Invertible field transformations with derivatives: necessary and sufficient conditions

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.

Automated summary

This article explicitly formulates the necessary and sufficient conditions for the local invertibility of a field transformation involving derivative terms, by applying the method of characteristics of differential equations. The obtained results generalize the widely used inverse function theorem and will have many useful applications in modern physics and mathematics where field transformations are common.