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.
Based on the hypersonic entry in the entry-descending-landing procedure,the three-dimensional Navier-Stokes equations are solved by a parallel code to analyze the flow field structures, the aerodynamic characteristics and their variation patterns of the Mars Science Laboratory entering the Martian atmosphere with and without the chemical reaction models. The analysis shows that a large number of CO2 is dissociated behind the shock and much energy is consumed, with the chemical non-equilibrium effect, the shock layer is strongly compressed. The comparisons between the real gas and the perfect gas show that the vortex in the wake flow is reduced, the positions of the separation lines and the singularity are distinctly different; under the chemical non-equilibrium condition, the lift coefficient almost keeps the same, the drag coefficient increases, the lift-to-drag ratio and the pitch moment are smaller than those of the perfect gas.
In this paper, a vibration testing method to determine the stiffness of the badminton rod is presented. The badminton rod is simplified as a mass-spring system of a single degree of freedom, the natural frequency of which is measured by the vibration testing method. Then the stiffness parameter of the rod can be calculated by the frequency formula. The suggested vibration testing method is more convenient and accurate than the current flexible test method, and can be used as a standard method of judging the soft and hard degree of a badminton rod.