A Topological Reassessment of Differential and Integral Calculus beyond Newtonian Frame Works

Abstract

The conventional subject of Newtonian calculus is purely mathematical and the concepts approximations and theorems employed within it have rarely been connected or correlated with the principles of physics and topology In this research article the entire domain of calculus has been revisited through a tripartite lens encompassing mathematics physics and topology This research article is based on the following 1 Topological interpretatio n of Calculus 2 Dimensional perspective of differential calculus how the variable changes for a mathematical function like xn when the value of n changes from 1 to 2 to 3 to 4 like this The variable does not remain to be x only 3 The physical significance of the differential co-efficient DC of a function which substantiates that the mathematical expression of DC should not retain the function itself since it is co-efficient 4 The dimensional invalidity of the tr igonometric theories at zeroor 90-degree values of angle of a right angled triangle 5 Differential co-efficient of trigonometric functions derived based on a novel concept of expansio n -contraction of a cuboid being composed of perpendicular base and hypotenuse of a right-angled triangle through very simple algebraical steps and without using limit theorem chain rule or quotient rule of the Newtonian Calculus 6 The problem of circularity of definition of the higher order differential co-efficient of the mathematical functions of Newtonian Calculus and offering a new mathematical expression of higher order differential co-efficient 7 The concept of integration has been established as a process of hybridizatio n of a mathematical variable in space through rotations or translations A genuine attempt has been made to unify these perspectives into a singular framework resulting in the development of a new field the reassessed topological version of calculus which is now presented to the global scientific community The limitations and flaws inherent in Newtonian calculus have been critically examined logically addressed and redefined within the broader scope of this newly proposed topological version of calculus