Mathematical Constraint on Functions with Continuous Second Partial Derivatives

TL;DR

This paper derives a new integral identity for functions with continuous second partial derivatives, showing their current values are fully determined by past values across space, under certain conditions, revealing fundamental constraints in physics-related functions.

Contribution

It introduces a novel integral identity that constrains functions with continuous second derivatives, linking current values to all past spatial points.

Findings

Functions with continuous second derivatives are fully determined by their past values.

The derived identity applies to many physical functions, indicating fundamental constraints.

Functions must vanish at infinity for the identity to hold.

Abstract

A new integral identity for functions with continuous second partial derivatives is derived. It is shown that the value of any function f(r,t) at position r and time t is completely determined by its previous values at all other locations r' and retarded times t'<t, provided that the function vanishes at infinity and has continuous second partial derivatives. Functions of this kind occur in many areas of physics and it seems somewhat surprising that they are constrained in this way.