Angular velocity and intensity change of the basic vectors of position vector tangent space of a material system kinetic point—four examples

Abstract

Chapter starts from author’s previous published results about nonlinear transformations of coordinate systems, from affine space to functional-nonlinear curvilinear coordinate system and corresponding geometrical and kinematical invariants along nonlinear transformations of their coordinates from one to other coordinate system. In a curvilinear coordinate system, coordinates of a geometrical or kinematical point are not equal as coordinates of its’ corresponding position vector. Expressions of basic vectors of tangent space of kinetic point vector position in generalized curvilinear coordinate systems for the cases of orthogonal curvilinear coordinate systems are derived and four examples are presented. Next, expressions of change of basic vectors of tangent space of kinetic point vector position with time, also, are done. In this chapter, new and original expressions of angular velocity and velocity of dilatations of each of the basic vectors of tangent space of kinetic point vector position, in four orthogonal curvilinear coordinate systems are presented. List of these curvilinear coordinate systems are: three-dimensional elliptical cylindrical curvilinear coordinate system; generalized cylindrical bipolar curvilinear coordinate system; generalized elliptical curvilinear coordinate system, and generalized oblate spheroidal curvilinear coordinate system.