In any case the dimension of a set is the number of numbers needed to define any element of the set.
Consider the set of points
\[\{ (x,x^2) : x \in \mathbb{R} \}\]
. This set defines the line \[y=x^2\]
which is a curve in the plane. The plane is two dimensional but the curve is one dimensional since only one point \[x\]
is needed to define each point on the curve.This example shows how it is possible for a space to be embedded in a higher dimensional subspace.
The example is illustrative.
\[x\]
and \[y\]
are coordinates on different axes. The coordinates are not independent if one coordinate is a function of the other, or if some coordinates are functions are some other coordinates. Some physical systems are defined by physics properties.One such is a confined ideal gas. A gas is defined totally by its internal heat energy
\[U\]
, its temperature \[T\]
, its pressure \[p\]
and volume \[V\]
.These quantities are not independent however.
\[U\]
is directly related to \[T\]
by the equation \[U= \frac{3}{2} kT\]
and we can define \[T\]
and hence \[U\]
in terms of \[p\]
and \[V\]
by the equation \[pV=nRT\]
where
\[n\]
is the number of mols and \[R=8.314 J/mol/K\]
is the Universal molar gas constant.\[U\]
and \[T\]
are not needed to describe the state of a gas. We only need \[p\]
and \[V\]
- or in fact any two of \[U, \: T, |; p, V\]
. The dimension of the state space of an ideal gas is 2 and we can plot any state of an ideal gas in the plane.