曲面物理和力学：最佳基本微分算子对

曲面物理和力学中有两个独立的基本微分算子(即"基本微分算子对"). 本文综述如下主题：在所有的基本微分算子对中,经典梯度▽(···) 和形状梯度▽ (···) 的配对▽,▽]] 是最佳的. 具体内容包括：(1)基本微分算子对的形式并不唯一;(2) 内积的可交换性确立了▽,▽]] 优于其他基本微分算子对的"最佳" 地位;(3) 基于▽,▽]] 可以最佳地构造曲面物理和力学的高阶标量微分算子,因而▽,▽]] 是构造曲面物理和力学微分方程的最佳"基本砖块";(4) ▽,▽]] 在软物质曲面物理和力学中普遍存在.

PHYSICS AND MECHANICS ON CURVED SURFACES: THE MOST OPTIMAL PAIRS OF FUNDAMENTAL DIFFERENTIAL OPERATORS

There are two independent fundamental differential operators (called the "fundamental differential operator pair") on curved surfaces. This paper focuses on the topic: Among all fundamental differential operator pairs, ▽,▽]], formed by the classical gradient ▽(···) and the shape gradient ▽ (···), is the optimal one. The following conclusions are included: (1) The paths for constructing the fundamental differential operator pairs are not unique. (2) The commutative nature of the inner-product of ▽,▽]] is the basis of its optimality and advantage over all other fundamental differential operator pairs. (3) Based on the inner-product of ▽,▽]], all higher order scalar differential operators for physics and mechanics on curved surfaces can be constructed optimally. In other words, ▽,▽]]is the optimal "fundamental brick" for establishing the differential equations of physics and mechanics on curved surfaces. (4) ▽,▽]] exists universally in physics and mechanics on soft matter curved surfaces.