Extending the Calculus of Moving Surfaces to Higher Orders

TL;DR

This paper extends the Calculus of Moving Surfaces to higher orders, introducing advanced tensorial tools for analyzing the geometry and curvature of time-dependent, moving surfaces.

Contribution

It develops higher-order tensorial methods within CMS, enhancing the analysis of curvature and commutation relations on evolving surfaces.

Findings

Extended CMS to higher-order tensors.

Derived new relations for surface curvature and commutation.

Improved mathematical tools for analyzing moving surfaces.

Abstract

In 2010, a book published on the work of Jaques Hadamard, entitled "Introduction to Tensor Analysis and the Calculus of Moving Surfaces" by Dr. Pavel Grinfeld, proposed an extension of Hadamard's work to ultimately allow principles of tensorial invariance on surfaces to be extended to notions of time-dependent and moving surfaces. Coined "The Calculus of Moving Surfaces" (CMS), notions of Invariant Time Derivatives on Arbitrary Surface/Ambient Tensors, Surface Velocities, and time derivatives of time-dependent Volume & Surface Integrals were introduced. This paper focuses on extending concepts found within CMS to other Surface Objects, to Higher Orders, and to further uncover Tensors which are powerful at representing Commutations and Curvature present on Surfaces which are moving in Time.