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The Specular Derivative

Kiyuob Jung, Jehan Oh

TL;DR

The paper introduces the specular derivative, a generalized one- and multi-dimensional derivative $f^{\wedge}$ that encodes left/right slope information via a phototangent, enabling analysis of nonsmooth behavior. It develops foundational results, including Quasi-Rolle's Theorem, Quasi-Mean Value Theorem, and a Fundamental Theorem of Calculus analogue in the specular setting, and then extends the framework to $\mathbb{R}^n$ with the specular gradient and specular partial derivatives. The authors apply the theory to ordinary and partial differential equations, notably a transport-type PDE, providing explicit solution strategies and illustrating with examples such as the ReLU and sign functions. An accompanying appendix supplies delayed proofs and notation to support the main results, highlighting the framework's potential for capturing and computing nonsmooth phenomena in analysis and differential equations.

Abstract

In this paper, we introduce a new generalized derivative, which we term the specular derivative. We establish the Quasi-Rolles' Theorem, the Quasi-Mean Value Theorem, and the Fundamental Theorem of Calculus in light of the specular derivative. We also investigate various analytic and geometric properties of specular derivatives and apply these properties to several differential equations.

The Specular Derivative

TL;DR

The paper introduces the specular derivative, a generalized one- and multi-dimensional derivative that encodes left/right slope information via a phototangent, enabling analysis of nonsmooth behavior. It develops foundational results, including Quasi-Rolle's Theorem, Quasi-Mean Value Theorem, and a Fundamental Theorem of Calculus analogue in the specular setting, and then extends the framework to with the specular gradient and specular partial derivatives. The authors apply the theory to ordinary and partial differential equations, notably a transport-type PDE, providing explicit solution strategies and illustrating with examples such as the ReLU and sign functions. An accompanying appendix supplies delayed proofs and notation to support the main results, highlighting the framework's potential for capturing and computing nonsmooth phenomena in analysis and differential equations.

Abstract

In this paper, we introduce a new generalized derivative, which we term the specular derivative. We establish the Quasi-Rolles' Theorem, the Quasi-Mean Value Theorem, and the Fundamental Theorem of Calculus in light of the specular derivative. We also investigate various analytic and geometric properties of specular derivatives and apply these properties to several differential equations.
Paper Structure (18 sections, 25 theorems, 205 equations, 11 figures, 1 table)

This paper contains 18 sections, 25 theorems, 205 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. Specular derivatives for single-variable functions
  3. Definitions and properties
  4. Application
  5. Higher order specular derivatives
  6. Specular derivatives for multi-variable functions
  7. Definitions and properties
  8. The specular gradient and specularly directional derivatives
  9. Differential equations with specular derivatives
  10. The Fundamental Theorem of Calculus with specular derivatives
  11. Ordinary differential equations with specular derivatives
  12. Partial differential equations with specular derivatives
  13. Appendix
  14. Delayed proofs
  15. Proof of Proposition \ref{['Prop : Calculating spd']}
  16. ...and 3 more sections

Key Result

Proposition 2.6

Let $f: I \to \mathbb{R}$ be a function on an open interval $I \subset \mathbb{R}$ and $x_0$ be a point in $I$. Suppose there exists a phototangent, say $\operatorname{pht}f$, of $f$ at $x_0$. Then $f$ is specularly differentiable at $x_0$ if and only if $\operatorname{pht}f$ is continuous at $x_0$.

Figures (11)

  • Figure 1: Motivation for specular derivatives
  • Figure 2: The blueprint for specular derivatives in one-dimension
  • Figure 3: Basic concepts concerning specular derivatives
  • Figure 4: The slope of the line GH converges the specular derivative of $f$ at $x$
  • Figure 5: Basic concepts concerning specularly partial derivatives
  • ...and 6 more figures

Theorems & Definitions (89)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Definition 2.5
  • Proposition 2.6
  • proof
  • Corollary 2.7
  • Example 2.8
  • Example 2.9
  • ...and 79 more