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:
There are many different types of norms on vectors. Some examples are:
(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.