Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 1806.02335

Extending the Calculus of Moving Surfaces to Higher Orders

Keith C. Afas
View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

math.GM

Keep this discovery

BibTeXRIS

Keith C. Afas. 2018-06-06. Extending the Calculus of Moving Surfaces to Higher Orders. https://arxiv.org/abs/1806.02335

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Average Chord Lengths in a Triangle

Let $P$ be a point inside a triangle $T$. We consider the average length of the chords of $T$ through $P$, where the direction of the chord is chosen uniformly. An elementary formula is obtained in terms of the distances from $P$ to the sides and vertices of the triangle. Several classical triangle centers give especially simple specializations. For example, if $I$ is the incenter, then \[ M_T(I)=\frac{2r}{\pi} \log\left(\cot\frac A4\cot\frac B4\cot\frac C4\right). \] Our main result is the sharp inequality \[ M_T(P)\le \frac{p}{\pi\sqrt3}\log(2+\sqrt3), \] valid simultaneously for every triangle of perimeter $p$ and every interior point $P$. Thus, among all such pairs $(T,P)$, the largest possible average chord length occurs only when $T$ is equilateral and $P$ is its center. The proof is an elementary symmetrization argument. We close with brief remarks relating the problem to the radial center of a convex body, the electrostatic potential center of a triangle, and dual quermassintegrals.

math.GM↗
arXiv2026

A Proof of Liu's Conjecture on the Fundamental Triangle Inequality

Let $a,b,c$ be the side lengths of a triangle, and let $R$ and $r$ denote its circumradius and inradius, respectively. We prove a conjecture of Liu stating that \[\sum_{\mathrm{cyc}} \left(\frac{a(b+c-a)}{bc}\right)^k \geq 2+\left(\frac{2r}{R}\right)^k,~~k>1, \] with the reverse inequality for $0<k<1$. The proof reduces the problem to three positive variables with fixed sum and product. We also determine the equality cases.

math.GM↗