On the relevance of the differential expressions $f^2+f'^2$, $f+f"$ and $f f"- f'^2$ for the geometrical and mechanical properties of curves

TL;DR

This paper explores how specific differential expressions relate to the geometric and mechanical properties of curves, providing a unified analytic framework for understanding their significance.

Contribution

It introduces a unified approach highlighting the importance of certain differential expressions in analyzing curves' properties, both known and new.

Findings

Demonstrates the relevance of $f^2+f'^2$, $f+f''$, and $f f'' - f'^2$ in curve analysis

Establishes connections between differential expressions and geometric properties

Provides new insights into the mechanical behavior of curves

Abstract

We present a unified approach to known and new properties of curves by showing the ubiquity of the expressions in the title in the analytic treatment of their mechanical and geometric properties