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.

从线性和非线性变换的概念出发，指出了坐标和坐标系的仿射和函数非线性映射，沿着线性或非线性变换的几何和运动不变量，它们的坐标从一个坐标系到另一个坐标系。在曲线坐标系中，几何或运动学点的坐标与其对应位置向量的坐标不相等。推导出正交曲线坐标系和非正交曲线坐标系情况下广义曲线坐标系中运动点向量位置切空间的基本向量表达式。极柱、球面运动学点向量位置切空间基本向量表达式实例 提出了曲线直角坐标系的抛物-圆柱和三维-三抛物线系统。接着，也给出了运动学点向量位置切空间的基本向量随时间变化的表达式。在广义曲线坐标系中，指出并确定了该动力学质点位置向量的几何（物理）、协变和逆变坐标，在由该动力学点向量位置的切空间基本向量确定的坐标系中。在广义曲线坐标系以及一系列特殊正交曲线坐标系中，推导出动点矢量位置切空间基本矢量的角速度和膨胀速度的原始表达式为该论文的作者和介绍。