The general basis-free spin and its concise proof

In this paper, the general basis-free spin expressions are presented. Combining with the representation theorem of a multivariable tensor function, a concise proof is given. First, the basic concept of spin is introduced, and several spins are given, which are Lagrangian spin, Eulerian spin, relative spin, and logarithmic spin. Then, the representation theorem of a multivariable tensor function is given, and based on this theorem and Rivlin identity, a tensor equation is solved. Finally, combining with the solution of the tensor equation, several basis-free spin expressions are presented. The solution process of the tensor equation is simple, avoiding the tedious process of determining coefficients, and the basis-free spin is simple and elegant. The simplicity and generality of the whole method are of great significance for understanding the nature of spin: any spin can be expressed as the double dot product of a fourth-order tensor of a certain stretch tensor and a certain deformation rate tensor. These various spins are eventually unified in form.