Integration by parts and by substitution unified, Green's Theorem and uniqueness for ODEs

TL;DR

This paper unifies integration by parts and substitution with Green's Theorem, revealing their equivalence and applying this insight to establish uniqueness results for ordinary differential equations.

Contribution

It introduces a lesser-known change of variables formula, demonstrating its equivalence to Green's Theorem and applying it to ODE uniqueness.

Findings

The change of variables formula is equivalent to Green's Theorem.

The formula has applications in proving ODE solution uniqueness.

Provides a new perspective linking integration techniques and differential equations.

Abstract

We present a rather unknown version of the change of variables formula for non-autonomous functions. We will show that this formula is equivalent to Green's Theorem for regions of the plane bounded by the graphs of two continuously differentiable functions. Besides, the formula has interesting applications in the uniqueness of solution of ordinary differential equations.