Property of Tensor Satisfying Binary Law 2
Koji Ichidayama

Abstract

I have already reported “Property of Tensor Satisfying Binary Law”. This article is the article that I revise the contents of “Property of Tensor Satisfying Binary Law”, and increase the report about new characteristics. We may arrive at the deeper understanding in this about “Property of Tensor Satisfying Binary Law”.

Keywords

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Ichidayama, K. (2020) Property of Tensor Satisfying Binary Law 2. Journal of Modern Physics, 11, 1649-1671. doi: 10.4236/jmp.2020.1110103.

1. Introduction

I have already reported “Property of Tensor Satisfying Binary Law” . This article is the article that I revise the contents of “Property of Tensor Satisfying Binary Law”, and increase the report about new characteristics. I show below it about the proposition supporting each for shifts from “Property of Tensor Satisfying Binary Law” to “[Property of Tensor Satisfying Binary Law 2]”.

Proposition 1→[Proposition 2], Proposition 3→[Proposition 6], Proposition 4→[Proposition 7, Proposition 15], Proposition 5→[Proposition 3].

2. Definition

Definition 1 $\stackrel{¯}{{x}^{\mu }}\text{\hspace{0.17em}}\ne {x}^{\mu }$, $\stackrel{¯}{{x}^{\nu }}\text{\hspace{0.17em}}\ne {x}^{\nu }$, $\stackrel{¯}{{x}^{\mu }}\text{\hspace{0.17em}}={x}^{\nu }$, $\stackrel{¯}{{x}^{\nu }}\text{\hspace{0.17em}}={x}^{\mu }$, $\left\{\stackrel{¯}{{x}_{\mu }}\text{\hspace{0.17em}}\ne {x}_{\mu }$, $\stackrel{¯}{{x}_{\nu }}\text{\hspace{0.17em}}\ne {x}_{\nu }$, $\stackrel{¯}{{x}_{\mu }}={x}_{\nu }$, $\stackrel{¯}{{x}_{\nu }}={x}_{\mu }\right\}$ is established . I named $\stackrel{¯}{{x}^{\mu }}\ne {x}^{\mu }$, $\stackrel{¯}{{x}^{\nu }}\ne {x}^{\nu }$, $\stackrel{¯}{{x}^{\mu }}={x}^{\nu }$, $\stackrel{¯}{{x}^{\nu }}={x}^{\mu }$, $\left\{\stackrel{¯}{{x}_{\mu }}\ne {x}_{\mu },\text{\hspace{0.17em}}\stackrel{¯}{{x}_{\nu }}\ne {x}_{\nu },\text{\hspace{0.17em}}\stackrel{¯}{{x}_{\mu }}={x}_{\nu },\text{\hspace{0.17em}}\stackrel{¯}{{x}_{\nu }}={x}_{\mu }\right\}$ “Binary Law” . $\left\{\stackrel{¯}{{x}_{\mu }}\ne {x}_{\mu },\stackrel{¯}{{x}_{\nu }}\ne {x}_{\nu },\stackrel{¯}{{x}_{\mu }}={x}_{\nu },\stackrel{¯}{{x}_{\nu }}={x}_{\mu }\right\}$ expresses a covariant form of Binary Law.

Definition 2 If $\stackrel{¯}{{x}^{\mu }}\ne {x}^{\mu },\stackrel{¯}{{x}^{\nu }}\ne {x}^{\nu },\stackrel{¯}{{x}^{\mu }}={x}^{\nu },\stackrel{¯}{{x}^{\nu }}={x}^{\mu }$ is established, ${x}_{\nu }={x}^{\mu }$ is established .

Definition 3 If $\stackrel{¯}{{x}^{\mu }}\ne {x}^{\mu },\stackrel{¯}{{x}^{\nu }}\ne {x}^{\nu },\stackrel{¯}{{x}^{\mu }}={x}^{\nu },\stackrel{¯}{{x}^{\nu }}={x}^{\mu }$ is established, ${x}_{\mu }={x}^{\nu }$ is established .

Definition 4 If $\stackrel{¯}{{x}^{\mu }}\ne {x}^{\mu },\stackrel{¯}{{x}^{\nu }}\ne {x}^{\nu },\stackrel{¯}{{x}^{\mu }}={x}^{\nu },\stackrel{¯}{{x}^{\nu }}={x}^{\mu }$ is established, ${x}_{\nu }=-{x}_{\mu }$ is established .

Definition 5 If $\stackrel{¯}{{x}^{\mu }}\ne {x}^{\mu },\stackrel{¯}{{x}^{\nu }}\ne {x}^{\nu },\stackrel{¯}{{x}^{\mu }}={x}^{\nu },\stackrel{¯}{{x}^{\nu }}={x}^{\mu }$ is established, ${x}^{\nu }=-{x}^{\mu }$ is established .

Definition 6 If all coordinate systems ${x}^{\mu },{x}^{\nu },{x}^{\sigma },{x}^{\lambda },\cdots$ satisfy $\stackrel{¯}{{x}^{\mu }}\ne {x}^{\mu }$, $\stackrel{¯}{{x}^{\nu }}\ne {x}^{\nu }$, $\stackrel{¯}{{x}^{\mu }}={x}^{\nu }$, $\stackrel{¯}{{x}^{\nu }}={x}^{\mu }$, all coordinate systems ${x}^{\mu },{x}^{\nu },{x}^{\sigma },{x}^{\lambda },\cdots$ shifts to only two of ${x}^{\mu },{x}^{\nu }$ .

Definition 7 ${g}_{\mu }^{\mu }=1$, ${g}_{\nu }^{\mu }=0:\left(\mu \ne \nu \right)$ is establishment .

Definition 8 $\frac{{\partial }^{3}{x}^{\mu }}{\partial {x}^{\nu }\partial {x}^{\nu }\partial {x}^{\nu }}=M$ is established for $\frac{{\partial }^{3}{x}^{\mu }}{\partial {x}^{\nu }\partial {x}^{\nu }\partial {x}^{\nu }}$.

Definition 9 ${m}_{;\nu }=\frac{\partial m}{\partial {x}^{\nu }}=0$ is established. “m” expresses Mass.

Hypothesis 1 $m\propto \text{\hspace{0.17em}}M$, $m=ϵM$ is established. “M” expresses $\frac{{\partial }^{3}{x}^{\mu }}{\partial {x}^{\nu }\partial {x}^{\nu }\partial {x}^{\nu }}=M$, “ $ϵ$ ” expresses Proportional constant, and “m” expresses Mass.

Definition 10 The first-order covariant derivative of the covariant vector satisfied ${x}_{\mu ;\nu }=\frac{\partial {x}_{\mu }}{\partial {x}^{\nu }}-{x}_{\tau }{\Gamma }_{\mu \nu }^{\tau }=\frac{\partial {x}_{\mu }}{\partial {x}^{\nu }}-{x}_{\tau }\frac{1}{2}{g}^{ϵ\tau }\left(\frac{\partial {g}_{\mu ϵ}}{\partial {x}^{\nu }}+\frac{\partial {g}_{\nu ϵ}}{\partial {x}^{\mu }}-\frac{\partial {g}_{\mu \nu }}{\partial {x}^{ϵ}}\right)$ .

Definition 11 The first-order covariant derivative of the contravariant vector satisfied ${x}_{;\nu }^{\mu }=\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}+{x}^{\tau }{\Gamma }_{\tau \nu }^{\mu }=\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}+{x}^{\tau }\frac{1}{2}{g}^{ϵ\mu }\left(\frac{\partial {g}_{\tau ϵ}}{\partial {x}^{\nu }}+\frac{\partial {g}_{\nu ϵ}}{\partial {x}^{\tau }}-\frac{\partial {g}_{\tau \nu }}{\partial {x}^{ϵ}}\right)$ .

Definition 12 The second-order covariant derivative of the contravariant vector satisfied

$\begin{array}{c}{x}_{;\nu ;\sigma }^{\mu }=\frac{\partial {x}_{;\nu }^{\mu }}{\partial {x}^{\sigma }}+{x}_{;\nu }^{\iota }{\Gamma }_{\iota \sigma }^{\mu }-{x}_{;\iota }^{\mu }{\Gamma }_{\nu \sigma }^{\iota }\\ =\frac{\partial }{\partial {x}^{\sigma }}\left(\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}+{x}^{\tau }{\Gamma }_{\tau \nu }^{\mu }\right)+\left(\frac{\partial {x}^{\iota }}{\partial {x}^{\nu }}+{x}^{\tau }{\Gamma }_{\tau \nu }^{\iota }\right){\Gamma }_{\iota \sigma }^{\mu }-\left(\frac{\partial {x}^{\mu }}{\partial {x}^{\iota }}+{x}^{\tau }{\Gamma }_{\tau \iota }^{\mu }\right){\Gamma }_{\nu \sigma }^{\iota }\\ =\frac{{\partial }^{2}{x}^{\mu }}{\partial {x}^{\nu }\partial {x}^{\sigma }}+\frac{\partial }{\partial {x}^{\sigma }}\left({x}^{\tau }\frac{1}{2}{g}^{ϵ\mu }\left(\frac{\partial {g}_{\tau ϵ}}{\partial {x}^{\nu }}+\frac{\partial {g}_{\nu ϵ}}{\partial {x}^{\tau }}-\frac{\partial {g}_{\tau \nu }}{\partial {x}^{ϵ}}\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{\partial {x}^{\iota }}{\partial {x}^{\nu }}\frac{1}{2}{g}^{ϵ\mu }\left(\frac{\partial {g}_{\iota ϵ}}{\partial {x}^{\sigma }}+\frac{\partial {g}_{\sigma ϵ}}{\partial {x}^{\iota }}-\frac{\partial {g}_{\iota \sigma }}{\partial {x}^{ϵ}}\right)\end{array}$

$\begin{array}{c}+{x}^{\tau }\frac{1}{2}{g}^{ϵ\iota }\left(\frac{\partial {g}_{\tau ϵ}}{\partial {x}^{\nu }}+\frac{\partial {g}_{\nu ϵ}}{\partial {x}^{\tau }}-\frac{\partial {g}_{\tau \nu }}{\partial {x}^{ϵ}}\right)\frac{1}{2}{g}^{ϵ\mu }\left(\frac{\partial {g}_{\iota ϵ}}{\partial {x}^{\sigma }}+\frac{\partial {g}_{\sigma ϵ}}{\partial {x}^{\iota }}-\frac{\partial {g}_{\iota \sigma }}{\partial {x}^{ϵ}}\right)\\ \text{ }-\frac{\partial {x}^{\mu }}{\partial {x}^{\iota }}\frac{1}{2}{g}^{ϵ\iota }\left(\frac{\partial {g}_{\nu ϵ}}{\partial {x}^{\sigma }}+\frac{\partial {g}_{\sigma ϵ}}{\partial {x}^{\nu }}-\frac{\partial {g}_{\nu \sigma }}{\partial {x}^{ϵ}}\right)\\ \text{ }-{x}^{\tau }\frac{1}{2}{g}^{ϵ\mu }\left(\frac{\partial {g}_{\tau ϵ}}{\partial {x}^{\iota }}+\frac{\partial {g}_{\iota ϵ}}{\partial {x}^{\tau }}-\frac{\partial {g}_{\tau \iota }}{\partial {x}^{ϵ}}\right)\frac{1}{2}{g}^{ϵ\iota }\left(\frac{\partial {g}_{\nu ϵ}}{\partial {x}^{\sigma }}+\frac{\partial {g}_{\sigma ϵ}}{\partial {x}^{\nu }}-\frac{\partial {g}_{\nu \sigma }}{\partial {x}^{ϵ}}\right).\end{array}$ 

Definition 13 The third-order covariant derivative of the contravariant vector satisfied

$\begin{array}{l}{x}_{;\nu ;\sigma ;\lambda }^{\mu }=\frac{\partial {x}_{;\nu ;\sigma }^{\mu }}{\partial {x}^{\lambda }}+{x}_{;\nu ;\sigma }^{\kappa }{\Gamma }_{\kappa \lambda }^{\mu }-{x}_{;\kappa ;\sigma }^{\mu }{\Gamma }_{\nu \lambda }^{\kappa }-{x}_{;\nu ;\kappa }^{\mu }{\Gamma }_{\sigma \lambda }^{\kappa }\\ =\frac{\partial }{\partial {x}^{\lambda }}\left\{\frac{\partial }{\partial {x}^{\sigma }}\left(\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}+{x}^{\tau }{\Gamma }_{\tau \nu }^{\mu }\right)+\left(\frac{\partial {x}^{\iota }}{\partial {x}^{\nu }}+{x}^{\tau }{\Gamma }_{\tau \nu }^{\iota }\right){\Gamma }_{\iota \sigma }^{\mu }-\left(\frac{\partial {x}^{\mu }}{\partial {x}^{\iota }}+{x}^{\tau }{\Gamma }_{\tau \iota }^{\mu }\right){\Gamma }_{\nu \sigma }^{\iota }\right\}\end{array}$

$\begin{array}{l}\text{ }+\left\{\frac{\partial }{\partial {x}^{\sigma }}\left(\frac{\partial {x}^{\kappa }}{\partial {x}^{\nu }}+{x}^{\tau }{\Gamma }_{\tau \nu }^{\kappa }\right)+\left(\frac{\partial {x}^{\iota }}{\partial {x}^{\nu }}+{x}^{\tau }{\Gamma }_{\tau \nu }^{\iota }\right){\Gamma }_{\iota \sigma }^{\kappa }-\left(\frac{\partial {x}^{\kappa }}{\partial {x}^{\iota }}+{x}^{\tau }{\Gamma }_{\tau \iota }^{\kappa }\right){\Gamma }_{\nu \sigma }^{\iota }\right\}{\Gamma }_{\kappa \lambda }^{\mu }\\ \text{ }-\left\{\frac{\partial }{\partial {x}^{\sigma }}\left(\frac{\partial {x}^{\mu }}{\partial {x}^{\kappa }}+{x}^{\tau }{\Gamma }_{\tau \kappa }^{\mu }\right)+\left(\frac{\partial {x}^{\iota }}{\partial {x}^{\kappa }}+{x}^{\tau }{\Gamma }_{\tau \kappa }^{\iota }\right){\Gamma }_{\iota \sigma }^{\mu }-\left(\frac{\partial {x}^{\mu }}{\partial {x}^{\iota }}+{x}^{\tau }{\Gamma }_{\tau \iota }^{\mu }\right){\Gamma }_{\kappa \sigma }^{\iota }\right\}{\Gamma }_{\nu \lambda }^{\kappa }\\ \text{ }-\left\{\frac{\partial }{\partial {x}^{\kappa }}\left(\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}+{x}^{\tau }{\Gamma }_{\tau \nu }^{\mu }\right)+\left(\frac{\partial {x}^{\iota }}{\partial {x}^{\nu }}+{x}^{\tau }{\Gamma }_{\tau \nu }^{\iota }\right){\Gamma }_{\iota \kappa }^{\mu }-\left(\frac{\partial {x}^{\mu }}{\partial {x}^{\iota }}+{x}^{\tau }{\Gamma }_{\tau \iota }^{\mu }\right){\Gamma }_{\nu \kappa }^{\iota }\right\}{\Gamma }_{\sigma \lambda }^{\kappa }\end{array}$

$\begin{array}{l}=\frac{{\partial }^{3}{x}^{\mu }}{\partial {x}^{\nu }\partial {x}^{\sigma }\partial {x}^{\lambda }}+\frac{{\partial }^{2}}{\partial {x}^{\sigma }\partial {x}^{\lambda }}\left({x}^{\tau }\frac{1}{2}{g}^{ϵ\mu }\left(\frac{\partial {g}_{\tau ϵ}}{\partial {x}^{\nu }}+\frac{\partial {g}_{\nu ϵ}}{\partial {x}^{\tau }}-\frac{\partial {g}_{\tau \nu }}{\partial {x}^{ϵ}}\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{\partial }{\partial {x}^{\lambda }}\left(\frac{\partial {x}^{\iota }}{\partial {x}^{\nu }}\frac{1}{2}{g}^{ϵ\mu }\left(\frac{\partial {g}_{\iota ϵ}}{\partial {x}^{\sigma }}+\frac{\partial {g}_{\sigma ϵ}}{\partial {x}^{\iota }}-\frac{\partial {g}_{\iota \sigma }}{\partial {x}^{ϵ}}\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{\partial }{\partial {x}^{\lambda }}\left({x}^{\tau }\frac{1}{2}{g}^{ϵ\iota }\left(\frac{\partial {g}_{\tau ϵ}}{\partial {x}^{\nu }}+\frac{\partial {g}_{\nu ϵ}}{\partial {x}^{\tau }}-\frac{\partial {g}_{\tau \nu }}{\partial {x}^{ϵ}}\right)\frac{1}{2}{g}^{ϵ\mu }\left(\frac{\partial {g}_{\iota ϵ}}{\partial {x}^{\sigma }}+\frac{\partial {g}_{\sigma ϵ}}{\partial {x}^{\iota }}-\frac{\partial {g}_{\iota \sigma }}{\partial {x}^{ϵ}}\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\frac{\partial }{\partial {x}^{\lambda }}\left(\frac{\partial {x}^{\mu }}{\partial {x}^{\iota }}\frac{1}{2}{g}^{ϵ\iota }\left(\frac{\partial {g}_{\nu ϵ}}{\partial {x}^{\sigma }}+\frac{\partial {g}_{\sigma ϵ}}{\partial {x}^{\nu }}-\frac{\partial {g}_{\nu \sigma }}{\partial {x}^{ϵ}}\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\frac{\partial }{\partial {x}^{\lambda }}\left({x}^{\tau }\frac{1}{2}{g}^{ϵ\mu }\left(\frac{\partial {g}_{\tau ϵ}}{\partial {x}^{\iota }}+\frac{\partial {g}_{\iota ϵ}}{\partial {x}^{\tau }}-\frac{\partial {g}_{\tau \iota }}{\partial {x}^{ϵ}}\right)\frac{1}{2}{g}^{ϵ\iota }\left(\frac{\partial {g}_{\nu ϵ}}{\partial {x}^{\sigma }}+\frac{\partial {g}_{\sigma ϵ}}{\partial {x}^{\nu }}-\frac{\partial {g}_{\nu \sigma }}{\partial {x}^{ϵ}}\right)\right)\end{array}$

$\begin{array}{l}\text{ }+\frac{{\partial }^{2}{x}^{\kappa }}{\partial {x}^{\nu }\partial {x}^{\sigma }}\frac{1}{2}{g}^{ϵ\mu }\left(\frac{\partial {g}_{\kappa ϵ}}{\partial {x}^{\lambda }}+\frac{\partial {g}_{\lambda ϵ}}{\partial {x}^{\kappa }}-\frac{\partial {g}_{\kappa \lambda }}{\partial {x}^{ϵ}}\right)\\ \text{ }+\frac{\partial }{\partial {x}^{\sigma }}\left({x}^{\tau }\frac{1}{2}{g}^{ϵ\kappa }\left(\frac{\partial {g}_{\tau ϵ}}{\partial {x}^{\nu }}+\frac{\partial {g}_{\nu ϵ}}{\partial {x}^{\tau }}-\frac{\partial {g}_{\tau \nu }}{\partial {x}^{ϵ}}\right)\right)\frac{1}{2}{g}^{ϵ\mu }\left(\frac{\partial {g}_{\kappa ϵ}}{\partial {x}^{\lambda }}+\frac{\partial {g}_{\lambda ϵ}}{\partial {x}^{\kappa }}-\frac{\partial {g}_{\kappa \lambda }}{\partial {x}^{ϵ}}\right)\\ \text{ }+\frac{\partial {x}^{\iota }}{\partial {x}^{\nu }}\frac{1}{2}{g}^{ϵ\kappa }\left(\frac{\partial {g}_{\iota ϵ}}{\partial {x}^{\sigma }}+\frac{\partial {g}_{\sigma ϵ}}{\partial {x}^{\iota }}-\frac{\partial {g}_{\iota \sigma }}{\partial {x}^{ϵ}}\right)\frac{1}{2}{g}^{ϵ\mu }\left(\frac{\partial {g}_{\kappa ϵ}}{\partial {x}^{\lambda }}+\frac{\partial {g}_{\lambda ϵ}}{\partial {x}^{\kappa }}-\frac{\partial {g}_{\kappa \lambda }}{\partial {x}^{ϵ}}\right)\\ \text{ }+{x}^{\tau }\frac{1}{2}{g}^{ϵ\iota }\left(\frac{\partial {g}_{\tau ϵ}}{\partial {x}^{\nu }}+\frac{\partial {g}_{\nu ϵ}}{\partial {x}^{\tau }}-\frac{\partial {g}_{\tau \nu }}{\partial {x}^{ϵ}}\right)\frac{1}{2}{g}^{ϵ\kappa }\left(\frac{\partial {g}_{\iota ϵ}}{\partial {x}^{\sigma }}+\frac{\partial {g}_{\sigma ϵ}}{\partial {x}^{\iota }}-\frac{\partial {g}_{\iota \sigma }}{\partial {x}^{ϵ}}\right)\\ \text{ }×\frac{1}{2}{g}^{ϵ\mu }\left(\frac{\partial {g}_{\kappa ϵ}}{\partial {x}^{\lambda }}+\frac{\partial {g}_{\lambda ϵ}}{\partial {x}^{\kappa }}-\frac{\partial {g}_{\kappa \lambda }}{\partial {x}^{ϵ}}\right)\end{array}$

$\begin{array}{l}\text{ }-\frac{\partial {x}^{\kappa }}{\partial {x}^{\iota }}\frac{1}{2}{g}^{ϵ\iota }\left(\frac{\partial {g}_{\nu ϵ}}{\partial {x}^{\sigma }}+\frac{\partial {g}_{\sigma ϵ}}{\partial {x}^{\nu }}-\frac{\partial {g}_{\nu \sigma }}{\partial {x}^{ϵ}}\right)\frac{1}{2}{g}^{ϵ\mu }\left(\frac{\partial {g}_{\kappa ϵ}}{\partial {x}^{\lambda }}+\frac{\partial {g}_{\lambda ϵ}}{\partial {x}^{\kappa }}-\frac{\partial {g}_{\kappa \lambda }}{\partial {x}^{ϵ}}\right)\\ \text{ }-{x}^{\tau }\frac{1}{2}{g}^{ϵ\kappa }\left(\frac{\partial {g}_{\tau ϵ}}{\partial {x}^{\iota }}+\frac{\partial {g}_{\iota ϵ}}{\partial {x}^{\tau }}-\frac{\partial {g}_{\tau \iota }}{\partial {x}^{ϵ}}\right)\frac{1}{2}{g}^{ϵ\iota }\left(\frac{\partial {g}_{\nu ϵ}}{\partial {x}^{\sigma }}+\frac{\partial {g}_{\sigma ϵ}}{\partial {x}^{\nu }}-\frac{\partial {g}_{\nu \sigma }}{\partial {x}^{ϵ}}\right)\\ \text{ }×\frac{1}{2}{g}^{ϵ\mu }\left(\frac{\partial {g}_{\kappa ϵ}}{\partial {x}^{\lambda }}+\frac{\partial {g}_{\lambda ϵ}}{\partial {x}^{\kappa }}-\frac{\partial {g}_{\kappa \lambda }}{\partial {x}^{ϵ}}\right)-\frac{{\partial }^{2}{x}^{\mu }}{\partial {x}^{\kappa }\partial {x}^{\sigma }}\frac{1}{2}{g}^{ϵ\kappa }\left(\frac{\partial {g}_{\nu ϵ}}{\partial {x}^{\lambda }}+\frac{\partial {g}_{\lambda ϵ}}{\partial {x}^{\nu }}-\frac{\partial {g}_{\nu \lambda }}{\partial {x}^{ϵ}}\right)\\ \text{ }-\frac{\partial }{\partial {x}^{\sigma }}\left({x}^{\tau }\frac{1}{2}{g}^{ϵ\mu }\left(\frac{\partial {g}_{\tau ϵ}}{\partial {x}^{\kappa }}+\frac{\partial {g}_{\kappa ϵ}}{\partial {x}^{\tau }}-\frac{\partial {g}_{\tau \kappa }}{\partial {x}^{ϵ}}\right)\right)\frac{1}{2}{g}^{ϵ\kappa }\left(\frac{\partial {g}_{\nu ϵ}}{\partial {x}^{\lambda }}+\frac{\partial {g}_{\lambda ϵ}}{\partial {x}^{\nu }}-\frac{\partial {g}_{\nu \lambda }}{\partial {x}^{ϵ}}\right)\\ \text{ }-\frac{\partial {x}^{\iota }}{\partial {x}^{\kappa }}\frac{1}{2}{g}^{ϵ\mu }\left(\frac{\partial {g}_{\iota ϵ}}{\partial {x}^{\sigma }}+\frac{\partial {g}_{\sigma ϵ}}{\partial {x}^{\iota }}-\frac{\partial {g}_{\iota \sigma }}{\partial {x}^{ϵ}}\right)\frac{1}{2}{g}^{ϵ\kappa }\left(\frac{\partial {g}_{\nu ϵ}}{\partial {x}^{\lambda }}+\frac{\partial {g}_{\lambda ϵ}}{\partial {x}^{\nu }}-\frac{\partial {g}_{\nu \lambda }}{\partial {x}^{ϵ}}\right)\end{array}$

$\begin{array}{l}\text{ }-{x}^{\tau }\frac{1}{2}{g}^{ϵ\iota }\left(\frac{\partial {g}_{\tau ϵ}}{\partial {x}^{\kappa }}+\frac{\partial {g}_{\kappa ϵ}}{\partial {x}^{\tau }}-\frac{\partial {g}_{\tau \kappa }}{\partial {x}^{ϵ}}\right)\frac{1}{2}{g}^{ϵ\mu }\left(\frac{\partial {g}_{\iota ϵ}}{\partial {x}^{\sigma }}+\frac{\partial {g}_{\sigma ϵ}}{\partial {x}^{\iota }}-\frac{\partial {g}_{\iota \sigma }}{\partial {x}^{ϵ}}\right)\\ \text{ }×\frac{1}{2}{g}^{ϵ\kappa }\left(\frac{\partial {g}_{\nu ϵ}}{\partial {x}^{\lambda }}+\frac{\partial {g}_{\lambda ϵ}}{\partial {x}^{\nu }}-\frac{\partial {g}_{\nu \lambda }}{\partial {x}^{ϵ}}\right)\end{array}$

$\begin{array}{l}\text{ }+\frac{\partial {x}^{\mu }}{\partial {x}^{\iota }}\frac{1}{2}{g}^{ϵ\iota }\left(\frac{\partial {g}_{\kappa ϵ}}{\partial {x}^{\sigma }}+\frac{\partial {g}_{\sigma ϵ}}{\partial {x}^{\kappa }}-\frac{\partial {g}_{\kappa \sigma }}{\partial {x}^{ϵ}}\right)\frac{1}{2}{g}^{ϵ\kappa }\left(\frac{\partial {g}_{\nu ϵ}}{\partial {x}^{\lambda }}+\frac{\partial {g}_{\lambda ϵ}}{\partial {x}^{\nu }}-\frac{\partial {g}_{\nu \lambda }}{\partial {x}^{ϵ}}\right)\\ \text{ }+{x}^{\tau }\frac{1}{2}{g}^{ϵ\mu }\left(\frac{\partial {g}_{\tau ϵ}}{\partial {x}^{\iota }}+\frac{\partial {g}_{\iota ϵ}}{\partial {x}^{\tau }}-\frac{\partial {g}_{\tau \iota }}{\partial {x}^{ϵ}}\right)\frac{1}{2}{g}^{ϵ\iota }\left(\frac{\partial {g}_{\kappa ϵ}}{\partial {x}^{\sigma }}+\frac{\partial {g}_{\sigma ϵ}}{\partial {x}^{\kappa }}-\frac{\partial {g}_{\kappa \sigma }}{\partial {x}^{ϵ}}\right)\\ \text{ }×\frac{1}{2}{g}^{ϵ\kappa }\left(\frac{\partial {g}_{\nu ϵ}}{\partial {x}^{\lambda }}+\frac{\partial {g}_{\lambda ϵ}}{\partial {x}^{\nu }}-\frac{\partial {g}_{\nu \lambda }}{\partial {x}^{ϵ}}\right)-\frac{{\partial }^{2}{x}^{\mu }}{\partial {x}^{\nu }\partial {x}^{\kappa }}\frac{1}{2}{g}^{ϵ\kappa }\left(\frac{\partial {g}_{\sigma ϵ}}{\partial {x}^{\lambda }}+\frac{\partial {g}_{\lambda ϵ}}{\partial {x}^{\sigma }}-\frac{\partial {g}_{\sigma \lambda }}{\partial {x}^{ϵ}}\right)\end{array}$

$\begin{array}{l}\text{ }-\frac{\partial }{\partial {x}^{\kappa }}\left({x}^{\tau }\frac{1}{2}{g}^{ϵ\mu }\left(\frac{\partial {g}_{\tau ϵ}}{\partial {x}^{\nu }}+\frac{\partial {g}_{\nu ϵ}}{\partial {x}^{\tau }}-\frac{\partial {g}_{\tau \nu }}{\partial {x}^{ϵ}}\right)\right)\frac{1}{2}{g}^{ϵ\kappa }\left(\frac{\partial {g}_{\sigma ϵ}}{\partial {x}^{\lambda }}+\frac{\partial {g}_{\lambda ϵ}}{\partial {x}^{\sigma }}-\frac{\partial {g}_{\sigma \lambda }}{\partial {x}^{ϵ}}\right)\\ \text{ }-\frac{\partial {x}^{\iota }}{\partial {x}^{\nu }}\frac{1}{2}{g}^{ϵ\mu }\left(\frac{\partial {g}_{\iota ϵ}}{\partial {x}^{\kappa }}+\frac{\partial {g}_{\kappa ϵ}}{\partial {x}^{\iota }}-\frac{\partial {g}_{\iota \kappa }}{\partial {x}^{ϵ}}\right)\frac{1}{2}{g}^{ϵ\kappa }\left(\frac{\partial {g}_{\sigma ϵ}}{\partial {x}^{\lambda }}+\frac{\partial {g}_{\lambda ϵ}}{\partial {x}^{\sigma }}-\frac{\partial {g}_{\sigma \lambda }}{\partial {x}^{ϵ}}\right)\\ \text{ }-{x}^{\tau }\frac{1}{2}{g}^{ϵ\iota }\left(\frac{\partial {g}_{\tau ϵ}}{\partial {x}^{\nu }}+\frac{\partial {g}_{\nu ϵ}}{\partial {x}^{\tau }}-\frac{\partial {g}_{\tau \nu }}{\partial {x}^{ϵ}}\right)\frac{1}{2}{g}^{ϵ\mu }\left(\frac{\partial {g}_{\iota ϵ}}{\partial {x}^{\kappa }}+\frac{\partial {g}_{\kappa ϵ}}{\partial {x}^{\iota }}-\frac{\partial {g}_{\iota \kappa }}{\partial {x}^{ϵ}}\right)\\ \text{ }×\frac{1}{2}{g}^{ϵ\kappa }\left(\frac{\partial {g}_{\sigma ϵ}}{\partial {x}^{\lambda }}+\frac{\partial {g}_{\lambda ϵ}}{\partial {x}^{\sigma }}-\frac{\partial {g}_{\sigma \lambda }}{\partial {x}^{ϵ}}\right)\end{array}$

$\begin{array}{l}\text{ }+\frac{\partial {x}^{\mu }}{\partial {x}^{\iota }}\frac{1}{2}{g}^{ϵ\iota }\left(\frac{\partial {g}_{\nu ϵ}}{\partial {x}^{\kappa }}+\frac{\partial {g}_{\kappa ϵ}}{\partial {x}^{\nu }}-\frac{\partial {g}_{\nu \kappa }}{\partial {x}^{ϵ}}\right)\frac{1}{2}{g}^{ϵ\kappa }\left(\frac{\partial {g}_{\sigma ϵ}}{\partial {x}^{\lambda }}+\frac{\partial {g}_{\lambda ϵ}}{\partial {x}^{\sigma }}-\frac{\partial {g}_{\sigma \lambda }}{\partial {x}^{ϵ}}\right)\\ \text{ }+{x}^{\tau }\frac{1}{2}{g}^{ϵ\mu }\left(\frac{\partial {g}_{\tau ϵ}}{\partial {x}^{\iota }}+\frac{\partial {g}_{\iota ϵ}}{\partial {x}^{\tau }}-\frac{\partial {g}_{\tau \iota }}{\partial {x}^{ϵ}}\right)\frac{1}{2}{g}^{ϵ\iota }\left(\frac{\partial {g}_{\nu ϵ}}{\partial {x}^{\kappa }}+\frac{\partial {g}_{\kappa ϵ}}{\partial {x}^{\nu }}-\frac{\partial {g}_{\nu \kappa }}{\partial {x}^{ϵ}}\right)\\ \text{ }×\frac{1}{2}{g}^{ϵ\kappa }\left(\frac{\partial {g}_{\sigma ϵ}}{\partial {x}^{\lambda }}+\frac{\partial {g}_{\lambda ϵ}}{\partial {x}^{\sigma }}-\frac{\partial {g}_{\sigma \lambda }}{\partial {x}^{ϵ}}\right).\end{array}$

Definition 14 When the next conversion equation is established, $\text{ }{x}_{\mu }^{\mu }$ is components of a tensor of rank zero. ${x}_{\mu }^{\mu }=\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}\frac{\partial {x}^{\nu }}{\partial {x}^{\mu }}{x}_{\nu }^{\nu }$

Definition 15 When the next conversion equation is established, ${x}^{\mu }$ is contravariant components of a tensor of the first rank . ${x}^{\mu }=\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}{x}^{\nu }$

Definition 16 When the next conversion equation is established, ${x}_{\mu }$ is covariant components of a tensor of the first rank . ${x}_{\mu }=\frac{\partial {x}^{\nu }}{\partial {x}^{\mu }}{x}_{\nu }$

Definition 17 When the next conversion equation is established, ${x}^{\mu \nu }$ is contravariant components of a tensor of the second rank . ${x}^{\mu \nu }=\frac{\partial {x}^{\mu }}{\partial {x}^{\sigma }}\frac{\partial {x}^{\nu }}{\partial {x}^{\lambda }}{x}^{\sigma \lambda }$

Definition 18 When the next conversion equation is established, ${x}_{\mu \nu }$ is covariant components of a tensor of the second rank . ${x}_{\mu \nu }=\frac{\partial {x}^{\sigma }}{\partial {x}^{\mu }}\frac{\partial {x}^{\lambda }}{\partial {x}^{\nu }}{x}_{\sigma \lambda }$

Definition 19 When the next conversion equation is established, ${x}_{\nu }^{\mu }$ is components of the mixed tensor of the second rank . ${x}_{\nu }^{\mu }=\frac{\partial {x}^{\mu }}{\partial {x}^{\sigma }}\frac{\partial {x}^{\lambda }}{\partial {x}^{\nu }}{x}_{\lambda }^{\sigma }$

Definition 20 When the next conversion equation is established, ${x}_{\nu \sigma }^{\mu }$ is components of the mixed tensor of the third rank of the second rank covariant in the first rank contravariant . ${x}_{\nu \sigma }^{\mu }=\frac{\partial {x}^{\mu }}{\partial {x}^{\lambda }}\frac{\partial {x}^{\iota }}{\partial {x}^{\nu }}\frac{\partial {x}^{ϵ}}{\partial {x}^{\sigma }}{x}_{\iota ϵ}^{\lambda }$

Definition 21 When the next conversion equation is established, ${x}_{\nu \sigma \lambda }^{\mu }$ is components of the mixed tensor of the fourth rank of the third rank covariant in the first rank contravariant. ${x}_{\nu \sigma \lambda }^{\mu }=\frac{\partial {x}^{\mu }}{\partial {x}^{\iota }}\frac{\partial {x}^{ϵ}}{\partial {x}^{\nu }}\frac{\partial {x}^{\alpha }}{\partial {x}^{\sigma }}\frac{\partial {x}^{\beta }}{\partial {x}^{\lambda }}{x}_{ϵ\alpha \beta }^{\iota }$

3. About Covariant Derivative for the Scalar in Tensor Satisfying Binary Law

Proposition 1 When all coordinate systems satisfy Binary Law, ${M}_{;\nu }=\frac{\partial M}{\partial {x}^{\nu }}=0$ is established for $\frac{{\partial }^{3}{x}^{\mu }}{\partial {x}^{\nu }\partial {x}^{\nu }\partial {x}^{\nu }}=M$.

Proof: I get

${M}_{;\nu }=\frac{\partial M}{\partial {x}^{\nu }}=0$ (1)

as $ϵ=1$ for Definition 9, Hypothesis 1.

-End Proof-

Proposition 2 When all coordinate systems satisfy Binary Law, $\frac{{\partial }^{2}{x}_{\mu }}{\partial {x}^{\nu }\partial {x}^{\nu }}=0$ is established.

Proof: I get

$\frac{{\partial }^{2}{x}_{\mu }}{\partial {x}^{\nu }\partial {x}^{\nu }}=0$ (2)

from (1), (77).

-End Proof-

Proposition 3 When all coordinate systems satisfy Binary Law, $\frac{{\partial }^{4}{x}^{\mu }}{\partial {x}^{\nu }\partial {x}^{\nu }\partial {x}^{\nu }\partial {x}^{\nu }}=0$ is established.

Proof: I get

$\frac{{\partial }^{4}{x}^{\mu }}{\partial {x}^{\nu }\partial {x}^{\nu }\partial {x}^{\nu }\partial {x}^{\nu }}=0$ (3)

from (1), Definition 8.

-End Proof-

4. About Covariant Derivative for the Vector in Tensor Satisfying Binary Law

Proposition 4 ${x}_{\mu ;\nu }=\frac{\partial {x}_{\mu }}{\partial {x}^{\nu }},{x}_{\mu }^{;\mu }=\frac{\partial {x}_{\mu }}{\partial {x}_{\mu }}-{x}_{\nu }\frac{1}{2}\left(\frac{\partial {g}^{\nu \mu }}{\partial {x}^{\mu }}\right)$ is established in tensor satisfying Binary Law.

Proof: If all coordinate systems satisfy Binary Law, I get

$\begin{array}{c}{x}_{\mu ;\nu }=\frac{\partial {x}_{\mu }}{\partial {x}^{\nu }}-{x}_{\nu }\frac{1}{2}{g}^{\nu \nu }\left(\frac{\partial {g}_{\mu \nu }}{\partial {x}^{\nu }}+\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\mu }}-\frac{\partial {g}_{\mu \nu }}{\partial {x}^{\nu }}\right)\\ =\frac{\partial {x}_{\mu }}{\partial {x}^{\nu }}-{x}_{\nu }\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\nu }}{\partial {x}^{\mu }}\right)\end{array}$ (4)

from Definition 10. (4) must rewrite it in

${x}_{\mu ;\nu }=\frac{\partial {x}_{\mu }}{\partial {x}^{\nu }}-{x}_{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\sigma }}{\partial {x}^{\mu }}\right)$ (5)

by (4) being a tensor equation. The dummy index has an invariable property for consideration of Binary Law. In other words, the index which was dummy index in Definition 10 is dummy index in (5). I get the conclusion that (5) doesn’t satisfy Binary Law from Definition 6. I get the conclusion that Definition 10 isn’t an equation of the tensor satisfying Binary Law because (5) doesn’t satisfy Binary Law.

I rewrite one existing index $\nu$ in each term of (5) in index $\mu$ using Definition 2 and get

${x}_{\mu }^{;\mu }=\frac{\partial {x}_{\mu }}{\partial {x}_{\mu }}-{x}_{\sigma }\frac{1}{2}\left(\frac{\partial {g}^{\sigma \mu }}{\partial {x}^{\mu }}\right),$

${x}_{\mu }^{;\mu }=\frac{\partial {x}_{\mu }}{\partial {x}_{\mu }}-{x}_{\nu }\frac{1}{2}\left(\frac{\partial {g}^{\nu \mu }}{\partial {x}^{\mu }}\right)$. (6)

I rewrite one existing index $\nu$ in each term of (5) in index $\mu$ using Definition 4 and get

$-{x}_{\mu ;\mu }=-\frac{\partial {x}_{\mu }}{\partial {x}^{\mu }}+{x}_{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\mu }^{\sigma }}{\partial {x}^{\mu }}\right).$ (7)

I get

$-{x}_{\mu ;\mu }=-\frac{\partial {x}_{\mu }}{\partial {x}^{\mu }}$ (8)

in consideration of Definition 7 for (7). Because the second term of the right side of (8) doesn’t exist,

${x}_{\mu ;\nu }=\frac{\partial {x}_{\mu }}{\partial {x}^{\nu }}$ (9)

can rewrite (8) using Definition 4. In addition, ${x}_{\mu ;\nu }$ can’t rewrite ${x}_{\mu }^{;\mu }$ of (6) using Definition 2 because the second term of the right side exists in (6).

-End Proof-

Proposition 5 ${x}_{;\nu }^{\mu }=\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}$ is established in tensor satisfying Binary Law.

Proof: If all coordinate systems satisfy Binary Law, I get

${x}_{;\nu }^{\mu }=\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}+{x}^{\nu }\frac{1}{2}{g}^{\nu \mu }\left(\frac{\partial \stackrel{^}{{g}_{\nu \nu }}}{\partial {x}^{\nu }}+\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}-\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}\right)$

$=\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}+{x}^{\nu }\frac{1}{2}\left(\frac{\partial \stackrel{^}{{g}_{\nu }^{\mu }}}{\partial {x}^{\nu }}\right)$ (10)

$=\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}+{x}^{\nu }\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\mu }}{\partial {x}^{\nu }}\right)$ (11)

from Definition 11. (10), (11) must rewrite it in

${x}_{;\nu }^{\mu }=\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}+{x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}^{\nu }}\right)$ (12)

$=\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}+{x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\mu }}{\partial {x}^{\sigma }}\right)$ (13)

by (10), (11) being a tensor equation.

The dummy index has an invariable property for consideration of Binary Law. In other words, the index which was dummy index in Definition 11 is dummy index in (12), (13). I get the conclusion that (12), (13) doesn’t satisfy Binary Law from Definition 6. I get the conclusion that Definition 11 isn’t an equation of the tensor satisfying Binary Law because (12), (13) doesn’t satisfy Binary Law.

I rewrite one existing index $\nu$ in each term of (12), (13) in index $\mu$ using Definition 4 and get

$\begin{array}{c}-{x}_{;\mu }^{\mu }=-\frac{\partial {x}^{\mu }}{\partial {x}^{\mu }}-{x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}^{\mu }}\right)\\ =-\frac{\partial {x}^{\mu }}{\partial {x}^{\mu }}-{x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\mu }^{\mu }}{\partial {x}^{\sigma }}\right).\end{array}$ (14)

I get

$-{x}_{;\mu }^{\mu }=-\frac{\partial {x}^{\mu }}{\partial {x}^{\mu }}$ (15)

in consideration of Definition 7 for (14). Because the second term of the right side of (15) doesn’t exist,

${x}_{;\nu }^{\mu }=\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}$ (16)

can rewrite (15) using Definition 4. I rewrite one existing index $\nu$ in each term of (12), (13) in index $\mu$ using Definition 2 and get

$\begin{array}{c}{x}^{\mu ;\mu }=\frac{\partial {x}^{\mu }}{\partial {x}_{\mu }}+{x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}_{\mu }}\right)\\ =\frac{\partial {x}^{\mu }}{\partial {x}_{\mu }}+{x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}^{\mu \mu }}{\partial {x}^{\sigma }}\right).\end{array}$

-End Proof-

Proposition 6 ${x}_{;\nu :\nu }^{\mu }=\frac{{\partial }^{2}{x}^{\mu }}{\partial {x}^{\nu }\partial {x}^{\nu }}$ is established in tensor satisfying Binary Law.

Proof: If all coordinate systems satisfy Binary Law, I get

$\begin{array}{c}{x}_{;\nu ;\nu }^{\mu }=\frac{{\partial }^{2}{x}^{\mu }}{\partial {x}^{\nu }\partial {x}^{\nu }}+\frac{\partial }{\partial {x}^{\nu }}\left({x}^{\nu }\frac{1}{2}{g}^{\nu \mu }\left(\frac{\partial \stackrel{^}{{g}_{\nu \nu }}}{\partial {x}^{\nu }}+\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}-\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{\partial {x}^{\nu }}{\partial {x}^{\nu }}\frac{1}{2}{g}^{\nu \mu }\left(\frac{\partial \stackrel{^}{{g}_{\nu \nu }}}{\partial {x}^{\nu }}+\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}-\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}\right)\end{array}$

$\begin{array}{c}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+{x}^{\nu }\frac{1}{2}{g}^{\nu \nu }\left(\frac{\partial \stackrel{^}{{g}_{\nu \nu }}}{\partial {x}^{\nu }}+\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}-\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}\right)\frac{1}{2}{g}^{\nu \mu }\left(\frac{\partial \stackrel{^}{{g}_{\nu \nu }}}{\partial {x}^{\nu }}+\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}-\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}\frac{1}{2}{g}^{\nu \nu }\left(\frac{\partial \stackrel{^}{{g}_{\nu \nu }}}{\partial {x}^{\nu }}+\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}-\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-{x}^{\nu }\frac{1}{2}{g}^{\nu \mu }\left(\frac{\partial \stackrel{^}{{g}_{\nu \nu }}}{\partial {x}^{\nu }}+\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}-\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}\right)\frac{1}{2}{g}^{\nu \nu }\left(\frac{\partial \stackrel{^}{{g}_{\nu \nu }}}{\partial {x}^{\nu }}+\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}-\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}\right)\end{array}$

$\begin{array}{l}=\frac{{\partial }^{2}{x}^{\mu }}{\partial {x}^{\nu }\partial {x}^{\nu }}+\frac{\partial }{\partial {x}^{\nu }}\left({x}^{\nu }\frac{1}{2}\left(\frac{\partial \stackrel{^}{{g}_{\nu }^{\mu }}}{\partial {x}^{\nu }}\right)\right)+\frac{\partial {x}^{\nu }}{\partial {x}^{\nu }}\frac{1}{2}\left(\frac{\partial \stackrel{^}{{g}_{\nu }^{\mu }}}{\partial {x}^{\nu }}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{x}^{\nu }\frac{1}{2}\left(\frac{\partial \stackrel{^}{{g}_{\nu }^{\nu }}}{\partial {x}^{\nu }}\right)\frac{1}{2}\left(\frac{\partial \stackrel{^}{{g}_{\nu }^{\mu }}}{\partial {x}^{\nu }}\right)-\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}\frac{1}{2}\left(\frac{\partial \stackrel{^}{{g}_{\nu }^{\nu }}}{\partial {x}^{\nu }}\right)-{x}^{\nu }\frac{1}{2}\left(\frac{\partial \stackrel{^}{{g}_{\nu }^{\mu }}}{\partial {x}^{\nu }}\right)\frac{1}{2}\left(\frac{\partial \stackrel{^}{{g}_{\nu }^{\nu }}}{\partial {x}^{\nu }}\right)\end{array}$ (17)

$\begin{array}{l}=\frac{{\partial }^{2}{x}^{\mu }}{\partial {x}^{\nu }\partial {x}^{\nu }}+\frac{\partial }{\partial {x}^{\nu }}\left({x}^{\nu }\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\mu }}{\partial {x}^{\nu }}\right)\right)+\frac{\partial {x}^{\nu }}{\partial {x}^{\nu }}\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\mu }}{\partial {x}^{\nu }}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{x}^{\nu }\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\nu }}{\partial {x}^{\nu }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\mu }}{\partial {x}^{\nu }}\right)-\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\nu }}{\partial {x}^{\nu }}\right)-{x}^{\nu }\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\mu }}{\partial {x}^{\nu }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\nu }}{\partial {x}^{\nu }}\right)\end{array}$ (18)

from Definition 12. (17), (18) must rewrite it in

$\begin{array}{c}{x}_{;\nu ;\nu }^{\mu }=\frac{{\partial }^{2}{x}^{\mu }}{\partial {x}^{\nu }\partial {x}^{\nu }}+\frac{\partial }{\partial {x}^{\nu }}\left({x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}^{\nu }}\right)\right)+\frac{\partial {x}^{\sigma }}{\partial {x}^{\nu }}\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}^{\nu }}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\sigma }}{\partial {x}^{\nu }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}^{\nu }}\right)-\frac{\partial {x}^{\mu }}{\partial {x}^{\sigma }}\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\sigma }}{\partial {x}^{\nu }}\right)-{x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\sigma }}{\partial {x}^{\nu }}\right)\end{array}$ (19)

$\begin{array}{l}=\frac{{\partial }^{2}{x}^{\mu }}{\partial {x}^{\nu }\partial {x}^{\nu }}+\frac{\partial }{\partial {x}^{\nu }}\left({x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\mu }}{\partial {x}^{\sigma }}\right)\right)+\frac{\partial {x}^{\sigma }}{\partial {x}^{\nu }}\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\mu }}{\partial {x}^{\sigma }}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\sigma }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\mu }}{\partial {x}^{\sigma }}\right)-\frac{\partial {x}^{\mu }}{\partial {x}^{\sigma }}\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\sigma }}{\partial {x}^{\nu }}\right)-{x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\sigma }}{\partial {x}^{\nu }}\right)\end{array}$ (20)

by (17), (18) being a tensor equation. The dummy index has an invariable property for consideration of Binary Law. In other words, the index which was dummy index in Definition 12 is dummy index in (19), (20). I get the conclusion that (19), (20) doesn’t satisfy Binary Law from Definition 6. I get the conclusion that Definition 12 isn’t an equation of the tensor satisfying Binary Law because (19), (20) doesn’t satisfy Binary Law.

I rewrite two existing index $\nu$ in each term of (19), (20) in index $\mu$ using Definition 4 and get

$\begin{array}{c}{x}_{;\mu ;\mu }^{\mu }=\frac{{\partial }^{2}{x}^{\mu }}{\partial {x}^{\mu }\partial {x}^{\mu }}+\frac{\partial }{\partial {x}^{\mu }}\left({x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}^{\mu }}\right)\right)+\frac{\partial {x}^{\sigma }}{\partial {x}^{\mu }}\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}^{\mu }}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\sigma }}{\partial {x}^{\mu }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}^{\mu }}\right)-\frac{\partial {x}^{\mu }}{\partial {x}^{\sigma }}\frac{1}{2}\left(\frac{\partial {g}_{\mu }^{\sigma }}{\partial {x}^{\mu }}\right)-{x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\mu }^{\sigma }}{\partial {x}^{\mu }}\right)\\ =\frac{{\partial }^{2}{x}^{\mu }}{\partial {x}^{\mu }\partial {x}^{\mu }}+\frac{\partial }{\partial {x}^{\mu }}\left({x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\mu }^{\mu }}{\partial {x}^{\sigma }}\right)\right)+\frac{\partial {x}^{\sigma }}{\partial {x}^{\mu }}\frac{1}{2}\left(\frac{\partial {g}_{\mu }^{\mu }}{\partial {x}^{\sigma }}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\mu }^{\sigma }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\mu }^{\mu }}{\partial {x}^{\sigma }}\right)-\frac{\partial {x}^{\mu }}{\partial {x}^{\sigma }}\frac{1}{2}\left(\frac{\partial {g}_{\mu }^{\sigma }}{\partial {x}^{\mu }}\right)-{x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\mu }^{\sigma }}{\partial {x}^{\mu }}\right).\end{array}$ (21)

I get

${x}_{;\mu ;\mu }^{\mu }=\frac{{\partial }^{2}{x}^{\mu }}{\partial {x}^{\mu }\partial {x}^{\mu }}$ (22)

in consideration of Definition 7 for (21). Because the second term of the right side of (22) doesn’t exist,

${x}_{;\nu :\nu }^{\mu }=\frac{{\partial }^{2}{x}^{\mu }}{\partial {x}^{\nu }\partial {x}^{\nu }}$ (23)

can rewrite (22) using Definition 4. I rewrite two existing index $\nu$ in each term of (19), (20) in index $\mu$ using Definition 2 and get

$\begin{array}{c}{x}^{\mu ;\mu ;\mu }=\frac{{\partial }^{2}{x}^{\mu }}{\partial {x}_{\mu }\partial {x}_{\mu }}+\frac{\partial }{\partial {x}_{\mu }}\left({x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}_{\mu }}\right)\right)+\frac{\partial {x}^{\sigma }}{\partial {x}_{\mu }}\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}_{\mu }}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\sigma }}{\partial {x}_{\mu }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}_{\mu }}\right)-\frac{\partial {x}^{\mu }}{\partial {x}^{\sigma }}\frac{1}{2}\left(\frac{\partial {g}^{\sigma \mu }}{\partial {x}_{\mu }}\right)-{x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}^{\sigma \mu }}{\partial {x}_{\mu }}\right)\\ =\frac{{\partial }^{2}{x}^{\mu }}{\partial {x}_{\mu }\partial {x}_{\mu }}+\frac{\partial }{\partial {x}_{\mu }}\left({x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}^{\mu \mu }}{\partial {x}^{\sigma }}\right)\right)+\frac{\partial {x}^{\sigma }}{\partial {x}_{\mu }}\frac{1}{2}\left(\frac{\partial {g}^{\mu \mu }}{\partial {x}^{\sigma }}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}^{\sigma \mu }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}^{\mu \mu }}{\partial {x}^{\sigma }}\right)-\frac{\partial {x}^{\mu }}{\partial {x}^{\sigma }}\frac{1}{2}\left(\frac{\partial {g}^{\sigma \mu }}{\partial {x}_{\mu }}\right)-{x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}^{\sigma \mu }}{\partial {x}_{\mu }}\right).\end{array}$

-End Proof-

Proposition 7 ${x}_{;\nu ;\nu ;\nu }^{\mu }=\frac{{\partial }^{3}{x}^{\mu }}{\partial {x}^{\nu }\partial {x}^{\nu }\partial {x}^{\nu }}$ is established in tensor satisfying Binary Law.

Proof: If all coordinate systems satisfy Binary Law, I get

$\begin{array}{c}{x}_{;\nu ;\nu ;\nu }^{\mu }=\frac{{\partial }^{3}{x}^{\mu }}{\partial {x}^{\nu }\partial {x}^{\nu }\partial {x}^{\nu }}+\frac{{\partial }^{2}}{\partial {x}^{\nu }\partial {x}^{\nu }}\left({x}^{\nu }\frac{1}{2}{g}^{\nu \mu }\left(\frac{\partial \stackrel{^}{{g}_{\nu \nu }}}{\partial {x}^{\nu }}+\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}-\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{\partial }{\partial {x}^{\nu }}\left(\frac{\partial {x}^{\nu }}{\partial {x}^{\nu }}\frac{1}{2}{g}^{\nu \mu }\left(\frac{\partial \stackrel{^}{{g}_{\nu \nu }}}{\partial {x}^{\nu }}+\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}-\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{\partial }{\partial {x}^{\nu }}\left({x}^{\nu }\frac{1}{2}{g}^{\nu \nu }\left(\frac{\partial \stackrel{^}{{g}_{\nu \nu }}}{\partial {x}^{\nu }}+\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}-\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}\right)\frac{1}{2}{g}^{\nu \mu }\left(\frac{\partial \stackrel{^}{{g}_{\nu \nu }}}{\partial {x}^{\nu }}+\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}-\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\frac{\partial }{\partial {x}^{\nu }}\left(\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}\frac{1}{2}{g}^{\nu \nu }\left(\frac{\partial \stackrel{^}{{g}_{\nu \nu }}}{\partial {x}^{\nu }}+\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}-\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\frac{\partial }{\partial {x}^{\nu }}\left({x}^{\nu }\frac{1}{2}{g}^{\nu \mu }\left(\frac{\partial \stackrel{^}{{g}_{\nu \nu }}}{\partial {x}^{\nu }}+\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}-\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}\right)\frac{1}{2}{g}^{\nu \nu }\left(\frac{\partial \stackrel{^}{{g}_{\nu \nu }}}{\partial {x}^{\nu }}+\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}-\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}\right)\right)\end{array}$

$\begin{array}{l}\text{ }+\frac{{\partial }^{2}{x}^{\nu }}{\partial {x}^{\nu }\partial {x}^{\nu }}\frac{1}{2}{g}^{\nu \mu }\left(\frac{\partial \stackrel{^}{{g}_{\nu \nu }}}{\partial {x}^{\nu }}+\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}-\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}\right)\\ \text{ }+\frac{\partial }{\partial {x}^{\nu }}\left({x}^{\nu }\frac{1}{2}{g}^{\nu \nu }\left(\frac{\partial \stackrel{^}{{g}_{\nu \nu }}}{\partial {x}^{\nu }}+\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}-\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}\right)\right)\frac{1}{2}{g}^{\nu \mu }\left(\frac{\partial \stackrel{^}{{g}_{\nu \nu }}}{\partial {x}^{\nu }}+\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}-\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}\right)\end{array}$

$\begin{array}{l}\text{ }+\frac{\partial {x}^{\nu }}{\partial {x}^{\nu }}\frac{1}{2}{g}^{\nu \nu }\left(\frac{\partial \stackrel{^}{{g}_{\nu \nu }}}{\partial {x}^{\nu }}+\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}-\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}\right)\frac{1}{2}{g}^{\nu \mu }\left(\frac{\partial \stackrel{^}{{g}_{\nu \nu }}}{\partial {x}^{\nu }}+\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}-\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}\right)\\ \text{ }+{x}^{\nu }\frac{1}{2}{g}^{\nu \nu }\left(\frac{\partial \stackrel{^}{{g}_{\nu \nu }}}{\partial {x}^{\nu }}+\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}-\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}\right)\frac{1}{2}{g}^{\nu \nu }\left(\frac{\partial \stackrel{^}{{g}_{\nu \nu }}}{\partial {x}^{\nu }}+\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}-\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}\right)\\ \text{ }×\frac{1}{2}{g}^{\nu \mu }\left(\frac{\partial \stackrel{^}{{g}_{\nu \nu }}}{\partial {x}^{\nu }}+\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}-\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}\right)\end{array}$

$\begin{array}{l}\text{ }-\frac{\partial {x}^{\nu }}{\partial {x}^{\nu }}\frac{1}{2}{g}^{\nu \nu }\left(\frac{\partial \stackrel{^}{{g}_{\nu \nu }}}{\partial {x}^{\nu }}+\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}-\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}\right)\frac{1}{2}{g}^{\nu \mu }\left(\frac{\partial \stackrel{^}{{g}_{\nu \nu }}}{\partial {x}^{\nu }}+\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}-\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}\right)\\ \text{ }-{x}^{\nu }\frac{1}{2}{g}^{\nu \nu }\left(\frac{\partial \stackrel{^}{{g}_{\nu \nu }}}{\partial {x}^{\nu }}+\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}-\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}\right)\frac{1}{2}{g}^{\nu \nu }\left(\frac{\partial \stackrel{^}{{g}_{\nu \nu }}}{\partial {x}^{\nu }}+\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}-\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}\right)\\ \text{ }×\frac{1}{2}{g}^{\nu \mu }\left(\frac{\partial \stackrel{^}{{g}_{\nu \nu }}}{\partial {x}^{\nu }}+\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}-\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}\right)-\frac{{\partial }^{2}{x}^{\mu }}{\partial {x}^{\nu }\partial {x}^{\nu }}\frac{1}{2}{g}^{\nu \nu }\left(\frac{\partial \stackrel{^}{{g}_{\nu \nu }}}{\partial {x}^{\nu }}+\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}-\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}\right)\\ \text{ }-\frac{\partial }{\partial {x}^{\nu }}\left({x}^{\nu }\frac{1}{2}{g}^{\nu \mu }\left(\frac{\partial \stackrel{^}{{g}_{\nu \nu }}}{\partial {x}^{\nu }}+\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}-\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}\right)\right)\frac{1}{2}{g}^{\nu \nu }\left(\frac{\partial \stackrel{^}{{g}_{\nu \nu }}}{\partial {x}^{\nu }}+\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}-\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}\right)\\ \text{ }-\frac{\partial {x}^{\nu }}{\partial {x}^{\nu }}\frac{1}{2}{g}^{\nu \mu }\left(\frac{\partial \stackrel{^}{{g}_{\nu \nu }}}{\partial {x}^{\nu }}+\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}-\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}\right)\frac{1}{2}{g}^{\nu \nu }\left(\frac{\partial \stackrel{^}{{g}_{\nu \nu }}}{\partial {x}^{\nu }}+\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}-\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}\right)\end{array}$

$\begin{array}{l}\text{ }-{x}^{\nu }\frac{1}{2}{g}^{\nu \nu }\left(\frac{\partial \stackrel{^}{{g}_{\nu \nu }}}{\partial {x}^{\nu }}+\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}-\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}\right)\frac{1}{2}{g}^{\nu \mu }\left(\frac{\partial \stackrel{^}{{g}_{\nu \nu }}}{\partial {x}^{\nu }}+\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}-\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}\right)\\ \text{ }×\frac{1}{2}{g}^{\nu \nu }\left(\frac{\partial \stackrel{^}{{g}_{\nu \nu }}}{\partial {x}^{\nu }}+\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}-\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}\right)\\ \text{ }+\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}\frac{1}{2}{g}^{\nu \nu }\left(\frac{\partial \stackrel{^}{{g}_{\nu \nu }}}{\partial {x}^{\nu }}+\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}-\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}\right)\frac{1}{2}{g}^{\nu \nu }\left(\frac{\partial \stackrel{^}{{g}_{\nu \nu }}}{\partial {x}^{\nu }}+\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}-\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}\right)\\ \text{ }+{x}^{\nu }\frac{1}{2}{g}^{\nu \mu }\left(\frac{\partial \stackrel{^}{{g}_{\nu \nu }}}{\partial {x}^{\nu }}+\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}-\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}\right)\frac{1}{2}{g}^{\nu \nu }\left(\frac{\partial \stackrel{^}{{g}_{\nu \nu }}}{\partial {x}^{\nu }}+\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}-\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}\right)\\ \text{ }×\frac{1}{2}{g}^{\nu \nu }\left(\frac{\partial \stackrel{^}{{g}_{\nu \nu }}}{\partial {x}^{\nu }}+\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}-\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}\right)-\frac{{\partial }^{2}{x}^{\mu }}{\partial {x}^{\nu }\partial {x}^{\nu }}\frac{1}{2}{g}^{\nu \nu }\left(\frac{\partial \stackrel{^}{{g}_{\nu \nu }}}{\partial {x}^{\nu }}+\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}-\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}\right)\end{array}$

$\begin{array}{l}\text{ }-\frac{\partial }{\partial {x}^{\nu }}\left({x}^{\nu }\frac{1}{2}{g}^{\nu \mu }\left(\frac{\partial \stackrel{^}{{g}_{\nu \nu }}}{\partial {x}^{\nu }}+\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}-\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}\right)\right)\frac{1}{2}{g}^{\nu \nu }\left(\frac{\partial \stackrel{^}{{g}_{\nu \nu }}}{\partial {x}^{\nu }}+\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}-\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}\right)\\ \text{ }-\frac{\partial {x}^{\nu }}{\partial {x}^{\nu }}\frac{1}{2}{g}^{\nu \mu }\left(\frac{\partial \stackrel{^}{{g}_{\nu \nu }}}{\partial {x}^{\nu }}+\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}-\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}\right)\frac{1}{2}{g}^{\nu \nu }\left(\frac{\partial \stackrel{^}{{g}_{\nu \nu }}}{\partial {x}^{\nu }}+\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}-\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}\right)\\ \text{ }-{x}^{\nu }\frac{1}{2}{g}^{\nu \nu }\left(\frac{\partial \stackrel{^}{{g}_{\nu \nu }}}{\partial {x}^{\nu }}+\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}-\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}\right)\frac{1}{2}{g}^{\nu \mu }\left(\frac{\partial \stackrel{^}{{g}_{\nu \nu }}}{\partial {x}^{\nu }}+\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}-\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}\right)\\ \text{ }×\frac{1}{2}{g}^{\nu \nu }\left(\frac{\partial \stackrel{^}{{g}_{\nu \nu }}}{\partial {x}^{\nu }}+\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}-\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}\right)\\ \text{ }+\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}\frac{1}{2}{g}^{\nu \nu }\left(\frac{\partial \stackrel{^}{{g}_{\nu \nu }}}{\partial {x}^{\nu }}+\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}-\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}\right)\frac{1}{2}{g}^{\nu \nu }\left(\frac{\partial \stackrel{^}{{g}_{\nu \nu }}}{\partial {x}^{\nu }}+\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}-\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}\right)\end{array}$

$\text{ }+{x}^{\nu }\frac{1}{2}{g}^{\nu \mu }\left(\frac{\partial \stackrel{^}{{g}_{\nu \nu }}}{\partial {x}^{\nu }}+\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}-\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}\right)\frac{1}{2}{g}^{\nu \nu }\left(\frac{\partial \stackrel{^}{{g}_{\nu \nu }}}{\partial {x}^{\nu }}+\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}-\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}\right)$

$\begin{array}{l}\text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}×\frac{1}{2}{g}^{\nu \nu }\left(\frac{\partial \stackrel{^}{{g}_{\nu \nu }}}{\partial {x}^{\nu }}+\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}-\frac{\partial {g}_{\nu \nu }}{\partial {x}^{\nu }}\right)\\ =\frac{{\partial }^{3}{x}^{\mu }}{\partial {x}^{\nu }\partial {x}^{\nu }\partial {x}^{\nu }}+\frac{{\partial }^{2}}{\partial {x}^{\nu }\partial {x}^{\nu }}\left({x}^{\nu }\frac{1}{2}\left(\frac{\partial \stackrel{^}{{g}_{\nu }^{\mu }}}{\partial {x}^{\nu }}\right)\right)+\frac{\partial }{\partial {x}^{\nu }}\left(\frac{\partial {x}^{\nu }}{\partial {x}^{\nu }}\frac{1}{2}\left(\frac{\partial \stackrel{^}{{g}_{\nu }^{\mu }}}{\partial {x}^{\nu }}\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{\partial }{\partial {x}^{\nu }}\left({x}^{\nu }\frac{1}{2}\left(\frac{\partial \stackrel{^}{{g}_{\nu }^{\nu }}}{\partial {x}^{\nu }}\right)\frac{1}{2}\left(\frac{\partial \stackrel{^}{{g}_{\nu }^{\mu }}}{\partial {x}^{\nu }}\right)\right)-\frac{\partial }{\partial {x}^{\nu }}\left(\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}\frac{1}{2}\left(\frac{\partial \stackrel{^}{{g}_{\nu }^{\nu }}}{\partial {x}^{\nu }}\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\frac{\partial }{\partial {x}^{\nu }}\left({x}^{\nu }\frac{1}{2}\left(\frac{\partial \stackrel{^}{{g}_{\nu }^{\mu }}}{\partial {x}^{\nu }}\right)\frac{1}{2}\left(\frac{\partial \stackrel{^}{{g}_{\nu }^{\nu }}}{\partial {x}^{\nu }}\right)\right)+\frac{{\partial }^{2}{x}^{\nu }}{\partial {x}^{\nu }\partial {x}^{\nu }}\frac{1}{2}\left(\frac{\partial \stackrel{^}{{g}_{\nu }^{\mu }}}{\partial {x}^{\nu }}\right)\end{array}$

$\begin{array}{l}\text{ }+\frac{\partial }{\partial {x}^{\nu }}\left({x}^{\nu }\frac{1}{2}\left(\frac{\partial \stackrel{^}{{g}_{\nu }^{\nu }}}{\partial {x}^{\nu }}\right)\right)\frac{1}{2}\left(\frac{\partial \stackrel{^}{{g}_{\nu }^{\mu }}}{\partial {x}^{\nu }}\right)+\frac{\partial {x}^{\nu }}{\partial {x}^{\nu }}\frac{1}{2}\left(\frac{\partial \stackrel{^}{{g}_{\nu }^{\nu }}}{\partial {x}^{\nu }}\right)\frac{1}{2}\left(\frac{\partial \stackrel{^}{{g}_{\nu }^{\mu }}}{\partial {x}^{\nu }}\right)\\ \text{ }+{x}^{\nu }\frac{1}{2}\left(\frac{\partial \stackrel{^}{{g}_{\nu }^{\nu }}}{\partial {x}^{\nu }}\right)\frac{1}{2}\left(\frac{\partial \stackrel{^}{{g}_{\nu }^{\nu }}}{\partial {x}^{\nu }}\right)\frac{1}{2}\left(\frac{\partial \stackrel{^}{{g}_{\nu }^{\mu }}}{\partial {x}^{\nu }}\right)-\frac{\partial {x}^{\nu }}{\partial {x}^{\nu }}\frac{1}{2}\left(\frac{\partial \stackrel{^}{{g}_{\nu }^{\nu }}}{\partial {x}^{\nu }}\right)\frac{1}{2}\left(\frac{\partial \stackrel{^}{{g}_{\nu }^{\mu }}}{\partial {x}^{\nu }}\right)\\ \text{ }-{x}^{\nu }\frac{1}{2}\left(\frac{\partial \stackrel{^}{{g}_{\nu }^{\nu }}}{\partial {x}^{\nu }}\right)\frac{1}{2}\left(\frac{\partial \stackrel{^}{{g}_{\nu }^{\nu }}}{\partial {x}^{\nu }}\right)\frac{1}{2}\left(\frac{\partial \stackrel{^}{{g}_{\nu }^{\mu }}}{\partial {x}^{\nu }}\right)-\frac{{\partial }^{2}{x}^{\mu }}{\partial {x}^{\nu }\partial {x}^{\nu }}\frac{1}{2}\left(\frac{\partial \stackrel{^}{{g}_{\nu }^{\nu }}}{\partial {x}^{\nu }}\right)\\ \text{ }-\frac{\partial }{\partial {x}^{\nu }}\left({x}^{\nu }\frac{1}{2}\left(\frac{\partial \stackrel{^}{{g}_{\nu }^{\mu }}}{\partial {x}^{\nu }}\right)\right)\frac{1}{2}\left(\frac{\partial \stackrel{^}{{g}_{\nu }^{\nu }}}{\partial {x}^{\nu }}\right)-\frac{\partial {x}^{\nu }}{\partial {x}^{\nu }}\frac{1}{2}\left(\frac{\partial \stackrel{^}{{g}_{\nu }^{\mu }}}{\partial {x}^{\nu }}\right)\frac{1}{2}\left(\frac{\partial \stackrel{^}{{g}_{\nu }^{\nu }}}{\partial {x}^{\nu }}\right)\\ \text{ }-{x}^{\nu }\frac{1}{2}\left(\frac{\partial \stackrel{^}{{g}_{\nu }^{\nu }}}{\partial {x}^{\nu }}\right)\frac{1}{2}\left(\frac{\partial \stackrel{^}{{g}_{\nu }^{\mu }}}{\partial {x}^{\nu }}\right)\frac{1}{2}\left(\frac{\partial \stackrel{^}{{g}_{\nu }^{\nu }}}{\partial {x}^{\nu }}\right)+\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}\frac{1}{2}\left(\frac{\partial \stackrel{^}{{g}_{\nu }^{\nu }}}{\partial {x}^{\nu }}\right)\frac{1}{2}\left(\frac{\partial \stackrel{^}{{g}_{\nu }^{\nu }}}{\partial {x}^{\nu }}\right)\end{array}$

$\begin{array}{l}\text{ }+{x}^{\nu }\frac{1}{2}\left(\frac{\partial \stackrel{^}{{g}_{\nu }^{\mu }}}{\partial {x}^{\nu }}\right)\frac{1}{2}\left(\frac{\partial \stackrel{^}{{g}_{\nu }^{\nu }}}{\partial {x}^{\nu }}\right)\frac{1}{2}\left(\frac{\partial \stackrel{^}{{g}_{\nu }^{\nu }}}{\partial {x}^{\nu }}\right)-\frac{{\partial }^{2}{x}^{\mu }}{\partial {x}^{\nu }\partial {x}^{\nu }}\frac{1}{2}\left(\frac{\partial \stackrel{^}{{g}_{\nu }^{\nu }}}{\partial {x}^{\nu }}\right)\\ \text{ }-\frac{\partial }{\partial {x}^{\nu }}\left({x}^{\nu }\frac{1}{2}\left(\frac{\partial \stackrel{^}{{g}_{\nu }^{\mu }}}{\partial {x}^{\nu }}\right)\right)\frac{1}{2}\left(\frac{\partial \stackrel{^}{{g}_{\nu }^{\nu }}}{\partial {x}^{\nu }}\right)-\frac{\partial {x}^{\nu }}{\partial {x}^{\nu }}\frac{1}{2}\left(\frac{\partial \stackrel{^}{{g}_{\nu }^{\mu }}}{\partial {x}^{\nu }}\right)\frac{1}{2}\left(\frac{\partial \stackrel{^}{{g}_{\nu }^{\nu }}}{\partial {x}^{\nu }}\right)\\ \text{ }-{x}^{\nu }\frac{1}{2}\left(\frac{\partial \stackrel{^}{{g}_{\nu }^{\nu }}}{\partial {x}^{\nu }}\right)\frac{1}{2}\left(\frac{\partial \stackrel{^}{{g}_{\nu }^{\mu }}}{\partial {x}^{\nu }}\right)\frac{1}{2}\left(\frac{\partial \stackrel{^}{{g}_{\nu }^{\nu }}}{\partial {x}^{\nu }}\right)+\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}\frac{1}{2}\left(\frac{\partial \stackrel{^}{{g}_{\nu }^{\nu }}}{\partial {x}^{\nu }}\right)\frac{1}{2}\left(\frac{\partial \stackrel{^}{{g}_{\nu }^{\nu }}}{\partial {x}^{\nu }}\right)\\ \text{ }+{x}^{\nu }\frac{1}{2}\left(\frac{\partial \stackrel{^}{{g}_{\nu }^{\mu }}}{\partial {x}^{\nu }}\right)\frac{1}{2}\left(\frac{\partial \stackrel{^}{{g}_{\nu }^{\nu }}}{\partial {x}^{\nu }}\right)\frac{1}{2}\left(\frac{\partial \stackrel{^}{{g}_{\nu }^{\nu }}}{\partial {x}^{\nu }}\right)\end{array}$ (24)

$\begin{array}{l}=\frac{{\partial }^{3}{x}^{\mu }}{\partial {x}^{\nu }\partial {x}^{\nu }\partial {x}^{\nu }}+\frac{{\partial }^{2}}{\partial {x}^{\nu }\partial {x}^{\nu }}\left({x}^{\nu }\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\mu }}{\partial {x}^{\nu }}\right)\right)+\frac{\partial }{\partial {x}^{\nu }}\left(\frac{\partial {x}^{\nu }}{\partial {x}^{\nu }}\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\mu }}{\partial {x}^{\nu }}\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{\partial }{\partial {x}^{\nu }}\left({x}^{\nu }\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\nu }}{\partial {x}^{\nu }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\mu }}{\partial {x}^{\nu }}\right)\right)-\frac{\partial }{\partial {x}^{\nu }}\left(\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\nu }}{\partial {x}^{\nu }}\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\frac{\partial }{\partial {x}^{\nu }}\left({x}^{\nu }\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\mu }}{\partial {x}^{\nu }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\nu }}{\partial {x}^{\nu }}\right)\right)+\frac{{\partial }^{2}{x}^{\nu }}{\partial {x}^{\nu }\partial {x}^{\nu }}\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\mu }}{\partial {x}^{\nu }}\right)\end{array}$

$\begin{array}{l}\text{ }+\frac{\partial }{\partial {x}^{\nu }}\left({x}^{\nu }\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\nu }}{\partial {x}^{\nu }}\right)\right)\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\mu }}{\partial {x}^{\nu }}\right)+\frac{\partial {x}^{\nu }}{\partial {x}^{\nu }}\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\nu }}{\partial {x}^{\nu }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\mu }}{\partial {x}^{\nu }}\right)\\ \text{ }+{x}^{\nu }\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\nu }}{\partial {x}^{\nu }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\nu }}{\partial {x}^{\nu }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\mu }}{\partial {x}^{\nu }}\right)-\frac{\partial {x}^{\nu }}{\partial {x}^{\nu }}\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\nu }}{\partial {x}^{\nu }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\mu }}{\partial {x}^{\nu }}\right)\end{array}$

$\begin{array}{l}\text{ }-{x}^{\nu }\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\nu }}{\partial {x}^{\nu }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\nu }}{\partial {x}^{\nu }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\mu }}{\partial {x}^{\nu }}\right)-\frac{{\partial }^{2}{x}^{\mu }}{\partial {x}^{\nu }\partial {x}^{\nu }}\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\nu }}{\partial {x}^{\nu }}\right)\\ \text{ }-\frac{\partial }{\partial {x}^{\nu }}\left({x}^{\nu }\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\mu }}{\partial {x}^{\nu }}\right)\right)\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\nu }}{\partial {x}^{\nu }}\right)-\frac{\partial {x}^{\nu }}{\partial {x}^{\nu }}\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\mu }}{\partial {x}^{\nu }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\nu }}{\partial {x}^{\nu }}\right)\\ \text{ }-{x}^{\nu }\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\nu }}{\partial {x}^{\nu }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\mu }}{\partial {x}^{\nu }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\nu }}{\partial {x}^{\nu }}\right)+\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\nu }}{\partial {x}^{\nu }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\nu }}{\partial {x}^{\nu }}\right)\\ \text{ }+{x}^{\nu }\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\mu }}{\partial {x}^{\nu }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\nu }}{\partial {x}^{\nu }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\nu }}{\partial {x}^{\nu }}\right)-\frac{{\partial }^{2}{x}^{\mu }}{\partial {x}^{\nu }\partial {x}^{\nu }}\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\nu }}{\partial {x}^{\nu }}\right)\end{array}$

$\begin{array}{l}\text{ }-\frac{\partial }{\partial {x}^{\nu }}\left({x}^{\nu }\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\mu }}{\partial {x}^{\nu }}\right)\right)\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\nu }}{\partial {x}^{\nu }}\right)-\frac{\partial {x}^{\nu }}{\partial {x}^{\nu }}\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\mu }}{\partial {x}^{\nu }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\nu }}{\partial {x}^{\nu }}\right)\\ \text{ }-{x}^{\nu }\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\nu }}{\partial {x}^{\nu }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\mu }}{\partial {x}^{\nu }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\nu }}{\partial {x}^{\nu }}\right)+\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\nu }}{\partial {x}^{\nu }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\nu }}{\partial {x}^{\nu }}\right)\\ \text{ }+{x}^{\nu }\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\mu }}{\partial {x}^{\nu }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\nu }}{\partial {x}^{\nu }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\nu }}{\partial {x}^{\nu }}\right)\end{array}$ (25)

from Definition 13. (24), (25) must rewrite it in

$\begin{array}{c}{x}_{;\nu ;\nu ;\nu }^{\mu }=\frac{{\partial }^{3}{x}^{\mu }}{\partial {x}^{\nu }\partial {x}^{\nu }\partial {x}^{\nu }}+\frac{{\partial }^{2}}{\partial {x}^{\nu }\partial {x}^{\nu }}\left({x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}^{\nu }}\right)\right)+\frac{\partial }{\partial {x}^{\nu }}\left(\frac{\partial {x}^{\sigma }}{\partial {x}^{\nu }}\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}^{\nu }}\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{\partial }{\partial {x}^{\nu }}\left({x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\sigma }}{\partial {x}^{\nu }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}^{\nu }}\right)\right)-\frac{\partial }{\partial {x}^{\nu }}\left(\frac{\partial {x}^{\mu }}{\partial {x}^{\sigma }}\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\sigma }}{\partial {x}^{\nu }}\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\frac{\partial }{\partial {x}^{\nu }}\left({x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\sigma }}{\partial {x}^{\nu }}\right)\right)+\frac{{\partial }^{2}{x}^{\sigma }}{\partial {x}^{\nu }\partial {x}^{\nu }}\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}^{\nu }}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{\partial }{\partial {x}^{\nu }}\left({x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\sigma }}{\partial {x}^{\nu }}\right)\right)\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}^{\nu }}\right)+\frac{\partial {x}^{\sigma }}{\partial {x}^{\nu }}\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\sigma }}{\partial {x}^{\nu }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}^{\nu }}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\sigma }}{\partial {x}^{\nu }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\sigma }}{\partial {x}^{\nu }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}^{\nu }}\right)-\frac{\partial {x}^{\sigma }}{\partial {x}^{\sigma }}\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\sigma }}{\partial {x}^{\nu }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}^{\nu }}\right)\end{array}$

$\begin{array}{l}\text{ }-{x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\sigma }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\sigma }}{\partial {x}^{\nu }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}^{\nu }}\right)-\frac{{\partial }^{2}{x}^{\mu }}{\partial {x}^{\sigma }\partial {x}^{\nu }}\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\sigma }}{\partial {x}^{\nu }}\right)\\ \text{ }-\frac{\partial }{\partial {x}^{\nu }}\left({x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}^{\sigma }}\right)\right)\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\sigma }}{\partial {x}^{\nu }}\right)-\frac{\partial {x}^{\sigma }}{\partial {x}^{\sigma }}\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}^{\nu }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\sigma }}{\partial {x}^{\nu }}\right)\\ \text{ }-{x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\sigma }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}^{\nu }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\sigma }}{\partial {x}^{\nu }}\right)+\frac{\partial {x}^{\mu }}{\partial {x}^{\sigma }}\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\sigma }}{\partial {x}^{\nu }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\sigma }}{\partial {x}^{\nu }}\right)\\ \text{ }+{x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\sigma }}{\partial {x}^{\nu }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\sigma }}{\partial {x}^{\nu }}\right)-\frac{{\partial }^{2}{x}^{\mu }}{\partial {x}^{\nu }\partial {x}^{\sigma }}\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\sigma }}{\partial {x}^{\nu }}\right)\end{array}$

$\begin{array}{l}\text{ }-\frac{\partial }{\partial {x}^{\sigma }}\left({x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}^{\nu }}\right)\right)\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\sigma }}{\partial {x}^{\nu }}\right)-\frac{\partial {x}^{\sigma }}{\partial {x}^{\nu }}\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\sigma }}{\partial {x}^{\nu }}\right)\\ \text{ }-{x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\sigma }}{\partial {x}^{\nu }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\sigma }}{\partial {x}^{\nu }}\right)+\frac{\partial {x}^{\mu }}{\partial {x}^{\sigma }}\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\sigma }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\sigma }}{\partial {x}^{\nu }}\right)\\ \text{ }+{x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\sigma }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\sigma }}{\partial {x}^{\nu }}\right)\end{array}$ (26)

$\begin{array}{l}=\frac{{\partial }^{3}{x}^{\mu }}{\partial {x}^{\nu }\partial {x}^{\nu }\partial {x}^{\nu }}+\frac{{\partial }^{2}}{\partial {x}^{\nu }\partial {x}^{\nu }}\left({x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\mu }}{\partial {x}^{\sigma }}\right)\right)+\frac{\partial }{\partial {x}^{\nu }}\left(\frac{\partial {x}^{\sigma }}{\partial {x}^{\nu }}\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\mu }}{\partial {x}^{\sigma }}\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{\partial }{\partial {x}^{\nu }}\left({x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\sigma }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\mu }}{\partial {x}^{\sigma }}\right)\right)-\frac{\partial }{\partial {x}^{\nu }}\left(\frac{\partial {x}^{\mu }}{\partial {x}^{\sigma }}\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\sigma }}{\partial {x}^{\nu }}\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\frac{\partial }{\partial {x}^{\nu }}\left({x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\sigma }}{\partial {x}^{\nu }}\right)\right)+\frac{{\partial }^{2}{x}^{\sigma }}{\partial {x}^{\nu }\partial {x}^{\nu }}\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\mu }}{\partial {x}^{\sigma }}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{\partial }{\partial {x}^{\nu }}\left({x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\sigma }}{\partial {x}^{\sigma }}\right)\right)\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\mu }}{\partial {x}^{\sigma }}\right)+\frac{\partial {x}^{\sigma }}{\partial {x}^{\nu }}\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\sigma }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\mu }}{\partial {x}^{\sigma }}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\sigma }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\sigma }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\mu }}{\partial {x}^{\sigma }}\right)-\frac{\partial {x}^{\sigma }}{\partial {x}^{\sigma }}\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\sigma }}{\partial {x}^{\nu }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\mu }}{\partial {x}^{\sigma }}\right)\end{array}$

$\begin{array}{l}\text{ }-{x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\sigma }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\sigma }}{\partial {x}^{\nu }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\mu }}{\partial {x}^{\sigma }}\right)-\frac{{\partial }^{2}{x}^{\mu }}{\partial {x}^{\sigma }\partial {x}^{\nu }}\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\sigma }}{\partial {x}^{\nu }}\right)\\ \text{ }-\frac{\partial }{\partial {x}^{\nu }}\left({x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}^{\sigma }}\right)\right)\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\sigma }}{\partial {x}^{\nu }}\right)-\frac{\partial {x}^{\sigma }}{\partial {x}^{\sigma }}\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\mu }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\sigma }}{\partial {x}^{\nu }}\right)\\ \text{ }-{x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\sigma }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\mu }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\sigma }}{\partial {x}^{\nu }}\right)+\frac{\partial {x}^{\mu }}{\partial {x}^{\sigma }}\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\sigma }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\sigma }}{\partial {x}^{\nu }}\right)\\ \text{ }+{x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\sigma }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\sigma }}{\partial {x}^{\nu }}\right)-\frac{{\partial }^{2}{x}^{\mu }}{\partial {x}^{\nu }\partial {x}^{\sigma }}\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\sigma }}{\partial {x}^{\nu }}\right)\end{array}$

$\begin{array}{l}\text{ }-\frac{\partial }{\partial {x}^{\sigma }}\left({x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\mu }}{\partial {x}^{\sigma }}\right)\right)\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\sigma }}{\partial {x}^{\nu }}\right)-\frac{\partial {x}^{\sigma }}{\partial {x}^{\nu }}\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\sigma }}{\partial {x}^{\nu }}\right)\\ \text{ }-{x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\sigma }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\sigma }}{\partial {x}^{\nu }}\right)+\frac{\partial {x}^{\mu }}{\partial {x}^{\sigma }}\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\sigma }}{\partial {x}^{\nu }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\sigma }}{\partial {x}^{\nu }}\right)\\ \text{ }+{x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\sigma }}{\partial {x}^{\nu }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\nu }^{\sigma }}{\partial {x}^{\nu }}\right)\end{array}$ (27)

by (24), (25) being a tensor equation. The dummy index has an invariable property for consideration of Binary Law. In other words, the index which was dummy index in Definition 13 is dummy index in (26), (27). I get the conclusion that (26), (27) doesn’t satisfy Binary Law from Definition 6. I get the conclusion that Definition 13 isn’t an equation of the tensor satisfying Binary Law because (26), (27) doesn’t satisfy Binary Law.

I rewrite three existing index $\nu$ in each term of (26), (27) in index $\mu$ using Definition 4 and get

$\begin{array}{c}-{x}_{;\mu ;\mu ;\mu }^{\mu }=-\frac{{\partial }^{3}{x}^{\mu }}{\partial {x}^{\mu }\partial {x}^{\mu }\partial {x}^{\mu }}-\frac{{\partial }^{2}}{\partial {x}^{\mu }\partial {x}^{\mu }}\left({x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}^{\mu }}\right)\right)-\frac{\partial }{\partial {x}^{\mu }}\left(\frac{\partial {x}^{\sigma }}{\partial {x}^{\mu }}\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}^{\mu }}\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\frac{\partial }{\partial {x}^{\mu }}\left({x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\sigma }}{\partial {x}^{\mu }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}^{\mu }}\right)\right)+\frac{\partial }{\partial {x}^{\mu }}\left(\frac{\partial {x}^{\mu }}{\partial {x}^{\sigma }}\frac{1}{2}\left(\frac{\partial {g}_{\mu }^{\sigma }}{\partial {x}^{\mu }}\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{\partial }{\partial {x}^{\mu }}\left({x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\mu }^{\sigma }}{\partial {x}^{\mu }}\right)\right)-\frac{{\partial }^{2}{x}^{\sigma }}{\partial {x}^{\mu }\partial {x}^{\mu }}\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}^{\mu }}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\frac{\partial }{\partial {x}^{\mu }}\left({x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\sigma }}{\partial {x}^{\mu }}\right)\right)\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}^{\mu }}\right)-\frac{\partial {x}^{\sigma }}{\partial {x}^{\mu }}\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\sigma }}{\partial {x}^{\mu }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}^{\mu }}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-{x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\sigma }}{\partial {x}^{\mu }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\sigma }}{\partial {x}^{\mu }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}^{\mu }}\right)+\frac{\partial {x}^{\sigma }}{\partial {x}^{\sigma }}\frac{1}{2}\left(\frac{\partial {g}_{\mu }^{\sigma }}{\partial {x}^{\mu }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}^{\mu }}\right)\end{array}$

$\begin{array}{l}\text{ }+{x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\sigma }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\mu }^{\sigma }}{\partial {x}^{\mu }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}^{\mu }}\right)+\frac{{\partial }^{2}{x}^{\mu }}{\partial {x}^{\sigma }\partial {x}^{\mu }}\frac{1}{2}\left(\frac{\partial {g}_{\mu }^{\sigma }}{\partial {x}^{\mu }}\right)\\ \text{ }+\frac{\partial }{\partial {x}^{\mu }}\left({x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}^{\sigma }}\right)\right)\frac{1}{2}\left(\frac{\partial {g}_{\mu }^{\sigma }}{\partial {x}^{\mu }}\right)+\frac{\partial {x}^{\sigma }}{\partial {x}^{\sigma }}\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}^{\mu }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\mu }^{\sigma }}{\partial {x}^{\mu }}\right)\\ \text{ }+{x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\sigma }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}^{\mu }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\mu }^{\sigma }}{\partial {x}^{\mu }}\right)-\frac{\partial {x}^{\mu }}{\partial {x}^{\sigma }}\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\sigma }}{\partial {x}^{\mu }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\mu }^{\sigma }}{\partial {x}^{\mu }}\right)\\ \text{ }-{x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\sigma }}{\partial {x}^{\mu }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\mu }^{\sigma }}{\partial {x}^{\mu }}\right)+\frac{{\partial }^{2}{x}^{\mu }}{\partial {x}^{\mu }\partial {x}^{\sigma }}\frac{1}{2}\left(\frac{\partial {g}_{\mu }^{\sigma }}{\partial {x}^{\mu }}\right)\\ \text{ }+\frac{\partial }{\partial {x}^{\sigma }}\left({x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}^{\mu }}\right)\right)\frac{1}{2}\left(\frac{\partial {g}_{\mu }^{\sigma }}{\partial {x}^{\mu }}\right)+\frac{\partial {x}^{\sigma }}{\partial {x}^{\mu }}\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\mu }^{\sigma }}{\partial {x}^{\mu }}\right)\end{array}$

$\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+{x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\sigma }}{\partial {x}^{\mu }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\mu }^{\sigma }}{\partial {x}^{\mu }}\right)-\frac{\partial {x}^{\mu }}{\partial {x}^{\sigma }}\frac{1}{2}\left(\frac{\partial {g}_{\mu }^{\sigma }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\mu }^{\sigma }}{\partial {x}^{\mu }}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-{x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\mu }^{\sigma }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\mu }^{\sigma }}{\partial {x}^{\mu }}\right)\\ =-\frac{{\partial }^{3}{x}^{\mu }}{\partial {x}^{\mu }\partial {x}^{\mu }\partial {x}^{\mu }}-\frac{{\partial }^{2}}{\partial {x}^{\mu }\partial {x}^{\mu }}\left({x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\mu }^{\mu }}{\partial {x}^{\sigma }}\right)\right)-\frac{\partial }{\partial {x}^{\mu }}\left(\frac{\partial {x}^{\sigma }}{\partial {x}^{\mu }}\frac{1}{2}\left(\frac{\partial {g}_{\mu }^{\mu }}{\partial {x}^{\sigma }}\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\frac{\partial }{\partial {x}^{\mu }}\left({x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\mu }^{\sigma }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\mu }^{\mu }}{\partial {x}^{\sigma }}\right)\right)+\frac{\partial }{\partial {x}^{\mu }}\left(\frac{\partial {x}^{\mu }}{\partial {x}^{\sigma }}\frac{1}{2}\left(\frac{\partial {g}_{\mu }^{\sigma }}{\partial {x}^{\mu }}\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{\partial }{\partial {x}^{\mu }}\left({x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\mu }^{\sigma }}{\partial {x}^{\mu }}\right)\right)-\frac{{\partial }^{2}{x}^{\sigma }}{\partial {x}^{\mu }\partial {x}^{\mu }}\frac{1}{2}\left(\frac{\partial {g}_{\mu }^{\mu }}{\partial {x}^{\sigma }}\right)\end{array}$

$\begin{array}{l}\text{ }-\frac{\partial }{\partial {x}^{\mu }}\left({x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\mu }^{\sigma }}{\partial {x}^{\sigma }}\right)\right)\frac{1}{2}\left(\frac{\partial {g}_{\mu }^{\mu }}{\partial {x}^{\sigma }}\right)-\frac{\partial {x}^{\sigma }}{\partial {x}^{\mu }}\frac{1}{2}\left(\frac{\partial {g}_{\mu }^{\sigma }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\mu }^{\mu }}{\partial {x}^{\sigma }}\right)\\ \text{ }-{x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\mu }^{\sigma }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\mu }^{\sigma }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\mu }^{\mu }}{\partial {x}^{\sigma }}\right)+\frac{\partial {x}^{\sigma }}{\partial {x}^{\sigma }}\frac{1}{2}\left(\frac{\partial {g}_{\mu }^{\sigma }}{\partial {x}^{\mu }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\mu }^{\mu }}{\partial {x}^{\sigma }}\right)\\ \text{ }+{x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\sigma }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\mu }^{\sigma }}{\partial {x}^{\mu }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\mu }^{\mu }}{\partial {x}^{\sigma }}\right)+\frac{{\partial }^{2}{x}^{\mu }}{\partial {x}^{\sigma }\partial {x}^{\mu }}\frac{1}{2}\left(\frac{\partial {g}_{\mu }^{\sigma }}{\partial {x}^{\mu }}\right)\end{array}$

$\begin{array}{l}\text{ }+\frac{\partial }{\partial {x}^{\mu }}\left({x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}^{\sigma }}\right)\right)\frac{1}{2}\left(\frac{\partial {g}_{\mu }^{\sigma }}{\partial {x}^{\mu }}\right)+\frac{\partial {x}^{\sigma }}{\partial {x}^{\sigma }}\frac{1}{2}\left(\frac{\partial {g}_{\mu }^{\mu }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\mu }^{\sigma }}{\partial {x}^{\mu }}\right)\\ \text{ }+{x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\sigma }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\mu }^{\mu }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\mu }^{\sigma }}{\partial {x}^{\mu }}\right)-\frac{\partial {x}^{\mu }}{\partial {x}^{\sigma }}\frac{1}{2}\left(\frac{\partial {g}_{\mu }^{\sigma }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\mu }^{\sigma }}{\partial {x}^{\mu }}\right)\end{array}$

$\begin{array}{l}\text{ }-{x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\mu }^{\sigma }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\mu }^{\sigma }}{\partial {x}^{\mu }}\right)+\frac{{\partial }^{2}{x}^{\mu }}{\partial {x}^{\mu }\partial {x}^{\sigma }}\frac{1}{2}\left(\frac{\partial {g}_{\mu }^{\sigma }}{\partial {x}^{\mu }}\right)\\ \text{ }+\frac{\partial }{\partial {x}^{\sigma }}\left({x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\mu }^{\mu }}{\partial {x}^{\sigma }}\right)\right)\frac{1}{2}\left(\frac{\partial {g}_{\mu }^{\sigma }}{\partial {x}^{\mu }}\right)+\frac{\partial {x}^{\sigma }}{\partial {x}^{\mu }}\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\mu }^{\sigma }}{\partial {x}^{\mu }}\right)\\ \text{ }+{x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\mu }^{\sigma }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\mu }^{\sigma }}{\partial {x}^{\mu }}\right)-\frac{\partial {x}^{\mu }}{\partial {x}^{\sigma }}\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\sigma }}{\partial {x}^{\mu }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\mu }^{\sigma }}{\partial {x}^{\mu }}\right)\\ \text{ }-{x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\sigma }}{\partial {x}^{\mu }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\mu }^{\sigma }}{\partial {x}^{\mu }}\right).\end{array}$ (28)

I get

$-{x}_{;\mu ;\mu ;\mu }^{\mu }=-\frac{{\partial }^{3}{x}^{\mu }}{\partial {x}^{\mu }\partial {x}^{\mu }\partial {x}^{\mu }}$ (29)

in consideration of Definition 7 for (28). Because the second term of the right side of (29) doesn’t exist,

${x}_{;\nu ;\nu ;\nu }^{\mu }=\frac{{\partial }^{3}{x}^{\mu }}{\partial {x}^{\nu }\partial {x}^{\nu }\partial {x}^{\nu }}$ (30)

can rewrite (29) using Definition 4. I rewrite three existing index $\nu$ in each term of (26), (27) in index $\mu$ using Definition 2 and get

$\begin{array}{c}{x}^{\mu ;\mu ;\mu ;\mu }=\frac{{\partial }^{3}{x}^{\mu }}{\partial {x}_{\mu }\partial {x}_{\mu }\partial {x}_{\mu }}+\frac{{\partial }^{2}}{\partial {x}_{\mu }\partial {x}_{\mu }}\left({x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}_{\mu }}\right)\right)+\frac{\partial }{\partial {x}_{\mu }}\left(\frac{\partial {x}^{\sigma }}{\partial {x}_{\mu }}\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}_{\mu }}\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{\partial }{\partial {x}_{\mu }}\left({x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\sigma }}{\partial {x}_{\mu }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}_{\mu }}\right)\right)-\frac{\partial }{\partial {x}_{\mu }}\left(\frac{\partial {x}^{\mu }}{\partial {x}^{\sigma }}\frac{1}{2}\left(\frac{\partial {g}^{\sigma \mu }}{\partial {x}_{\mu }}\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\frac{\partial }{\partial {x}_{\mu }}\left({x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}^{\sigma \mu }}{\partial {x}_{\mu }}\right)\right)+\frac{{\partial }^{2}{x}^{\sigma }}{\partial {x}_{\mu }\partial {x}_{\mu }}\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}_{\mu }}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{\partial }{\partial {x}_{\mu }}\left({x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\sigma }}{\partial {x}_{\mu }}\right)\right)\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}_{\mu }}\right)+\frac{\partial {x}^{\sigma }}{\partial {x}_{\mu }}\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\sigma }}{\partial {x}_{\mu }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}_{\mu }}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\sigma }}{\partial {x}_{\mu }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\sigma }}{\partial {x}_{\mu }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}_{\mu }}\right)-\frac{\partial {x}^{\sigma }}{\partial {x}^{\sigma }}\frac{1}{2}\left(\frac{\partial {g}^{\sigma \mu }}{\partial {x}_{\mu }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}_{\mu }}\right)\end{array}$

$\begin{array}{l}\text{ }-{x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\sigma }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}^{\sigma \mu }}{\partial {x}_{\mu }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}_{\mu }}\right)-\frac{{\partial }^{2}{x}^{\mu }}{\partial {x}^{\sigma }\partial {x}_{\mu }}\frac{1}{2}\left(\frac{\partial {g}^{\sigma \mu }}{\partial {x}_{\mu }}\right)\\ \text{ }-\frac{\partial }{\partial {x}_{\mu }}\left({x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}^{\sigma }}\right)\right)\frac{1}{2}\left(\frac{\partial {g}^{\sigma \mu }}{\partial {x}_{\mu }}\right)-\frac{\partial {x}^{\sigma }}{\partial {x}^{\sigma }}\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}_{\mu }}\right)\frac{1}{2}\left(\frac{\partial {g}^{\sigma \mu }}{\partial {x}_{\mu }}\right)\\ \text{ }-{x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\sigma }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}_{\mu }}\right)\frac{1}{2}\left(\frac{\partial {g}^{\sigma \mu }}{\partial {x}_{\mu }}\right)+\frac{\partial {x}^{\mu }}{\partial {x}^{\sigma }}\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\sigma }}{\partial {x}_{\mu }}\right)\frac{1}{2}\left(\frac{\partial {g}^{\sigma \mu }}{\partial {x}_{\mu }}\right)\end{array}$

$\begin{array}{l}\text{ }+{x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\sigma }}{\partial {x}_{\mu }}\right)\frac{1}{2}\left(\frac{\partial {g}^{\sigma \mu }}{\partial {x}_{\mu }}\right)-\frac{{\partial }^{2}{x}^{\mu }}{\partial {x}_{\mu }\partial {x}^{\sigma }}\frac{1}{2}\left(\frac{\partial {g}^{\sigma \mu }}{\partial {x}_{\mu }}\right)\\ \text{ }-\frac{\partial }{\partial {x}^{\sigma }}\left({x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}_{\mu }}\right)\right)\frac{1}{2}\left(\frac{\partial {g}^{\sigma \mu }}{\partial {x}_{\mu }}\right)-\frac{\partial {x}^{\sigma }}{\partial {x}_{\mu }}\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}^{\sigma \mu }}{\partial {x}_{\mu }}\right)\end{array}$

$\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-{x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\sigma }}{\partial {x}_{\mu }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}^{\sigma \mu }}{\partial {x}_{\mu }}\right)+\frac{\partial {x}^{\mu }}{\partial {x}^{\sigma }}\frac{1}{2}\left(\frac{\partial {g}^{\sigma \mu }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}^{\sigma \mu }}{\partial {x}_{\mu }}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}^{\sigma \mu }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}^{\sigma \mu }}{\partial {x}_{\mu }}\right)\\ =\frac{{\partial }^{3}{x}^{\mu }}{\partial {x}_{\mu }\partial {x}_{\mu }\partial {x}_{\mu }}+\frac{{\partial }^{2}}{\partial {x}_{\mu }\partial {x}_{\mu }}\left({x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}^{\mu \mu }}{\partial {x}^{\sigma }}\right)\right)+\frac{\partial }{\partial {x}_{\mu }}\left(\frac{\partial {x}^{\sigma }}{\partial {x}_{\mu }}\frac{1}{2}\left(\frac{\partial {g}^{\mu \mu }}{\partial {x}^{\sigma }}\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{\partial }{\partial {x}_{\mu }}\left({x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}^{\sigma \mu }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}^{\mu \mu }}{\partial {x}^{\sigma }}\right)\right)-\frac{\partial }{\partial {x}_{\mu }}\left(\frac{\partial {x}^{\mu }}{\partial {x}^{\sigma }}\frac{1}{2}\left(\frac{\partial {g}^{\sigma \mu }}{\partial {x}_{\mu }}\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\frac{\partial }{\partial {x}_{\mu }}\left({x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}^{\sigma \mu }}{\partial {x}_{\mu }}\right)\right)+\frac{{\partial }^{2}{x}^{\sigma }}{\partial {x}_{\mu }\partial {x}_{\mu }}\frac{1}{2}\left(\frac{\partial {g}^{\mu \mu }}{\partial {x}^{\sigma }}\right)\end{array}$

$\begin{array}{l}\text{ }+\frac{\partial }{\partial {x}_{\mu }}\left({x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}^{\sigma \mu }}{\partial {x}^{\sigma }}\right)\right)\frac{1}{2}\left(\frac{\partial {g}^{\mu \mu }}{\partial {x}^{\sigma }}\right)+\frac{\partial {x}^{\sigma }}{\partial {x}_{\mu }}\frac{1}{2}\left(\frac{\partial {g}^{\sigma \mu }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}^{\mu \mu }}{\partial {x}^{\sigma }}\right)\\ \text{ }+{x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}^{\sigma \mu }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}^{\sigma \mu }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}^{\mu \mu }}{\partial {x}^{\sigma }}\right)-\frac{\partial {x}^{\sigma }}{\partial {x}^{\sigma }}\frac{1}{2}\left(\frac{\partial {g}^{\sigma \mu }}{\partial {x}_{\mu }}\right)\frac{1}{2}\left(\frac{\partial {g}^{\mu \mu }}{\partial {x}^{\sigma }}\right)\\ \text{ }-{x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\sigma }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}^{\sigma \mu }}{\partial {x}_{\mu }}\right)\frac{1}{2}\left(\frac{\partial {g}^{\mu \mu }}{\partial {x}^{\sigma }}\right)-\frac{{\partial }^{2}{x}^{\mu }}{\partial {x}^{\sigma }\partial {x}_{\mu }}\frac{1}{2}\left(\frac{\partial {g}^{\sigma \mu }}{\partial {x}_{\mu }}\right)\\ \text{ }-\frac{\partial }{\partial {x}_{\mu }}\left({x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}^{\sigma }}\right)\right)\frac{1}{2}\left(\frac{\partial {g}^{\sigma \mu }}{\partial {x}_{\mu }}\right)-\frac{\partial {x}^{\sigma }}{\partial {x}^{\sigma }}\frac{1}{2}\left(\frac{\partial {g}^{\mu \mu }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}^{\sigma \mu }}{\partial {x}_{\mu }}\right)\\ \text{ }-{x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\sigma }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}^{\mu \mu }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}^{\sigma \mu }}{\partial {x}_{\mu }}\right)+\frac{\partial {x}^{\mu }}{\partial {x}^{\sigma }}\frac{1}{2}\left(\frac{\partial {g}^{\sigma \mu }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}^{\sigma \mu }}{\partial {x}_{\mu }}\right)\end{array}$

$\begin{array}{l}\text{ }+{x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}^{\sigma \mu }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}^{\sigma \mu }}{\partial {x}_{\mu }}\right)-\frac{{\partial }^{2}{x}^{\mu }}{\partial {x}_{\mu }\partial {x}^{\sigma }}\frac{1}{2}\left(\frac{\partial {g}^{\sigma \mu }}{\partial {x}_{\mu }}\right)\\ \text{ }-\frac{\partial }{\partial {x}^{\sigma }}\left({x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}^{\mu \mu }}{\partial {x}^{\sigma }}\right)\right)\frac{1}{2}\left(\frac{\partial {g}^{\sigma \mu }}{\partial {x}_{\mu }}\right)-\frac{\partial {x}^{\sigma }}{\partial {x}_{\mu }}\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}^{\sigma \mu }}{\partial {x}_{\mu }}\right)\\ \text{ }-{x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}^{\sigma \mu }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}^{\sigma \mu }}{\partial {x}_{\mu }}\right)+\frac{\partial {x}^{\mu }}{\partial {x}^{\sigma }}\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\sigma }}{\partial {x}_{\mu }}\right)\frac{1}{2}\left(\frac{\partial {g}^{\sigma \mu }}{\partial {x}_{\mu }}\right)\\ \text{ }+{x}^{\sigma }\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\mu }}{\partial {x}^{\sigma }}\right)\frac{1}{2}\left(\frac{\partial {g}_{\sigma }^{\sigma }}{\partial {x}_{\mu }}\right)\frac{1}{2}\left(\frac{\partial {g}^{\sigma \mu }}{\partial {x}_{\mu }}\right).\end{array}$

-End Proof-

5. About a Coordinate Transformations Equation in Tensor Satisfying Binary Law

Proposition 8 When all coordinate systems satisfy Binary Law, ${x}_{\mu }^{\mu }=\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}\frac{\partial {x}^{\nu }}{\partial {x}^{\mu }}{x}_{\nu }^{\nu }={x}_{\nu }^{\nu }$ is established for ${x}_{\mu }^{\mu }$ components of a tensor satisfying Binary law of rank zero.

Proof: When all coordinate systems satisfy Binary Law, I get

${x}_{\mu }^{\mu }=\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}\frac{\partial {x}^{\nu }}{\partial {x}^{\mu }}{x}_{\nu }^{\nu }={x}_{\nu }^{\nu }$ (31)

from Definition 14. Because (31) accords in Definition 14, the components of a tensor of rank zero are equivalent with components of a tensor satisfying Binary law of rank zero. I rewrite (31) by consideration of ${x}_{\mu }^{\mu }\to {x}_{\mu }^{;\mu },{x}_{\nu }^{\nu }\to {x}_{\nu }^{;\nu }$, (6), $\mu -\nu$ inversion form of (6) and get

$\begin{array}{c}\left(\frac{\partial {x}_{\mu }}{\partial {x}_{\mu }}-{x}_{\nu }\frac{1}{2}\left(\frac{\partial {g}^{\nu \mu }}{\partial {x}^{\mu }}\right)\right)=\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}\frac{\partial {x}^{\nu }}{\partial {x}^{\mu }}\left(\frac{\partial {x}_{\nu }}{\partial {x}_{\nu }}-{x}_{\mu }\frac{1}{2}\left(\frac{\partial {g}^{\mu \nu }}{\partial {x}^{\nu }}\right)\right)\\ =\left(\frac{\partial {x}_{\nu }}{\partial {x}_{\nu }}-{x}_{\mu }\frac{1}{2}\left(\frac{\partial {g}^{\mu \nu }}{\partial {x}^{\nu }}\right)\right).\end{array}$ (32)

-End Proof-

Proposition 9 When all coordinate systems satisfy Binary Law, ${x}^{\mu }=\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}{x}^{\nu }$ is established for ${x}^{\mu }$ contravariant components of a tensor satisfying Binary law of the first rank.

Proof: When all coordinate systems satisfy Binary Law, I get

${x}^{\mu }=\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}{x}^{\nu }$ (33)

from Definition 15. Because (33) accords in Definition 15, the contravariant components of a tensor of the first rank are equivalent with contravariant components of a tensor satisfying Binary law of the first rank.

-End Proof-

Proposition 10 When all coordinate systems satisfy Binary Law, ${x}_{\mu }=\frac{\partial {x}^{\nu }}{\partial {x}^{\mu }}{x}_{\nu }$ is established for ${x}_{\mu }$ covariant components of a tensor satisfying Binary law of the first rank.

Proof: When all coordinate systems satisfy Binary Law, I get

${x}_{\mu }=\frac{\partial {x}^{\nu }}{\partial {x}^{\mu }}{x}_{\nu }$ (34)

from Definition 16. Because (34) accords in Definition 16, the covariant components of a tensor of the first rank are equivalent with covariant components of a tensor satisfying Binary law of the first rank.

-End Proof-

Proposition 11 When all coordinate systems satisfy Binary Law, ${x}^{\mu \nu }=\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}\frac{\partial {x}^{\nu }}{\partial {x}^{\mu }}{x}^{\nu \mu }={x}^{\nu \mu }$ is established for ${x}^{\mu \nu }$ contravariant components of a tensor satisfying Binary law of the second rank.

Proof: When all coordinate systems satisfy Binary Law, I get

${x}^{\mu \nu }=\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}\frac{\partial {x}^{\nu }}{\partial {x}^{\nu }}{x}^{\nu \nu }=\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}{x}^{\nu \nu }$ (35)

from Definition 17. ${x}^{\mu \nu }$ isn’t contravariant components of a tensor satisfying Binary law of the second rank than (35). This is a problem. The dummy index has an invariable property for consideration of Binary Law. In other words, the index which was dummy index in Definition 17 is dummy index in (35). Therefore,

I rewrite dummy index $\nu$ in $\frac{\partial {x}^{\nu }}{\partial {x}^{\nu }}{x}^{\nu \nu }$ of (35) in $\mu$ and get

${x}^{\mu \nu }=\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}\frac{\partial {x}^{\nu }}{\partial {x}^{\mu }}{x}^{\nu \mu }={x}^{\nu \mu }$. (36)

The kind of the optional dummy index is only two kinds of $\nu ,\mu$ in consideration of Definition 6 here. A problem in (35) is solved in (36). If I assume establishment of

${x}^{\mu \nu }={x}_{\mu }^{\mu },{x}^{\nu \mu }={x}_{\nu }^{\nu }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(\text{False}\right)$. (37)

I get

${x}_{\mu }^{\mu }=\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}\frac{\partial {x}^{\nu }}{\partial {x}^{\mu }}{x}_{\nu }^{\nu }={x}_{\nu }^{\nu }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(\text{False}\right)$ (38)

from (36), (37). I get

${x}_{\mu }^{\mu }={x}_{\mu }^{\mu }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(\text{False}\right)$ (39)

from (31), (38). Because (39) isn’t established,

${x}^{\mu \nu }={x}_{\mu }^{\mu },{x}^{\nu \mu }={x}_{\nu }^{\nu }$ (40)

is established.

-End Proof-

Proposition 12 When all coordinate systems satisfy Binary Law, ${x}_{\mu \nu }=\frac{\partial {x}^{\nu }}{\partial {x}^{\mu }}\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}{x}_{\nu \mu }={x}_{\nu \mu }$ is established for ${x}_{\mu \nu }$ covariant components of a tensor satisfying Binary law of the second rank.

Proof: When all coordinate systems satisfy Binary Law, I get

${x}_{\mu \nu }=\frac{\partial {x}^{\nu }}{\partial {x}^{\mu }}\frac{\partial {x}^{\nu }}{\partial {x}^{\nu }}{x}_{\nu \nu }=\frac{\partial {x}^{\nu }}{\partial {x}^{\mu }}{x}_{\nu \nu }$ (41)

from Definition 18. ${x}_{\mu \nu }$ isn’t covariant components of a tensor satisfying Binary law of the second rank than (41). This is a problem. The dummy index has an invariable property for consideration of Binary Law. In other words, the index which was dummy index in Definition 18 is dummy index in (41). Therefore,

I rewrite dummy index $\nu$ in $\frac{\partial {x}^{\nu }}{\partial {x}^{\nu }}{x}_{\nu \nu }$ of (41) in $\mu$ and get

${x}_{\mu \nu }=\frac{\partial {x}^{\nu }}{\partial {x}^{\mu }}\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}{x}_{\nu \mu }={x}_{\nu \mu }$. (42)

The kind of the optional dummy index is only two kinds of $\nu ,\mu$ in consideration of Definition 6 here. A problem in (41) is solved in (42). If I assume establishment of

${x}_{\mu \nu }={x}_{\nu }^{\nu },{x}_{\nu \mu }={x}_{\mu }^{\mu }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(\text{False}\right)$. (43)

I get

${x}_{\nu }^{\nu }=\frac{\partial {x}^{\nu }}{\partial {x}^{\mu }}\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}{x}_{\mu }^{\mu }={x}_{\mu }^{\mu }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(\text{False}\right)$ (44)

from (42), (43). I get

${x}_{\mu }^{\mu }={x}_{\mu }^{\mu }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(\text{False}\right)$ (45)

from (31), (44). Because (45) isn’t established,

${x}_{\mu \nu }={x}_{\nu }^{\nu },{x}_{\nu \mu }={x}_{\mu }^{\mu }$ (46)

is established. I rewrite (42) by consideration of ${x}_{\mu \nu }\to {x}_{\mu ;\nu },{x}_{\nu \mu }\to {x}_{\nu ;\mu }$, (9), $\mu -\nu$ inversion form of (9) and get

$\left(\frac{\partial {x}_{\mu }}{\partial {x}^{\nu }}\right)=\frac{\partial {x}^{\nu }}{\partial {x}^{\mu }}\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}\left(\frac{\partial {x}_{\nu }}{\partial {x}^{\mu }}\right)=\left(\frac{\partial {x}_{\nu }}{\partial {x}^{\mu }}\right)$. (47)

I rewrite (46) by consideration of ${x}_{\mu \nu }\to {x}_{\mu ;\nu },{x}_{\nu }^{\nu }\to {x}_{\nu }^{;\nu }$, (9), $\mu -\nu$ inversion form of (6) and get

$\frac{\partial {x}_{\mu }}{\partial {x}^{\nu }}=\left(\frac{\partial {x}_{\nu }}{\partial {x}_{\nu }}-{x}_{\mu }\frac{1}{2}\left(\frac{\partial {g}^{\mu \nu }}{\partial {x}^{\nu }}\right)\right)$. (48)

-End Proof-

Proposition 13 When all coordinate systems satisfy Binary Law, ${x}_{\nu }^{\mu }=\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}{x}_{\mu }^{\nu }$ is established for ${x}_{\nu }^{\mu }$ components of the mixed tensor satisfying Binary law of the second rank.

Proof: When all coordinate systems satisfy Binary Law, I get

${x}_{\nu }^{\mu }=\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}\frac{\partial {x}^{\nu }}{\partial {x}^{\nu }}{x}_{\nu }^{\nu }=\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}{x}_{\nu }^{\nu }$ (49)

from Definition 19. ${x}_{\nu }^{\mu }$ isn’t components of the mixed tensor satisfying Binary law of the second rank than (49). This is a problem. The dummy index has an invariable property for consideration of Binary Law. In other words, the index which was dummy index in Definition 19 is dummy index in (49). Therefore, I

rewrite dummy index $\nu$ in $\frac{\partial {x}^{\nu }}{\partial {x}^{\nu }}{x}_{\nu }^{\nu }$ of (49) in $\mu$ and get

${x}_{\nu }^{\mu }=\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}{x}_{\mu }^{\nu }$. (50)

The kind of the optional dummy index is only two kinds of $\nu ,\mu$ in consideration of Definition 6 here. A problem in (49) is solved in (50). I rewrite (50) by consideration of ${x}_{\nu }^{\mu }\to {x}_{;\nu }^{\mu },{x}_{\mu }^{\nu }\to {x}_{;\mu }^{\nu }$, (16), $\mu -\nu$ inversion form of (16) and get

$\left(\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}\right)=\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}\left(\frac{\partial {x}^{\nu }}{\partial {x}^{\mu }}\right)$. (51)

-End Proof-

Proposition 14 When all coordinate systems satisfy Binary Law, ${x}_{\nu \nu }^{\mu }=\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}\frac{\partial {x}^{\nu }}{\partial {x}^{\mu }}\frac{\partial {x}^{\nu }}{\partial {x}^{\mu }}{x}_{\mu \mu }^{\nu }=\frac{\partial {x}^{\nu }}{\partial {x}^{\mu }}{x}_{\mu \mu }^{\nu }$ is established for ${x}_{\nu \nu }^{\mu }$ components of the mixed tensor satisfying Binary law of the third rank of the second rank covariant in the first rank contravariant.

Proof: When all coordinate systems satisfy Binary Law, I get

${x}_{\nu \nu }^{\mu }=\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}\frac{\partial {x}^{\nu }}{\partial {x}^{\nu }}\frac{\partial {x}^{\nu }}{\partial {x}^{\nu }}{x}_{\nu \nu }^{\nu }=\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}{x}_{\nu \nu }^{\nu }$ (52)

from Definition 20. ${x}_{\nu \nu }^{\mu }$ isn’t components of the mixed tensor satisfying Binary law of the third rank of the second rank covariant in the first rank contravariant than (52). This is a problem. The dummy index has an invariable property for consideration of Binary Law.

In other words, the index which was dummy index in Definition 20 is dummy index in (52). Therefore, I rewrite dummy index $\nu$ in $\frac{\partial {x}^{\nu }}{\partial {x}^{\nu }}\frac{\partial {x}^{\nu }}{\partial {x}^{\nu }}{x}_{\nu \nu }^{\nu }$ of (52) in $\mu$ and get

${x}_{\nu \nu }^{\mu }=\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}{x}_{\mu \mu }^{\nu }$. (53)

The kind of the optional dummy index is only two kinds of $\nu ,\mu$ in consideration of Definition 6 here. A problem in (52) is solved in (53). If I assume establishment of

${x}_{\nu \nu }^{\mu }=\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}\frac{\partial {x}^{\nu }}{\partial {x}^{\mu }}\frac{\partial {x}^{\nu }}{\partial {x}^{\mu }}{x}_{\mu \mu }^{\nu }=\frac{\partial {x}^{\nu }}{\partial {x}^{\mu }}{x}_{\mu \mu }^{\nu }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(\text{False}\right)$. (54)

I get

$\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}{x}_{\mu \mu }^{\nu }=\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}\frac{\partial {x}^{\nu }}{\partial {x}^{\mu }}\frac{\partial {x}^{\nu }}{\partial {x}^{\mu }}{x}_{\mu \mu }^{\nu }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(\text{False}\right)$ (55)

from (53), (54). I rewrite the right side of (55) using Definition 4, Definition 5 and get

$\begin{array}{c}\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}{x}_{\mu \mu }^{\nu }=\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}\frac{\partial \left(-{x}^{\mu }\right)}{\partial \left(-{x}^{\nu }\right)}\frac{\partial \left(-{x}^{\mu }\right)}{\partial \left(-{x}^{\nu }\right)}{x}_{\mu \mu }^{\nu }\\ =\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}{x}_{\mu \mu }^{\nu }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(\text{False}\right).\end{array}$ (56)

Because (56) isn’t established,

${x}_{\nu \nu }^{\mu }=\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}\frac{\partial {x}^{\nu }}{\partial {x}^{\mu }}\frac{\partial {x}^{\nu }}{\partial {x}^{\mu }}{x}_{\mu \mu }^{\nu }=\frac{\partial {x}^{\nu }}{\partial {x}^{\mu }}{x}_{\mu \mu }^{\nu }$ (57)

is established. If I assume establishment of

${x}_{\nu \nu }^{\mu }={x}_{\mu },{x}_{\mu \mu }^{\nu }={x}_{\nu }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(\text{False}\right)$. (58)

I get

${x}_{\mu }=\frac{\partial {x}^{\nu }}{\partial {x}^{\mu }}{x}_{\nu }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(\text{False}\right)$ (59)

from (57), (58). I get

${x}_{\mu }={x}_{\mu }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(\text{False}\right)$ (60)

from (34), (59). Because (60) isn’t established,

${x}_{\nu \nu }^{\mu }={x}_{\mu },{x}_{\mu \mu }^{\nu }={x}_{\nu }$ (61)

is established. I rewrite (57) by consideration of ${x}_{\nu \nu }^{\mu }\to {x}_{;\nu ;\nu }^{\mu },{x}_{\mu \mu }^{\nu }\to {x}_{;\mu ;\mu }^{\nu }$, (23), $\mu -\nu$ inversion form of (23) and get

$\frac{{\partial }^{2}{x}^{\mu }}{\partial {x}^{\nu }\partial {x}^{\nu }}=\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}\frac{\partial {x}^{\nu }}{\partial {x}^{\mu }}\frac{\partial {x}^{\nu }}{\partial {x}^{\mu }}\frac{{\partial }^{2}{x}^{\nu }}{\partial {x}^{\mu }\partial {x}^{\mu }}=\frac{\partial {x}^{\nu }}{\partial {x}^{\mu }}\frac{{\partial }^{2}{x}^{\nu }}{\partial {x}^{\mu }\partial {x}^{\mu }}$. (62)

I rewrite (61) by consideration of ${x}_{\nu \nu }^{\mu }\to {x}_{;\nu ;\nu }^{\mu }$, (23) and get

$\frac{{\partial }^{2}{x}^{\mu }}{\partial {x}^{\nu }\partial {x}^{\nu }}={x}_{\mu }$. (63)

-End Proof-

Proposition 15 When all coordinate systems satisfy Binary Law, ${x}_{\nu \nu \nu }^{\mu }=\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}\frac{\partial {x}^{\nu }}{\partial {x}^{\mu }}\frac{\partial {x}^{\nu }}{\partial {x}^{\mu }}{x}_{\mu \mu \mu }^{\nu }={x}_{\mu \mu \mu }^{\nu }$ is established for ${x}_{\nu \nu \nu }^{\mu }$ components of the mixed tensor satisfying Binary law of the fourth rank of the third rank covariant in the first rank contravariant.

Proof: When all coordinate systems satisfy Binary Law, I get

${x}_{\nu \nu \nu }^{\mu }=\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}\frac{\partial {x}^{\nu }}{\partial {x}^{\nu }}\frac{\partial {x}^{\nu }}{\partial {x}^{\nu }}\frac{\partial {x}^{\nu }}{\partial {x}^{\nu }}{x}_{\nu \nu \nu }^{\nu }=\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}{x}_{\nu \nu \nu }^{\nu }$. (64)

from Definition 21. ${x}_{\nu \nu \nu }^{\mu }$ isn’t components of the mixed tensor satisfying Binary law of the fourth rank of the third rank covariant in the first rank contravariant than (64). This is a problem. The dummy index has an invariable property for consideration of Binary Law.

In other words, the index which was dummy index in Definition 21 is dummy index in (64). Therefore, I rewrite dummy index $\nu$ in $\frac{\partial {x}^{\nu }}{\partial {x}^{\nu }}\frac{\partial {x}^{\nu }}{\partial {x}^{\nu }}\frac{\partial {x}^{\nu }}{\partial {x}^{\nu }}{x}_{\nu \nu \nu }^{\nu }$ of (64) in $\mu$ and get

${x}_{\nu \nu \nu }^{\mu }=\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}{x}_{\mu \mu \mu }^{\nu }$. (65)

The kind of the optional dummy index is only two kinds of $\nu ,\mu$ in consideration of Definition 6 here. A problem in (64) is solved in (65). If I assume establishment of

${x}_{\nu \nu \nu }^{\mu }=\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}\frac{\partial {x}^{\nu }}{\partial {x}^{\mu }}\frac{\partial {x}^{\nu }}{\partial {x}^{\mu }}{x}_{\mu \mu \mu }^{\nu }={x}_{\mu \mu \mu }^{\nu }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(\text{False}\right)$. (66)

I get

$\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}{x}_{\mu \mu \mu }^{\nu }=\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}\frac{\partial {x}^{\nu }}{\partial {x}^{\mu }}\frac{\partial {x}^{\nu }}{\partial {x}^{\mu }}{x}_{\mu \mu \mu }^{\nu }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(\text{False}\right)$ (67)

from (65), (66). I rewrite the right side of (67) using Definition 4, Definition 5 and get

$\begin{array}{c}\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}{x}_{\mu \mu \mu }^{\nu }=\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}\frac{\partial \left(-{x}^{\mu }\right)}{\partial \left(-{x}^{\nu }\right)}\frac{\partial \left(-{x}^{\mu }\right)}{\partial \left(-{x}^{\nu }\right)}{x}_{\mu \mu \mu }^{\nu }\\ =\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}{x}_{\mu \mu \mu }^{\nu }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(\text{False}\right).\end{array}$ (68)

Because (68) isn’t established,

${x}_{\nu \nu \nu }^{\mu }=\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}\frac{\partial {x}^{\nu }}{\partial {x}^{\mu }}\frac{\partial {x}^{\nu }}{\partial {x}^{\mu }}{x}_{\mu \mu \mu }^{\nu }={x}_{\mu \mu \mu }^{\nu }$ (69)

is established. If I assume establishment of

${x}_{\nu \nu \nu }^{\mu }={x}_{\mu \nu },{x}_{\mu \mu \mu }^{\nu }={x}_{\nu \mu }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(\text{False}\right)$. (70)

I get

${x}_{\mu \nu }=\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}\frac{\partial {x}^{\nu }}{\partial {x}^{\mu }}\frac{\partial {x}^{\nu }}{\partial {x}^{\mu }}{x}_{\nu \mu }={x}_{\nu \mu }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(\text{False}\right)$ (71)

from (69), (70). I get

${x}_{\mu \nu }={x}_{\mu \nu }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(\text{False}\right)$ (72)

from (42), (71). Because (72) isn’t established,

${x}_{\nu \nu \nu }^{\mu }={x}_{\mu \nu },{x}_{\mu \mu \mu }^{\nu }={x}_{\nu \mu }$ (73)

is established. I get

${x}_{\nu \nu \nu }^{\mu }={x}_{\mu \nu }={x}_{\nu }^{\nu },{x}_{\mu \mu \mu }^{\nu }={x}_{\nu \mu }={x}_{\mu }^{\mu }$ (74)

from (46), (73). I rewrite (69) by consideration of ${x}_{\nu \nu \nu }^{\mu }\to {x}_{;\nu ;\nu ;\nu }^{\mu }$, ${x}_{\mu \mu \mu }^{\nu }\to {x}_{;\mu ;\mu ;\mu }^{\nu }$, (30), $\mu -\nu$ inversion form of (30) and get

$\frac{{\partial }^{3}{x}^{\mu }}{\partial {x}^{\nu }\partial {x}^{\nu }\partial {x}^{\nu }}=\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}\frac{\partial {x}^{\mu }}{\partial {x}^{\nu }}\frac{\partial {x}^{\nu }}{\partial {x}^{\mu }}\frac{\partial {x}^{\nu }}{\partial {x}^{\mu }}\frac{{\partial }^{3}{x}^{\nu }}{\partial {x}^{\mu }\partial {x}^{\mu }\partial {x}^{\mu }}=\frac{{\partial }^{3}{x}^{\nu }}{\partial {x}^{\mu }\partial {x}^{\mu }\partial {x}^{\mu }}$. (75)

I rewrite (74) by consideration of ${x}_{\nu \nu \nu }^{\mu }\to {x}_{;\nu ;\nu ;\nu }^{\mu },{x}_{\mu \nu }\to {x}_{\mu ;\nu },{x}_{\nu }^{\nu }\to {x}_{\nu }^{;\nu }$, (9), (30), $\mu -\nu$ inversion form of (6) and get

$\frac{{\partial }^{3}{x}^{\mu }}{\partial {x}^{\nu }\partial {x}^{\nu }\partial {x}^{\nu }}=\frac{\partial {x}_{\mu }}{\partial {x}^{\nu }}=\left(\frac{\partial {x}_{\nu }}{\partial {x}_{\nu }}-{x}_{\mu }\frac{1}{2}\left(\frac{\partial {g}^{\mu \nu }}{\partial {x}^{\nu }}\right)\right)$. (76)

I get

$\frac{{\partial }^{3}{x}^{\mu }}{\partial {x}^{\nu }\partial {x}^{\nu }\partial {x}^{\nu }}=\frac{\partial {x}_{\mu }}{\partial {x}^{\nu }}=\left(\frac{\partial {x}_{\nu }}{\partial {x}_{\nu }}-{x}_{\mu }\frac{1}{2}\left(\frac{\partial {g}^{\mu \nu }}{\partial {x}^{\nu }}\right)\right)=M$ (77)

from (76), Definition 8.

-End Proof-

6. Discussion

Because (31) accords in Definition 14, components of a tensor of rank zero accord in components of a tensor satisfying Binary law of rank zero.

Because (33) accords in Definition 15, contravariant components of a tensor of the first rank accord in contravariant components of a tensor satisfying Binary law of the first rank.

Because (34) accords in Definision16, covariant components of a tensor of the first rank accord in covariant components of a tensor satisfying Binary law of the first rank.

I get

${x}_{\left[1\right]}^{1}=\frac{\partial {x}^{1}}{\partial {x}^{\left[1\right]}}\frac{\partial {x}^{1}}{\partial {x}^{\left[1\right]}}{x}_{1}^{\left[1\right]}$, ${x}_{\left[2\right]}^{1}=\frac{\partial {x}^{1}}{\partial {x}^{\left[2\right]}}\frac{\partial {x}^{1}}{\partial {x}^{\left[2\right]}}{x}_{1}^{\left[2\right]}$,

${x}_{\left[1\right]}^{2}=\frac{\partial {x}^{2}}{\partial {x}^{\left[1\right]}}\frac{\partial {x}^{2}}{\partial {x}^{\left[1\right]}}{x}_{2}^{\left[1\right]}$, ${x}_{\left[2\right]}^{2}=\frac{\partial {x}^{2}}{\partial {x}^{\left[2\right]}}\frac{\partial {x}^{2}}{\partial {x}^{\left[2\right]}}{x}_{2}^{\left[2\right]}$

from (50) if I assume a dimensional number 2.

Conflicts of Interest

The author declares no conflicts of interest regarding the publication of this paper.

  Ichidayama, K. (2017) Journal of Modern Physics, 8, 944-963. https://doi.org/10.4236/jmp.2017.86060  Ichidayama, K. (2017) Journal of Modern Physics, 8, 126-132. https://doi.org/10.4236/jmp.2017.81011  Dirac, P.A.M. (1975) General Theory of Relativity. John Wiley and Sons, Inc., New York.  Fleisch, D. (2012) A Student’s Guide to Vectors and Tensors. Cambridge University Press, Cambridge.     customer@scirp.org +86 18163351462(WhatsApp) 1655362766  Paper Publishing WeChat 