Sunday, 23 December 2012

Calculus- The Integral Differentiation

Its incalculable complexity reduces arithmetic to child's play while its unsurpassed applicability belittles algebra with the rest of its regularity. A mathematical scheme bent on the diversity of change, the correlation of the input with its output and the continuity of functions. Since its inception via Newton and Leibniz, it has emerged as a union between the celestial and the terrestrial; a language of fluxions and fluents and the manifestation of the laws of nature. Calculus is a dimorphic synthesis of integration and differentiation linked via its fundamental theorem. The former of the two, seeks the area between the curve of a function f(x) while the latter pursues the way a function behaves as its input differs such as the of instantaneous rate of change. While the regularity, uniformity and discreteness of institutions such as algebra remain most apparent, calculus intervenes through the annihilation of such regularity and uniformity,assuming the curliness of slopes and curves. Calculus remains the gateway to realising the spread of vector and tensor fields permeating and changing through space and time, the medium towards the fathoming of continuums beyond that of the discrete. For instance, in realising limits for functions f(x), we may realise their behavior near certain values and extrapolate them for understanding the behavior of particle frequencies near magnetic fields, dummy variables and their jump discontinuity or even the interference pattern of photons via double slits. The elliptical orbit of the celestial bodies, the logarithmic spiral of a chambered nautilus shell and Zeno's paradox; whereby an infinite number of halves may be crossed in a finite amount of time, all share this common denomination of change and fluctuation as quantified by calculus. Trigonometric functions  included, as their differentiation reveals the rate of change of such variables relative to one another.  But the applicability of calculus exceeds that of the two dimensional, path integrals give rise to higher dimensions of application such as integration above curves while volume integrals consider the everyday reality of three dimensional spatial freedom. It is a ever encompassing framework, played by certain simple rules yet fulfilled by a nature's laws and governing dynamics; a formulation of maxima and minima, of input and output and of limit and continuity.