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                                      Michael Lee's Mathematics and Physics

Analytic Geometry


Zeno of Elea (430 BCE), a follower of Parmenides, questioned how an infinite plurality of many parts can compose a finite whole.  Well suppose we have a line segment of finite length and it is divided by two to create two equal pieces.  After the first division, we would have two pieces that are exactly half the length of the original line segment.  Next, take the two pieces and divide each of them by two that results in four pieces each a quarter of the length of the original line segment.  Again, take the four pieces and divide them by two resulting in eight pieces that are an eighth of the original line segment.  Continue this process of dividing each piece by two many, many times.  In mathematics though, this process of division does not proceed at a finite rate but rather happens instantaneously (there is no speed limit in mathematics).  Likewise, their  summation, that also happens instantaneously, equals the length of the original finite line segment.  In other words, as the length of the segments diminishes to being infinitely small, the number of pieces tends towards being infinity large.


So are differentials (like dx for example) meaningless or do they have some kind of useful purpose?  I believe the latter to be true.  Take the area under a function for example; doesn't it consist of taking values of "dx" multiplied by the value of  f(x) at some particular point on the x axis and then adding them up between the range of the definite integral?   By thinking this way I derived formulas for the surface areas of a sphere, dome, cone (and its frustums),  pyramid (and its frustums) and the torus (and its "frustums"), by finding relationships between differentials and the shape of the body thought about and being measured. The idea is to hold the some size variables constant, such as lengths and measure other things, such as areas, in a universal way, but to be quite honest with you, I haven't a clue as to why it works.