On a Generalization of the Newton Derivative via Mapping Functions
Description
This paper introduces a formal generalization of the classical New tonian derivative by incorporating an auxiliary mapping function, ϵ(x), to redefine the limit-based difference quotient. While traditional calculus evaluates rates of change over a linear identity domain, the proposed General Derivative characterizes the sensitivity of a function f relative to a non-linear transformation of its independent variable. We establish the theoretical framework for this operator and demon strate that the standard derivative emerges as a specific realization where ϵ is the identity mapping. Furthermore, through various com plex transcendental cases, we show that this formulation effectively decouples functional complexity from domain geometry. The study concludes by discussing potential extensions into functional analysis, specifically regarding Fr´echet and Gˆateaux differentiability.
Files
Steps to reproduce
References [1] Apostol, T. M. (1974). Mathematical Analysis. Addison-Wesley. [2] Rudin, W. (1976). Principles of Mathematical Analysis. McGraw-Hill. [3] Spivak, M. (2008). Calculus. Publish or Perish. [4] Courant, R., & John, F. (1989). Introduction to Calculus and Analysis. Springer. [5] Bartle, R. G., & Sherbert, D. R. (2011). Introduction to Real Analysis. Wiley. [6] Hardy, G. H. (1908). A Course of Pure Mathematics. Cambridge Uni versity Press. [7] Loomis, L. H., & Sternberg, S. (1968). Advanced Calculus. Addison Wesley. [8] Lang, S. (1997). Undergraduate Analysis. Springer. [9] Abbott, S. (2015). Understanding Analysis. Springer. [10] Ross, K. A. (2013). Elementary Analysis: The Theory of Calculus. Springer. [11] Brezis, H. (2011). Functional Analysis, Sobolev Spaces and Partial Dif ferential Equations. Springer. [12] Kreyszig, E. (1978). Introductory Functional Analysis with Applications. Wiley. [13] Lax, P. D. (2002). Functional Analysis. Wiley-Interscience. [14] Yosida, K. (1980). Functional Analysis. Springer. [15] Conway, J. B. (1990). A Course in Functional Analysis. Springer. [16] Reed, M., & Simon, B. (1980). Methods of Modern Mathematical Physics: Functional Analysis. Academic Press. 6 [17] Zeidler, E. (1986). Nonlinear Functional Analysis and its Applications. Springer. [18] Kantorovich, L. V., & Akilov, G. P. (1982). Functional Analysis. Perg amon Press. [19] Vainberg, M. M. (1964). Variational Methods for the Study of Nonlinear Operators. Holden-Day. [20] Cioranescu, I. (1990). Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems. Kluwer. [21] Miller, K. S., & Ross, B. (1993). An Introduction to the Fractional Cal culus and Fractional Differential Equations. Wiley. [22] Podlubny, I. (1998). Fractional Differential Equations. Academic Press. [23] Samko, S. G., Kilbas, A. A., & Marichev, O. I. (1993). Fractional Inte grals and Derivatives. Gordon and Breach. [24] Kilbas, A. A., Srivastava, H. M., & Trujillo, J. J. (2006). Theory and Applications of Fractional Differential Equations. Elsevier. [25] Tarasov, V. E. (2010). Fractional Dynamics. Springer.