596 VIII. ON THE STRUCTURE OF MATHEMATICS |
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becomes zero. In other words, the value of our function is momentarily constant, it has a stationary value.
Quite naturally, differential equations which involve derivatives involve implicitly and explicitly the whole fundamental structural framework of the calculus as explained in this chapter by expressing the 'rate of change' of some natural process. If the rate of change is zero, it might express some 'natural law', or some uniformity as found in nature. In other words, differential equations express differential laws, which in turn express the momentary tendencies of processes whose outcomes are given by the process of integration.
From what has already been said here, it is obvious that differential equations and the differential laws which they express are of extreme structural importance. They formulate not only the uniformities and tendencies found in nature, but also of necessity somehow involve causality. Besides which, they are also in accord with the physical structure and function of the nervous system. We shall return to this most important subject in the next chapter, in which we shall analyse the physical significance and aspects of what has been explained here.
3.
METHODS OF APPROXIMATIONIn discussing the above fundamental notions of the calculus we considered a portion
AB, of the curve given by the equation, (Fig. 13) and twopoints on this curve Pi with co-ordinates
(xi, yi) and Pi with co-ordinates(*2,
yd moving along the curve, the secant, or chord, P1P2 rotates about Pi, its length steadily diminishing, and in the limit as the length of the chord PiPs tends toward zero, the slope of the secant approaches the slope of the tangent Pi7\ We saw that the slope of this tangent was given by the value of the first derivative of the function which represented the curve. We were trying to get some knowledge of the direction of our curve at a given point by considering the slope of a. straight line of smaller and smaller length. When we studied the curvature of our curve we considered the rate of change of the slope of our tangent and so, by the aid of a second derivative, we found the curvature. In this case we approximated our curve to a circle of radius equal to the radius of curvature of the curve at a given point.In attempting to determine the length of a portion of our curve a point cannot be regarded as a piece of the curve but only as marking a position on it. For the purpose of determining the length of an arc it is convenient to replace each small element of the arc by its chord, a
lineal element. By definition the length of an arc of a curve is the limit, if such limit exists, toward which the |
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