Notion of a virtual derivative

TL;DR

The paper introduces a diagrammatic approach to derivatives, called virtual derivatives, which simplifies the calculation of derivatives for composed functions by visualizing and summing monomic components, aiding in differential geometry and differential equations.

Contribution

It proposes a novel graphical method for representing derivatives as virtual derivatives, reducing the need to explicitly compute all components of the derivative.

Findings

Diagrammatic representation of derivatives simplifies calculations.

Virtual derivatives allow for selective focus on needed components.

Potential applications in differential geometry and asymptotic expansions.

Abstract

Diagrams as a graphic expresion of derivatives is proposed for calculation of derivatives for composed function. The concret diagram is understood as a virtual derivative in contrast of concret derivative. In polynomial expression of functions derivative the concret derivative will be every monomic member, and the virtual derivative represent the sum of similar monomic members. The word virtual denotes that we dont need to know every virtual derivative, we don't write all the sequence of these virtual derivatives, and simply pick the needed one. This is in contrast of tradition to write the whole algebraic expresion as a denotion of whole function's derivative. Such graphic expresion can be helpful in the problems of differential geometry, in the various asymptotic expantions, also in the solution of some differential equations.