Linear and non-linear transformation of coordinates and angular velocity and intensity change of basic vectors of tangent space of a position vector of a material system kinetic point

Abstract

Starting from the notion of linear and nonlinear transformations, affine and functional-nonlinear mappings of coordinates and coordinate systems, geometrical and kinematical invariants along linear or nonlinear transformations their coordinates from one to other coordinate system are pointed out. 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 kinematical point vector positions in generalized curvilinear coordinate systems for the cases of orthogonal and nonorthogonal curvilinear coordinate systems are derived. Examples of expressions of basic vectors of tangent space of kinematical point vector position in polar-cylindrical, spherical, parabolic-cylindrical and three-dimensional-three-parabolic system of curvilinear orthogonal coordinates are presented. Next, expressions of change of basic vectors of tangent space of kinematical point vector position with time, also, are done. Geometrical (physical), covariant and contra-variant coordinates of position vector of a kinetic mass point, in a coordinate system determined by basic vectors of tangent space of this kinetic point vector position, in generalized curvilinear coordinate systems, are pointed out and determined. Original expressions of angular velocity and velocity of dilatations of basic vectors of tangent space of kinetic point vector position, in generalized curvilinear coordinate systems as well as in series of special orthogonal curvilinear coordinate system are derived by author of this paper and presented.