Dimensional Homogeneity

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.

❶

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.

function of n-independent variables with operator f and dependent variable y
The definition of a function is such that, 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.

❷

Mathematical definition

function of n-independent variables with operator f and dependent variable y
Let us consider that there exist a function of n-variables such that some operator ƒ is applied to independent variables x1, x2, …, xn to yield dependent variable y.

We denote this consideration as

y = ƒ (x1, x2, …, xn)

Furthermore, if the original units are subject to change, the variables x1, x2, …, xn, and y will be transformed into x̄1, x̄2, …, x̄n, and ȳ.

The change is expressed as

ȳ = ƒ (x̄1, x̄2, …, x̄n)
such that the transformed coordinate of magnitude(ibid. 1.) for each variable is
ȳ = y ⋅ Aa ⋅ Bb ⋅ Cc ⋅ Dd ⋅ Ee ⋅ Ff ⋅ Gg = y ⋅ K
x̄1 = x1 ⋅ Aa1 ⋅ Bb1 ⋅ Cc1 ⋅ Dd1 ⋅ Ee1 ⋅ Ff1 ⋅ Gg1 = x1 ⋅ K1
x̄2 = x2 ⋅ Aa2 ⋅ Bb2 ⋅ Cc2 ⋅ Dd2 ⋅ Ee2 ⋅ Ff2 ⋅ Gg2 = x2 ⋅ K2
⋮
x̄n = xn ⋅ Aan ⋅ Bbn ⋅ Ccn ⋅ Ddn ⋅ Een ⋅ Ffn ⋅ Ggn = xn ⋅ Kn
where
K = Aa ⋅ Bb ⋅ Cc ⋅ Dd ⋅ Ee ⋅ Ff ⋅ Gg
Ki = Aai ⋅ Bbi ⋅ Cci ⋅ Ddi ⋅ Eei ⋅ Ffi ⋅ Ggi    for all i = 1, 2, …, n

Notice that the operator ƒ remains the same.

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

The expressions
y = ƒ (x1, x2, …, xn), and
ȳ = ƒ (x̄1, x̄2, …, x̄n)
are the mathematical expressions for stating that the equation
y = ƒ (x1, x2, …, xn)
is dimensionally homogeneous if, and only if,
ȳ = ƒ (x̄1, x̄2, …, x̄n)
is invariant under the group of transformations generated by all possible changes of the units of the fundamental dimensions.

❸

Mathematical definition with more rigor

The dimensions of the variables (independent x̄1, x̄2, …, x̄n, and dependent ȳ) can be designated as the dimensional matrix

 yx1 x2…xn
Maa1 a2…an
Lbb1 b2…bn
Tcc1 c2…cn
Add1 d2…dn
Kee1 e2…en
cdff1 f2…fn
molgg1 g2…gn
=y ⋅ K x1 ⋅ K1 x2 ⋅ K2… xn ⋅ Kn
=ȳx̄1 x̄2…x̄n

Substituting

y = ƒ (x1, x2, …, xn)
ȳ = ƒ (x̄1, x̄2, …, x̄n)
into
K ⋅ y = ȳ
we get,
K ⋅ ƒ (x1, x2, …, xn) = ƒ (x̄1, x̄2, …, x̄n)
Furthermore, substituting for the x̄i's we get,
K ⋅ ƒ (x1, x2, …, xn) = ƒ (K1x1, K2x2, …, Knxn)

The function ƒ (x1, x2, …, xn) is dimensionally homogeneous if, and only if,
K ⋅ ƒ (x1, x2, …, xn) = ƒ (K1x1, K2x2, …, Knxn)
is an identity in the variables (x1, x2, …, xn, A, B, C, D, E, F, G).

❹

Dimensional homogeneity from view of a graph

A function defined by a graph is dimensionally homogeneous if, and only if, the curves remain unchanged when the basic unit of measurements are changed in any way.
For example, for a square of length ℓ with area, A = ℓ2 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.
length and area of square as function of length
Now, assume the case of changing the units for length from decimeter to meter. Passing these transformed lengths into the function for the area of a square we observe another parabolic curve.

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 = ℓ2 is dimensionally homogeneous.

❺

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.

graphs of stress vs strain
Comparing the two curves we notice that they are not the same. That is, the function is not invariant and hence the relationship between the stress and strain (indicated by the graph) is not dimensionally homogeneous. This is because the form of the ordinary tension stress-strain curve depends on the unit of stress.

❻

Next:

Dimensional Homogeneity on Some Arithmetic Operations (p:3) ➽

✪