Uniqueness of Transformation based on Jacobian Determinant and curl-Vector

TL;DR

This paper investigates the conditions under which a transformation is uniquely determined by its Jacobian determinant and curl-vector, proposing an analytical approach to prove such uniqueness.

Contribution

It introduces a conjecture that combining Jacobian determinant and curl-vector ensures transformation uniqueness and provides an analytical framework to support this claim.

Findings

Numerical examples show Jacobian determinant alone does not guarantee uniqueness.

The paper proposes a new approach to prove transformation uniqueness using both Jacobian and curl-vector.

Analytical results support the conjecture that combined Jacobian and curl-vector determine a transformation uniquely.

Abstract

Numerical examples demonstrated that a prescribed positive Jacobian determinant alone can not uniquely determine a diffeomorphism. It is conjectured that the uniqueness of a transformation can be assured by its Jacobian determinant and the curl-vector. In this work, we study the uniqueness problem analytically and propose an approach to the proof of the uniqueness of a transformation with prescribed Jacobian determinant and curl-vector.