We begin our description of gradient flows by reviewing familiar
concepts for
vector quantities for points and then generalizing this to fields.
Let 
and 
be vectors with components
and let t be a scalar.
Let 
be a
differentiable function which operates on a vector and
yields a scalar ( 
): f is a field.
It may be useful to think of , as a space-like variable
and t as a time-like quantity and 
as a local
thermodynamic quantity like free energy density.
The norm or magnitude of a vector
( 
, is the norm of )
must have the
following properties:
for all 
, and
for 
alone.
, where |t| is the absolute value
of the real number t.
There are many different types of norms on vectors. Some examples are:
: this is the
familiar Euclidean norm.
: the
Manhattan
metric, how one would measure the distance walked if one can only
make 90 degree turns in two dimensions (n=2). When 
is the number of
moles of the ith component this is the norm that describes the molar size of
a system[4].
: a
weighted rescaling of the Euclidean norm, as would be useful
in rescaling the diffusion equation for an
anisotropic single crystal.
(1) - (3) are examples of 
-norms
with 
respectively.
Norms often have a natural inner (dot, or scalar) product
where the
operator `` 
, 
''
maps two vectors to a scalar ( 
)
and has additional properties to those
defined by the norm above such as symmetry, associativity and distributivity.
In multivariable calculus, the chain rule is used to compute
This is the gradient of f evaluated at
projected onto the direction of 
and multiplied by the magnitude
of .
From the properties of the inner product, the fastest increase of f is
for a parallel to the gradient.
If one chose a different inner product than the standard one,
then there would be a different gradient 
, in order that
continue to equal
.
Equation 1 is thus a relation between the gradient
operator and the inner product: the gradient of a field
depends on the choice of the inner product.
This is natural because the inner product defines both a magnitude
and
a projection; the properties of the gradient depend on both of these
operations through equation 1.
For any inner product the gradient, thus defined, points in the direction of fastest increase.
Next we will show that an analogous treatment of inner products on functions defines the gradient of a functional.
