{a2}⋅

*B*

^{b2}⋅

*C*

^{c2}⋅

*D*

^{d2}⋅

*E*

^{e2}⋅

*F*

^{f2}⋅

*G*

^{g2}=

*x*

_{2}⋅

*K*

_{2}

⋮

*x̄*=

_{n}*x*⋅

_{n}*A*⋅

^{an}*B*⋅

^{bn}*C*⋅

^{cn}*D*⋅

^{dn}*E*⋅

^{en}*F*⋅

^{fn}*G*=

^{gn}*x*⋅

_{n}*K*

_{n}*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*

*ƒ*remains the same.

What was just described was dimensional homogeneity of equation which can be summarized as

*y*=

*ƒ*(

*x*

_{1},

*x*

_{2}, …,

*x*), and

_{n}*ȳ*=

*ƒ*(

*x̄*

_{1},

*x̄*

_{2}, …,

*x̄*)

_{n}*y*=

*ƒ*(

*x*

_{1},

*x*

_{2}, …,

*x*)

_{n}*ȳ*=

*ƒ*(

*x̄*

_{1},

*x̄*

_{2}, …,

*x̄*)

_{n}*invariant under the group of transformations*generated by all possible changes of the units of the fundamental dimensions.

## Mathematical definition with more rigor

The dimensions of the variables (independent *x̄*_{1},
*x̄*_{2}, …, *x̄ _{n}*, and dependent

*ȳ*) can be designated as the dimensional matrix

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

M | a | a_{1} |
a_{2} | … | a_{n} |

L | b | b_{1} |
b_{2} | … | b_{n} |

T | c | c_{1} |
c_{2} | … | c_{n} |

A | d | d_{1} |
d_{2} | … | d_{n} |

K | e | e_{1} |
e_{2} | … | e_{n} |

cd | f | f_{1} |
f_{2} | … | f_{n} |

mol | g | g_{1} |
g_{2} | … | g_{n} |

= | y ⋅ K |
x_{1} ⋅ K_{1} |
x_{2} ⋅ K_{2} | … | x ⋅ _{n}K_{n} |

= | ȳ | x̄_{1} |
x̄_{2} | … | x̄_{n} |

Substituting

*y*=

*ƒ*(

*x*

_{1},

*x*

_{2}, …,

*x*)

_{n}*ȳ*=

*ƒ*(

*x̄*

_{1},

*x̄*

_{2}, …,

*x̄*)

_{n}*K*⋅

*y*=

*ȳ*

*K*⋅

*ƒ*(

*x*

_{1},

*x*

_{2}, …,

*x*) =

_{n}*ƒ*(

*x̄*

_{1},

*x̄*

_{2}, …,

*x̄*)

_{n}*x̄*'s we get,

_{i}*K*⋅

*ƒ*(

*x*

_{1},

*x*

_{2}, …,

*x*) =

_{n}*ƒ*(

*K*

_{1}

*x*

_{1},

*K*

_{2}

*x*

_{2}, …,

*K*

_{n}*x*)

_{n}*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*).

## Dimensional homogeneity from view of a graph

*ℓ*with area,

*A*=

*ℓ*

^{2}we can see that

*A*is a function of

*ℓ*. Consider the case of values of

*ℓ*from 1 to 10 originally measured in decimeter (

*dm*). Each value of

*ℓ*is worked on by the operator to return the area of its square. Let us also say that the coordinate of an area is ploted in a 2-D graph with abscissa

*ℓ*and ordinate

*A*. We observe that the area of a square,

*A*as a function of

*ℓ*is represented by a parabolic curve for lengths measured in decimeter.

Comparing the two parabolic curves we notice that the graphs/parabolic curve is valid, irrespective of the unit of lengths.

Since, we know that the graph is a representation of function *A* and the *unchanged curve implies an
invariant function*. Therefore, the relationship *A* = *ℓ*^{2} is dimensionally homogeneous.

A graph may not always be valid when the unit is changed.

Consider the case of stress-strain curve with the original unit of stress measured in pounds per square inch (psi). The curve indicates a relation between strain and stress but is stress as the independent variable sufficient to explain the dependent variable strain? In other words, is the function represented by the graph dimensionally homogeneous?

To answer this question, let us transform the orginial unit (of psi) to Pascal (pa) and compute the coordinate of magnitude for the transformed unit.