Invertible field transformations with derivatives: necessary and sufficient conditions

TL;DR

This paper establishes explicit necessary and sufficient conditions for the local invertibility of field transformations involving derivatives, generalizing the inverse function theorem and applicable across physics and mathematics.

Contribution

It provides a rigorous framework using the method of characteristics to determine invertibility of derivative-involving field transformations, extending classical theorems.

Findings

Derived explicit invertibility criteria for derivative-based field transformations

Generalized the inverse function theorem for complex field transformations

Applicable to various fields in physics and mathematics

Abstract

We formulate explicitly the necessary and sufficient conditions for the local invertibility of a field transformation involving derivative terms. Our approach is to apply the method of characteristics of differential equations, by treating such a transformation as differential equations that give new variables in terms of original ones. The obtained results generalise the well-known and widely used inverse function theorem. Taking into account that field transformations are ubiquitous in modern physics and mathematics, our criteria for invertibility will find many useful applications.