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We begin by extending the functions 
, 
, and 
from
quantities
which are refer to chemical energy per unit mole, surface
energy
per unit area, and distance traveled per unit time to
quantities
which refer to the `free energy' of a system containing a specified
number of moles or a surface with an specified area, or the `distance'
that an interface has moved in a specified time.
In thermodynamics these are called extensive variables.[45]
In the mathematics literature such functions are called
homogeneous degree one (HD1)
[39]
and are defined by the property:
We limit ourselves to positive homogeneity where 
is a
real and positive scalar.
When we homogeneously extend the three functions (, , ),
by
letting 
be the number of moles 
, the area 
,
or the time 
, we
obtain:
where 
is the number of moles, 
, and 
is a vector that represents a surface. Its direction is
along the outward normal 
and its length is the
area 
; 
, or 
.
Its components
are the projected areas along the three coordinate
axes.
represents the
distance the interface with orientation will travel in time 
in
a direction parallel to 
.
Note that 
is the familiar extensive function from
solution thermodynamics. We do not change the symbol for the
functions, e.g.
and are the 
function, but the later is
restricted to a restricted set (
) of the space
of systems of all sizes and compositions, parameterized by 
.
Note also that 
, where 
is
the amount of component 
, while 
. The difference in form between these two expressions
will be seen to have important consequences.
The three extended functions , 
, and 
are actually what one
would infer from a particular experiment.
For instance: in a calorimetry experiment, the enthalpy of
a closed system is measured, but the value is reported as what
would have been measured if one mole (or one Kg) were present; the
surface
free energy is unlikely to be measured for a square meter, but is reported that
way;
the positions of a moving surface are rarely measured at one second intervals.
Any HD1 function is fully determined if its value is
known along some curve which intersects all rays emanating from
the origin.
We use this property in equation (2)
to extend , and homogeneously to vectors of arbitrary
magnitude and to compute their values on the plane 
, and on
the spheres 
, and 
.
Another way an HD1 function can be reconstructed is from one of their
level sets, i. e. the set of points 
for which 
Then 
where 
and 
are in the same
direction.
The gradients of any HD1 function 
depend only on the
direction of 
, but not its magnitude, 
. For this gradient is a vector; since 
the 
component of this vector is the chemical
potential of the 
species. Consistent with this principle,
chemical potentials depend only on the composition.
Any HD1 function can be written as the dot-product of its gradient
and its argument vector, and its argument vector is perpendicular to
the differential of its gradient:
For these are the familiar integral expression for the Gibbs
free energy 
and the Gibbs-Duhem
equation[45]
. The gradient
of , called the vector 
, has these properties, which
have been used for anisotropic surfaces.[47][46] The
gradient of is
the characteristic of the motion of the surface.[36][49][48]
We next review the mathematics of convex functions.[39] A scalar
function 
of 
variables, or of a -dimensional vector, is said to
be convex if it is bounded
from below, it is not everywhere infinite, and if
If is also HD1 this inequality can be simplified. Setting 
and 
, and
making use of Eq. 1 gives:
The definitions in Eqs. 4 and 5
can be extended to partitions of vectors into a sum of an arbitrary
number of terms: i.e., 
for
a convex HD1 .
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