Conventionally speaking is defined as the number of times the diameter of a circle will fit around its circumference. This seems somewhat reasonable at first glance, but if you really consider it there is a more intuitive definition.

Consider the following geometric argument. You are standing on an isolated beach and you decide to build a circular hut using only primitive tools. First you must lay out your foundation. You dig into your tool chest and pull out two stakes, a hammer and a rope. You drive one stake into the sand at the point where you want the center of your hut. Using the rope tied to the central stake you walk the perimeter with the other stake tied to the rope and dragging in the sand at the desired radius. This is quite possibly the easiest way to construct a circle. In fact you might be hard pressed to create a circle by specifying its diameter (aside from dividing it in half to get the radius) or its circumference. Speaking from a constructive point of view the radius is the primal measure of a circle.

From a more mathematical perspective, consider the “natural” unit of angular measure, the radian. The name hints that its fundamental origin of measure is derived from the radius. The unit angular distance around a circle is . Why should the fundamental unit of angular measure be two of something? Shouldn't it be one of something?

In ether case, the concept of the ratio of the circle to its radius seems more reasonable. The actual value of should be 6.28318531… This might also keep you from carrying around the algebraic baggage of an extra 2 in many equations.

The integral formulation of Greens theorem and Stokes theorem imply that there might exist a general formulation for going from an n dimensional to an (n+1) dimensional representation.

In calculus we are introduced to the formula for calculating the arc length of a path in integral form. Namely;

The integrand is actually only an approximation when calculating the length of any intrinsically curved path, ie. not a straight line or multiple straight connected segments.

Consider the curved path that lies along the periphery of a circle of finite fixed radius r. In plane polar coordinates the arc length is an exact differential, ie.

Converting to Cartesian coordinates based on the geometry indicate in the image to the right we take the following steps;

or, in a slightly more manageable form;

Now lets expand the sin function in a Mclaurin Series;

if we keep only the first term, we get

or after just a litle algebraic fiddling;

the Cartesian differential arc length formula, as we should expect. But note, this is only the first term in an expansion for the arc length of a circle. If instead we take only the first two terms of the sine expansion;

This is a classic example of why a differential element can't be treated as an algebraic quantity.

Actually, the definition of an integral shows why this fails;

In physics we all get introduced to the formula for distance as a function of constant acceleration,

Working with the simplified form,

Have you ever said Hmmm, can this be expressed as an iterative formula? It can if we let be some arbitrary but constant time step, then we can rewrite it as (call it The Equation)

Where the formula represents the displacement after n time steps, we could set the initial time to zero but we will see that it goes away naturally. The equation implies that

Note that The Equation can be solved for the number of steps n to give

Now expand the incremented form of The Equation as follows

recognising the terms in common with The Equation

Now substitute in for n and simplify to get

Where we have chosen the positive root in n because the negative root causes the formula to stutter (try it, you will see).

The result for the unsimplified constant acceleration equation is

Although this form requires information about the initial state, so is not strictly in iterative form.

We could continue on by defining and then eliminate in order to derive a completely iterative equation without respect to time.

Say you have a function, and you want to do interpolation with a function of N parameters. The points that give you the 'best' fit can be determined in a couple of ways. For instance you may wish to minimize the fitting error over the entire interpolation interval. You may also want to find the points so that the sum of the difference between the fitting polynomial and the original function at the points are minimized. In physics you often want to conserve boundary conditions and things like location of extrema in the function and its (anti)derivatives or the value of the integrated area.

These points can be found by picking N+1 points on your interval and solving for the coefficients of the Nth order polynomial in terms of those N+1 points then minimizing the appropriate error difference equations.

Minimize the integral of the difference between the fitted function and the fitting function with respect to the fitting function coefficients.

Minimize the sum of the following: the difference between the fitted function and the fitting function with respect to the fitting function parameters at each point.

WORK IN PROGRESS

For example, suppose the function we wish to interpolate is sin(x) and we are going to use quadratic interpolation. First solve for the coefficints like so

, ,

Leads to

, ,

Now for each point minimize the function which in this instance can be done by setting the derivative to zero;

and the three points can be found by solving the following system of equations

In order to maintain a particular waveform with a set of time evolving components, one must look at deconvolving the phases of the frequency deconvolution as well. Perhaps a variational minimization of the parametric evolution function or its expansion would suffice.

The ability of fractional calculus to modify Taylors expansion using non-integer derivatives should lead to a method of fitting the expansion to the function being expanded allowing the low order terms to optimally represent the function being expanded.

Measurement grid spacing for a uniform grid has a hard bandwidth representation limit ( Nyquist frequency). Use specially constructed non-uniform spacing (perhaps a perfect Golumb ruler) to allow a probabilistic measurement of the reliability of the under sampled bandwidth and so retain added data with a measurable degree of reliability.

A plot of the two variable function

Conjecture:

Proof:

⇒
⇒
⇒
⇒

Proof of follows trivially.