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.