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Inner Products and Norms for Vectors

We begin our description of gradient flows by reviewing familiar concepts for vector quantities for points and then generalizing this to fields. Let tex2html_wrap_inline938 and tex2html_wrap_inline940 be vectors with components tex2html_wrap_inline942 and let t be a scalar. Let tex2html_wrap_inline946 be a differentiable function which operates on a vector and yields a scalar ( tex2html_wrap_inline948 ): f is a field. It may be useful to think of tex2html_wrap_inline938 , tex2html_wrap_inline940 as a space-like variable tex2html_wrap_inline928 and t as a time-like quantity and tex2html_wrap_inline960 as a local thermodynamic quantity like free energy density.

The norm or magnitude of a vector ( tex2html_wrap_inline962 , is the norm of tex2html_wrap_inline938 ) must have the following properties:

non-negative
tex2html_wrap_inline966 for all tex2html_wrap_inline1209 , and tex2html_wrap_inline970 for tex2html_wrap_inline972 alone.
scale invariance
tex2html_wrap_inline974 , where |t| is the absolute value of the real number t.
Schwarz (or triangle) inequality
tex2html_wrap_inline980

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

  1. tex2html_wrap_inline982 : this is the familiar Euclidean norm.
  2. tex2html_wrap_inline984 : the Manhattan metric, how one would measure the distance walked if one can only make 90 degree turns in two dimensions (n=2). When tex2html_wrap_inline988 is the number of moles of the ith component this is the norm that describes the molar size of a system[4].
  3. tex2html_wrap_inline992
  4. tex2html_wrap_inline994 : 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 tex2html_wrap_inline996 -norms

displaymath998

with tex2html_wrap_inline1000 respectively. Norms often have a natural inner (dot, or scalar) product tex2html_wrap_inline1002 where the operator `` tex2html_wrap_inline1004 , tex2html_wrap_inline1006 '' maps two vectors to a scalar ( tex2html_wrap_inline1008 ) and has additional properties to those defined by the norm above such as symmetry, associativity and distributivity. gif

In multivariable calculus, the chain rule is used to compute

  equation661

This is the gradient of f evaluated at tex2html_wrap_inline1209 projected onto the direction of tex2html_wrap_inline1030 and multiplied by the magnitude of tex2html_wrap_inline1030 . From the properties of the inner product, the fastest increase of f is for a tex2html_wrap_inline1030 parallel to the gradient. If one chose a different inner product than the standard one, then there would be a different gradient tex2html_wrap_inline1038 , in order that tex2html_wrap_inline1040 continue to equal tex2html_wrap_inline1042 . 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.


next up previous
Next: Inner Products on Fields Up: Introduction Previous: Introduction

W. Craig Carter
Tue Sep 30 16:07:27 EDT 1997