mean curvature and 2nd variation

[AntonioLao]

It has occurred when reading Goldstein’s ‘Classical Mechanics’ Page 146, Section 5-2 that a tensor is really the quotient of vector division. Here, he said about quotients in general belonging to a different class, a much more complicated class of mathematical objects, meaning that the quotient of two integers is in general not an integer but rather a rational number. Consequently, the quotient of two vectors, as is well known in studying physics, cannot be well defined with any consistency among the classes of physical vectors: force, velocity, acceleration, and momentum, etc. It is therefore logical to define the quotient of vector division as a new type of quantity properly called a second rank tensor, reserving first rank tensor as vector and zero rank tensor as dimensioned scalar. Nevertheless, what rank tensor is a dimensionless scalar? Is there a dimensionless vector?

[Guille]

I can't answer those questions. I think that this part of math (scalars, vectors, tensors, matrix, spinors and twistors) are still big for me, this six months I'm going to be very profoundly studying them, and I hope I can cover some gaps in it, like the questions you ponder.

[AntonioLao]

I can't answer those questions.

Although most physicists as well as some mathematicians believed that the physical theory of fluids could be based entirely on vector analysis, the theory of elasticity demanded exclusive working knowledge of tensor analysis.

However, in anisotropic fluids, such as liquid crystals (LCs), the physics of these substances have characteristic features of ordinary liquids and elastic media. Hence, LCs are intermediate between fluid mechanics and elastic mechanics and must then be analyzed using both vectors and tensors.

LCs can be classified into three main categories (1) nematics, (2) cholesterics, and (3) smectics. Nevertheless, a common feature shared among these classes of LCs is the existence of an axis of symmetry unique for each particle within a given LC domain (analogous to time axis and space axis), of which the positive and negative axes are directionally equivalent. That is to say, there is no distinction between them in physical equations.

Controlling these axes of symmetry compared to the feasibility of controlling the time axis and space axis inherent in all space-time events. This control is possible if and only if space-time is a model of a perfect liquid crystal.

Footnote: It seems clear that the theory of general relativity is a theory of elasticity applied to the entire universe because of its reliance on tensor analysis. On the other hand, quantum mechanics is a theory of fluids applied to the infinitesimal universe relying on the concept of eigenvectors or orthogonal vectors. The introduction of spinors was the beginning of a first step synthesis between vectors and tensors into a theory of quantized anisotropic fluid field of liquid crystals. In this case, the global homogeneity and isotropy of the true vacuum merged with the local inhomogeneity and anisotropy of stars and galaxies.