APPLICATION OF THE SECOND COVARIANT DERIVATIVE FROM THE VECTOR FUNCTION TO THE PROBLEMS OF HYDRODYNAMICS AND ELASTICITY THEORY

Abstract

Based on the method described in our paper “Alternative calculation of covariant derivatives with an application to the problems of mechanics, physics and geometry”, we have calculated the result of the action of the Laplace operator on covariant and contravariant vector functions. The projections of the vectors B and C, defined as Bi = (∆A)i and Ci = (∆A)i, on the corresponding curvilinear orthonormal bases ei and ei are found. The general covariant expressions for the operators ∆A and graddivA that are valid in any curvilinear coordinate system are presented. As a reference material, the projections of the vector Ci = (∆A)i in a parabolic coordinate system are calculated. As an illustrative example, the problem of torsion of an elastically deformable vertically standing cylindrical body under the conditions of its uneven radial heating, provided that its lower end is rigidly fixed, is solved.

Keywords

• orthonormal unit basis
• covariant differentiation
• Christoffel symbol
• metric tensor
• parabolic coordinates
• volume expansion