The Anageometric Calculus: Theory, Fundamental Results, and Connections With Classical Analysis

In this study, we develop a systematic framework for anageometric calculus as an extension of non-Newtonian analysis. Fundamental concepts, including limits, continuity, differentiation, and higher order derivatives, are established, and the connection to classical calculus is clarified through a logarithmic transformation. Within this framework, we derive the properties of the anageometric derivative, inverse function relations, partial derivatives, and mean value theorems. The theory is extended to integration by establishing the first and second fundamental theorems of anageometric calculus. Furthermore, the structural implications of the anageometric derivative are investigated, demonstrating that the equation D∼f=λf naturally generates power-law solutions rather than classical exponential ones. This highlights the suitability of anageometric analysis for scale-invariant phenomena and proportional growth processes, supported by a practical heat-conduction example.