Translation:On the spacetime lines of a Minkowski world/Paragraph 2

by Friedrich Kottler

$n=3$, $p=2$. Theorem of Gauss.

By (1):

${\begin{aligned}\int \int du^{(1)}du^{(2)}\left[{\frac {\partial \left(x^{(2)}x^{(3)}\right)}{\partial \left(u^{(1)}u^{(2)}\right)}}A_{23}+{\frac {\partial \left(x^{(3)}x^{(1)}\right)}{\partial \left(u^{(1)}u^{(2)}\right)}}A_{31}+{\frac {\partial \left(x^{(1)}x^{(2)}\right)}{\partial \left(u^{(1)}u^{(2)}\right)}}A_{12}\right]=\\=\int \int \int dv^{(1)}dv^{(2)}dv^{(3)}{\frac {\partial \left(x^{(1)}x^{(2)}x^{(3)}\right)}{\partial \left(v^{(1)}v^{(2)}v^{(3)}\right)}}A_{123},\end{aligned}}$,

$A_{123}={\frac {\partial }{\partial x^{(3)}}}A_{23}-{\frac {\partial }{\partial x^{(2)}}}A_{13}+{\frac {\partial }{\partial x^{(1)}}}A_{23},\quad A_{\alpha \beta }=-A_{\beta \alpha }$

Introducing the supplements

${\sqrt {\frac {1}{c}}}A_{23}=B^{(1)},\quad {\sqrt {\frac {1}{c}}}A_{31}=B^{(2)},\quad {\sqrt {\frac {1}{c}}}A_{12}=B^{(3)},$

${\sqrt {\frac {c}{b}}}{\frac {\partial \left(x^{(2)}x^{(3)}\right)}{\partial \left(u^{(1)}u^{(2)}\right)}}=N_{1},\quad {\sqrt {\frac {c}{b}}}{\frac {\partial \left(x^{(3)}x^{(1)}\right)}{\partial \left(u^{(1)}u^{(2)}\right)}}=N_{2},\quad {\sqrt {\frac {c}{b}}}{\frac {\partial \left(x^{(1)}x^{(2)}\right)}{\partial \left(u^{(1)}u^{(2)}\right)}}=N_{3},$

where

$b=\left|{\begin{matrix}b_{11}&b_{12}\\b_{21}&b_{22}\end{matrix}}\right|=\left|{\begin{matrix}\sum c_{\alpha \beta }{\frac {\partial x^{(\alpha )}}{\partial u^{(1)}}}{\frac {\partial x^{(\beta )}}{\partial u^{(1)}}}&\sum c_{\alpha \beta }{\frac {\partial x^{(\alpha )}}{\partial u^{(1)}}}{\frac {\partial x^{(\beta )}}{\partial u^{(1)}}}\\\sum c_{\alpha \beta }{\frac {\partial x^{(\alpha )}}{\partial u^{(2)}}}{\frac {\partial x^{(\beta )}}{\partial u^{(1)}}}&\sum c_{\alpha \beta }{\frac {\partial x^{(\alpha )}}{\partial u^{(2)}}}{\frac {\partial x^{(\beta )}}{\partial u^{(2)}}}\end{matrix}}\right|$

is the discriminant of the arc on $M_{2}$,^[1] thus because of

$dS_{2}={\sqrt {b}}du^{(1)}du^{(2)},\quad dS_{3}={\sqrt {c}}dx^{(1)}dx^{(2)}dx^{(3)}$:

it follows

${\begin{aligned}&\int \int dS_{2}\left[N_{1}B^{(1)}+N_{2}B^{(2)}+N_{3}B^{(3)}\right]=\\&\quad =\int \int \int dS_{3}{\frac {1}{\sqrt {c}}}\left[{\frac {\partial }{\partial x^{(1)}}}\left({\sqrt {c}}B^{(1)}\right)+{\frac {\partial }{\partial x^{(2)}}}\left({\sqrt {c}}B^{(2)}\right)+{\frac {\partial }{\partial x^{(3)}}}\left({\sqrt {c}}B^{(3)}\right)\right]\end{aligned}}$

Here, the normal $N$ goes to the exterior. For the generalized divergence:

$\mathrm {div} \left(B^{(1)}B^{(2)}B^{(3)}\right)={\frac {1}{\sqrt {c}}}\sum _{\alpha }{\frac {\partial }{\partial x^{(\alpha )}}}\left({\sqrt {c}}B^{(\alpha )}\right)$

or introducing the system reciprocal to $B^{(\alpha )}$

$B^{(\alpha )}=\sum _{\beta }c^{(\alpha \beta )}B_{\beta }:$:

it follows

$\mathrm {div} \left(B_{1}B_{2}B_{3}\right)={\frac {1}{\sqrt {c}}}\sum _{\alpha }{\frac {\partial }{\partial x^{(\alpha )}}}\left({\sqrt {c}}\sum _{\beta }c^{(\alpha \beta )}B_{\beta }\right)$

or eventually, if $B_{\beta }=\mathrm {B_{\beta }} ={\frac {\partial B}{\partial x^{(\beta )}}}$ can be represented as gradient of a scalar (invariant) quantity $B$:

$\mathrm {div} \left(\mathrm {B} _{1}\mathrm {B} _{2}\mathrm {B} _{3}\right)=\Delta B={\frac {1}{\sqrt {c}}}\sum _{\alpha }{\frac {\partial }{\partial x^{(\alpha )}}}\left({\sqrt {c}}\sum _{\beta }c^{(\alpha \beta )}{\frac {\partial B}{\partial x^{(\beta )}}}\right)$^[2]

$n=3$, $p=1$. Theorem of Stokes.

By (1):

${\begin{aligned}&\int du\left[{\frac {\partial x^{(1)}}{\partial u}}A_{1}+{\frac {\partial x^{(2)}}{\partial u}}A_{2}+{\frac {\partial x^{(3)}}{\partial u}}A_{3}\right]=\\&\quad =\int \int dv^{(1)}dv^{(2)}\left[{\frac {\partial \left(x^{(1)}x^{(3)}\right)}{\partial \left(v^{(1)}v^{(2)}\right)}}\mathrm {A} _{23}+{\frac {\partial \left(x^{(3)}x^{(1)}\right)}{\partial \left(v^{(1)}v^{(2)}\right)}}\mathrm {A} _{31}+{\frac {\partial \left(x^{(1)}x^{(2)}\right)}{\partial \left(v^{(1)}v^{(2)}\right)}}\mathrm {A} _{12}\right]\end{aligned}}$

$\mathrm {A} _{23}={\frac {\partial A_{2}}{\partial x^{(3)}}}-{\frac {\partial A_{3}}{\partial x^{(2)}}},\quad \mathrm {A} _{31}={\frac {\partial A_{3}}{\partial x^{(1)}}}-{\frac {\partial A_{1}}{\partial x^{(3)}}},\quad \mathrm {A} _{12}={\frac {\partial A_{1}}{\partial x^{(2)}}}-{\frac {\partial A_{2}}{\partial x^{(1)}}}$

or

${\begin{aligned}&\int ds\left[{\frac {dx^{(1)}}{ds}}A_{1}+{\frac {dx^{(2)}}{ds}}A_{2}+{\frac {dx^{(3)}}{ds}}A_{3}\right]=\\&\quad =\int \int dS_{2}\left[N_{1}{\frac {1}{\sqrt {c}}}\left({\frac {\partial A_{2}}{\partial x^{(3)}}}-{\frac {\partial A_{3}}{\partial x^{(2)}}}\right)+N_{2}{\frac {1}{\sqrt {c}}}\left({\frac {\partial A_{3}}{\partial x^{(1)}}}-{\frac {\partial A_{1}}{\partial x^{(3)}}}\right)+N_{3}{\frac {1}{\sqrt {c}}}\left({\frac {\partial A_{1}}{\partial x^{(2)}}}-{\frac {\partial A_{2}}{\partial x^{(1)}}}\right)\right]\end{aligned}}$

Here, the line integral is orbiting around the normal $N$ in the negative sense, thus clock-wise if the coordinate system is a right-system; because by § 1 the directions ${\frac {dx}{ds}}$, $N'$, $N$ are following each other like the coordinate axes, where $N'$ is the normal (which is directed outwards of the area framed on $M_{2}$) of the framing-$M_{1}$. Ordinarily, one prefers a positive sense of circulation and therefore the generalized rotation:

$\mathrm {rot} ^{(1)}\left(A_{1}A_{2}A_{3}\right)={\frac {1}{\sqrt {c}}}\left({\frac {\partial A_{3}}{\partial x^{(2)}}}-{\frac {\partial A_{2}}{\partial x^{(3)}}}\right)$ etc.

$\mathrm {rot} ^{(1)}\left(A^{(1)}A^{(2)}A^{(3)}\right)={\frac {1}{\sqrt {c}}}\left({\frac {\partial }{\partial x^{(2)}}}\left(\sum _{\beta }c_{3\beta }A^{(\beta )}\right)-{\frac {\partial }{\partial x^{(3)}}}\left(\sum _{\beta }c_{2\beta }A^{(\beta )}\right)\right)$ etc.

$n=4$, $p=3$.

${\begin{aligned}&\int \int \int du^{(1)}du^{(2)}du^{(3)}\left[{\frac {\partial \left(x^{(2)}x^{(3)}x^{(4)}\right)}{\partial \left(u^{(1)}u^{(2)}u^{(3)}\right)}}A_{234}+{\frac {\partial \left(x^{(1)}x^{(3)}x^{(4)}\right)}{\partial \left(u^{(1)}u^{(2)}u^{(3)}\right)}}A_{134}+\right.\\&\left.+{\frac {\partial \left(x^{(1)}x^{(2)}x^{(4)}\right)}{\partial \left(u^{(1)}u^{(2)}u^{(3)}\right)}}A_{124}+{\frac {\partial \left(x^{(1)}x^{(2)}x^{(3)}\right)}{\partial \left(u^{(1)}u^{(2)}u^{(3)}\right)}}A_{123}\right]=\\&\quad =\int \int \int dv^{(1)}dv^{(2)}dv^{(3)}dv^{(4)}{\frac {\partial \left(x^{(1)}x^{(2)}x^{(3)}x^{(4)}\right)}{\partial \left(v^{(1)}v^{(2)}v^{(3)}v^{(4)}\right)}}\mathrm {A} _{1234}\end{aligned}}$

$\mathrm {A} _{1234}={\frac {\partial }{\partial x^{(4)}}}A_{123}-{\frac {\partial }{\partial x^{(3)}}}A_{124}+{\frac {\partial }{\partial x^{(2)}}}A_{134}-{\frac {\partial }{\partial x^{(1)}}}A_{234},$

$A_{123}=-A_{131}=-A_{321}=A_{231}=A_{312}=-A_{213}$ etc.

Introducing the supplements

$-{\sqrt {\frac {1}{c}}}A_{234}=B^{(1)},\quad {\sqrt {\frac {1}{c}}}A_{134}=B^{(2)},\quad -{\sqrt {\frac {1}{c}}}A_{124}=B^{(3)},\quad {\sqrt {\frac {1}{c}}}A_{123}=B^{(4)},$

${\begin{aligned}-{\sqrt {\frac {c}{b}}}{\frac {\partial \left(x^{(2)}x^{(3)}x^{(4)}\right)}{\partial \left(u^{(1)}u^{(2)}u^{(3)}\right)}}&=N_{1},&{\sqrt {\frac {c}{b}}}{\frac {\partial \left(x^{(1)}x^{(3)}x^{(4)}\right)}{\partial \left(u^{(1)}u^{(2)}u^{(3)}\right)}}&=N_{2},\\-{\sqrt {\frac {c}{b}}}{\frac {\partial \left(x^{(1)}x^{(2)}x^{(4)}\right)}{\partial \left(u^{(1)}u^{(2)}u^{(3)}\right)}}&=N_{3},&{\sqrt {\frac {c}{b}}}{\frac {\partial \left(x^{(1)}x^{(2)}x^{(3)}\right)}{\partial \left(u^{(1)}u^{(2)}u^{(3)}\right)}}&=N_{4},\end{aligned}}$

where $b$ is the discriminant of the arc-element on $M_{3}$, so because of

$dS_{3}={\sqrt {b}}du^{(1)}du^{(2)}du^{(3)},\quad dS_{4}={\sqrt {c}}dx^{(1)}dx^{(2)}dx^{(3)}dx^{(4)}$:

it is given

${\begin{aligned}&\int \int dS_{3}\left[N_{1}B^{(1)}+N_{2}B^{(2)}+N_{3}B^{(3)}+N_{4}B^{(4)}\right]=\\&\quad =\int \int \int dS_{4}{\frac {1}{\sqrt {c}}}\left[{\frac {\partial }{\partial x^{(1)}}}\left({\sqrt {c}}B^{(1)}\right)+{\frac {\partial }{\partial x^{(2)}}}\left({\sqrt {c}}B^{(2)}\right)+{\frac {\partial }{\partial x^{(3)}}}\left({\sqrt {c}}B^{(3)}\right)+{\frac {\partial }{\partial x^{(4)}}}\left({\sqrt {c}}B^{(4)}\right)\right]\end{aligned}}$

where $N$ goes to the exterior. Therefore it is given for the generalized four-dimensional divergence:^[3]

$\mathrm {Div} \left(B^{(1)}B^{(2)}B^{(3)}B^{(4)}\right)={\frac {1}{\sqrt {c}}}\sum _{\alpha }{\frac {\partial }{\partial x^{(\alpha )}}}\left({\sqrt {c}}B^{(\alpha )}\right),$

$\mathrm {div} \left(B_{1}B_{2}B_{3}B_{4}\right)={\frac {1}{\sqrt {c}}}\sum _{\alpha }{\frac {\partial }{\partial x^{(\alpha )}}}\left({\sqrt {c}}\sum _{\beta }c^{(\alpha \beta )}B_{\beta }\right),$,

$\mathrm {div} \left(\mathrm {B} _{1}\mathrm {B} _{2}\mathrm {B} _{3}\mathrm {B} _{4}\right)=\square B={\frac {1}{\sqrt {c}}}\sum _{\alpha }{\frac {\partial }{\partial x^{(\alpha )}}}\left({\sqrt {c}}\sum _{\beta }c^{(\alpha \beta )}{\frac {\partial B}{\partial x^{(\beta )}}}\right).$

$n=4$, $p=2$.

${\begin{aligned}&\int \int du^{(1)}du^{(2)}\left[{\frac {\partial \left(x^{(1)}x^{(2)}\right)}{\partial \left(u^{(1)}u^{(2)}\right)}}A_{12}+{\frac {\partial \left(x^{(1)}x^{(3)}\right)}{\partial \left(u^{(1)}u^{(2)}\right)}}A_{13}+{\frac {\partial \left(x^{(1)}x^{(4)}\right)}{\partial \left(u^{(1)}u^{(2)}\right)}}A_{14}+\right.\\&\quad \left.+{\frac {\partial \left(x^{(2)}x^{(3)}\right)}{\partial \left(u^{(1)}u^{(2)}\right)}}A_{23}+{\frac {\partial \left(x^{(2)}x^{(4)}\right)}{\partial \left(u^{(1)}u^{(2)}\right)}}A_{24}+{\frac {\partial \left(x^{(3)}x^{(4)}\right)}{\partial \left(u^{(1)}u^{(2)}\right)}}A_{34}\right]=\\&=\int \int \int dv^{(1)}dv^{(2)}dv^{(3)}\left[{\frac {\partial \left(x^{(2)}x^{(3)}x^{(4)}\right)}{\partial \left(v^{(1)}v^{(2)}v^{(3)}\right)}}\mathrm {A} _{234}+{\frac {\partial \left(x^{(1)}x^{(3)}x^{(4)}\right)}{\partial \left(v^{(1)}v^{(2)}v^{(3)}\right)}}\mathrm {A} _{134}+\right.\\&\quad \left.+{\frac {\partial \left(x^{(1)}x^{(2)}x^{(4)}\right)}{\partial \left(v^{(1)}v^{(2)}v^{(3)}\right)}}\mathrm {A} _{124}+{\frac {\partial \left(x^{(1)}x^{(2)}x^{(3)}\right)}{\partial \left(v^{(1)}v^{(2)}v^{(3)}\right)}}\mathrm {A} _{123}\right]\end{aligned}}$

Introducing the supplements

${\begin{aligned}&{\sqrt {\frac {1}{c}}}A_{12}=B^{(34)},\quad {\sqrt {\frac {1}{c}}}A_{13}=B^{(42)},\quad {\sqrt {\frac {1}{c}}}A_{14}=B^{(23)},\\&{\sqrt {\frac {1}{c}}}A_{23}=B^{(14)},\quad {\sqrt {\frac {1}{c}}}A_{24}=B^{(31)},\quad {\sqrt {\frac {1}{c}}}A_{34}=B^{(12)},\end{aligned}}$

${\sqrt {\frac {c}{a}}}{\frac {\partial \left(x^{(1)}x^{(2)}\right)}{\partial \left(u^{(1)}u^{(2)}\right)}}=N_{34},$ etc.

$-{\sqrt {\frac {c}{b}}}{\frac {\partial \left(x^{(2)}x^{(3)}x^{(4)}\right)}{\partial \left(v^{(1)}v^{(2)}v^{(3)}\right)}}=N_{1},$ etc.

where $a$ or $b$ are the discriminants of the arc-element of $M_{2}$ or $M_{3}$:

${\begin{aligned}\int \int dS_{2}\left[N_{12}B^{(11)}+N_{13}B^{(13)}+N_{14}B^{(14)}+N_{23}B^{(23)}+N_{24}B^{(24)}+N_{34}B^{(34)}\right]=\\=-\int \int \int dS_{3}\left\{N_{1}{\frac {1}{\sqrt {c}}}\left[^{\ast }+{\frac {\partial }{\partial x^{(2)}}}\left({\sqrt {c}}B^{(12)}\right)+{\frac {\partial }{\partial x^{(3)}}}\left({\sqrt {c}}B^{(13)}\right)+{\frac {\partial }{\partial x^{(4)}}}\left({\sqrt {c}}B^{(14)}\right)\right]\right.\\N_{2}{\frac {1}{\sqrt {c}}}\left[{\frac {\partial }{\partial x^{(1)}}}\left({\sqrt {c}}B^{(21)}\right)+{}^{\ast }+{\frac {\partial }{\partial x^{(3)}}}\left({\sqrt {c}}B^{(23)}\right)+{\frac {\partial }{\partial x^{(4)}}}\left({\sqrt {c}}B^{(24)}\right)\right]\\N_{3}{\frac {1}{\sqrt {c}}}\left[{\frac {\partial }{\partial x^{(1)}}}\left({\sqrt {c}}B^{(31)}\right)+{\frac {\partial }{\partial x^{(2)}}}\left({\sqrt {c}}B^{(32)}\right)+{}^{\ast }+{\frac {\partial }{\partial x^{(4)}}}\left({\sqrt {c}}B^{(34)}\right)\right]\\\left.N_{4}{\frac {1}{\sqrt {c}}}\left[{\frac {\partial }{\partial x^{(1)}}}\left({\sqrt {c}}B^{(41)}\right)+{\frac {\partial }{\partial x^{(2)}}}\left({\sqrt {c}}B^{(42)}\right)+{\frac {\partial }{\partial x^{(3)}}}\left({\sqrt {c}}B^{(43)}\right)+{}^{\ast }\right]\right\}\end{aligned}}$

where the normal plane $N_{\alpha \beta }$ is given by $[N'N]$^[4] and $N'$, which is the normal of $M_{2}$ directed outwards of the area limited on $M_{3}$ as mentioned in § 1. By that, the generalized vector divergence becomes:^[5]

${\mathfrak {Div}}^{(1)}\left(B^{(12)}B^{(13)}B^{(14)}B^{(23)}B^{(24)}B^{(34)}\right)={\frac {1}{\sqrt {c}}}\sum {\frac {\partial }{\partial x^{(\alpha )}}}\left({\sqrt {c}}B^{(1\alpha )}\right)$ etc.

${\mathfrak {Div}}^{(1)}\left(B_{12}B_{13}B_{14}B_{23}B_{24}B_{34}\right)={\frac {1}{\sqrt {c}}}\sum _{\alpha }{\frac {\partial }{\partial x^{(\alpha )}}}\left({\sqrt {c}}\sum _{\beta ,\gamma }c^{(1\beta )}c^{(\alpha \gamma )}B_{\beta \gamma }\right)$ etc.

The system

$B_{\alpha _{3}\alpha _{4}}=\sum _{\beta _{3},\beta _{4}}c_{\alpha _{3}\beta _{3}}c_{\alpha _{4}\beta _{4}}B^{\left(\beta _{3}\beta _{4}\right)}={\frac {1}{\sqrt {c}}}\sum c_{\alpha _{3}\beta _{3}}c_{\alpha _{4}\beta _{4}}A_{\beta _{1}\beta _{2}}$

shall be called the system dual to $A_{\alpha \beta }$ and be denoted as $A_{\alpha _{3}\alpha _{4}}^{\ast }$. So it follows

${\mathfrak {Div}}^{(1)}\left(A_{12}^{\ast }A_{13}^{\ast }A_{14}^{\ast }A_{23}^{\ast }A_{24}^{\ast }A_{34}^{\ast }\right)={\frac {1}{\sqrt {c}}}\mathrm {A} _{234},$

${\mathfrak {Div}}^{(2)}\left(A_{12}^{\ast }A_{13}^{\ast }A_{14}^{\ast }A_{23}^{\ast }A_{24}^{\ast }A_{34}^{\ast }\right)=-{\frac {1}{\sqrt {c}}}\mathrm {A} _{134}$ etc.

and

${\mathfrak {Div}}^{(1)}\left(A_{12}^{\text{ }}A_{13}A_{14}A_{23}A_{24}A_{34}\right)={\frac {1}{\sqrt {c}}}\mathrm {A} _{234}^{\ast },$

${\mathfrak {Div}}^{(2)}\left(A_{12}^{\text{ }}A_{13}A_{14}A_{23}A_{24}A_{34}\right)=-{\frac {1}{\sqrt {c}}}\mathrm {A} _{134}^{\ast }$ etc.

where $\mathrm {A} _{234}^{\ast }$ etc. are to be formed from $A_{\alpha \beta }^{\ast }$, as $\mathrm {A} _{234}$ etc. from $A_{\alpha \beta }$.

In the case

$\mathrm {Div} \left(A_{\alpha \beta }^{\ast }\right)\equiv 0$

it therefore follows

$A_{\alpha \beta }=\mathrm {A} _{\alpha \beta }={\frac {\partial A_{\alpha }}{\partial x^{(\beta )}}}-{\frac {\partial A_{\beta }}{\partial x^{(\alpha )}}}$.

$n=4$, $p=1$.

${\begin{aligned}&\int du\left({\frac {\partial x^{(1)}}{\partial u}}A_{1}+{\frac {\partial x^{(2)}}{\partial u}}A_{2}+{\frac {\partial x^{(3)}}{\partial u}}A_{3}+{\frac {\partial x^{(4)}}{\partial u}}A_{4}\right)=\\&\quad =\int \int dv^{(1)}dv^{(2)}\left({\frac {\partial \left(x^{(1)}x^{(2)}\right)}{\partial \left(v^{(1)}v^{(2)}\right)}}A_{12}+{\frac {\partial \left(x^{(1)}x^{(3)}\right)}{\partial \left(v^{(1)}v^{(2)}\right)}}A_{13}+\right.\\&\qquad \left.{\frac {\partial \left(x^{(1)}x^{(4)}\right)}{\partial \left(v^{(1)}v^{(2)}\right)}}A_{14}+{\frac {\partial \left(x^{(2)}x^{(3)}\right)}{\partial \left(v^{(1)}v^{(2)}\right)}}A_{23}+{\frac {\partial \left(x^{(2)}x^{(4)}\right)}{\partial \left(v^{(1)}v^{(2)}\right)}}A_{24}+{\frac {\partial \left(x^{(3)}x^{(4)}\right)}{\partial \left(v^{(1)}v^{(2)}\right)}}A_{34}\right)\end{aligned}}$

or

$\int ds\sum {\frac {dx^{(\alpha )}}{ds}}A_{\alpha }=\int \int dS_{2}\sum N_{\alpha _{3}\alpha _{4}}{\frac {1}{\sqrt {c}}}A_{\alpha _{1}\alpha _{2}}$

with the corresponding orientation (by § 1). Therefore, it is given for the generalized rotation^[6] with the common signs:

$\mathrm {Rot} ^{(12)}\left(A_{1}A_{2}A_{3}A_{4}\right)={\frac {1}{\sqrt {c}}}\left({\frac {\partial A_{4}}{\partial x^{(3)}}}-{\frac {\partial A_{3}}{\partial x^{(4)}}}\right)$ etc.

$\mathrm {Rot} _{12}\left(A_{1}A_{2}A_{3}A_{4}\right)={\frac {\partial A_{2}}{\partial x^{(1)}}}-{\frac {\partial A_{1}}{\partial x^{(2)}}}$ etc.

The integral forms $n=4$ in the notation of the absolute differential calculus.

As appendix, the methods of the already mentioned absolute differential calculus shall be demonstrated, because it will be applied later; while it is less suited for the transformation of the actual integral form, it can hardly be avoided in connection with other vectorial formations which are more combined. In the mentioned work,^[7] Christoffel shows, based on the differential equations for the second derivative ${\frac {\partial ^{2}x^{(\alpha )}}{\partial {\bar {x}}^{(\lambda )}\partial {\bar {x}}^{(\mu )}}}$, that from a covariant system $A_{\alpha _{1}\alpha _{2}\dots \alpha _{p}}$ of $p$-the order, a system of $p+1$-th order emerges as follows:

$A_{\alpha _{1}\alpha _{2}\dots \alpha _{p}/\alpha _{p+1}}={\frac {\partial }{\partial x^{\left(\alpha _{p+1}\right)}}}A_{\alpha _{1}\alpha _{2}\dots \alpha _{p}}-\sum _{h=1}^{p}\sum _{\beta =1}^{n}\left\{{\begin{matrix}\alpha _{p+1}\alpha _{h}\\\beta \end{matrix}}\right\}A_{\alpha _{1}\dots \alpha _{h-1}\beta \alpha _{h+1}\dots \alpha _{p}},$

where the Christoffel symbols of second order with triple-indices arise, which are defined as follows:

$\left\{{\begin{matrix}\alpha \beta \\\gamma \end{matrix}}\right\}=\sum _{\delta =1}^{n}c^{(\gamma \delta )}{\begin{bmatrix}\alpha \beta \\\delta \end{bmatrix}}=\sum _{\delta =1}^{n}c^{(\gamma \delta )}\cdot {\frac {1}{2}}\left({\frac {\partial c_{\alpha \delta }}{\partial x^{(\beta )}}}+{\frac {\partial c_{\beta \delta }}{\partial x^{(\alpha )}}}-{\frac {\partial c_{\alpha \beta }}{\partial x^{(\delta )}}}\right).$

Ricci and Levi-Cività denote this as the covariant differential quotient of $A_{\alpha _{1}\alpha _{2}\dots \alpha _{p}}$ with respect to $x^{\left(\alpha _{p+1}\right)}$. The prime separates the indices added by differentiation from the others. For the contravariant differential quotient it is given:

$A^{\left(\alpha _{1}\alpha _{2}\dots \alpha _{p}/\alpha _{p+1}\right)}=\sum _{\beta =1}^{n}c^{\left(\alpha _{p+1}\beta \right)}\left[{\frac {\partial }{\partial x^{(\beta )}}}A^{\left(\alpha _{1}\alpha _{2}\dots \alpha _{p}\right)}+\sum _{h=1}^{p}\sum _{\gamma =1}^{n}\left\{{\begin{matrix}\beta \gamma \\\alpha _{h}\end{matrix}}\right\}A^{\left(\alpha _{1}\alpha _{2}\dots \alpha _{h-1}\gamma \alpha _{h+1}\dots \alpha _{p}\right)}\right].$

Then we have, as it can be easily shown:

${\begin{aligned}\mathrm {A} _{1234}&=A_{123/4}-A_{124/3}+A_{134/2}-A_{234/1}\\&={\sqrt {c}}\mathrm {Div} \left(B^{(1)}B^{(2)}B^{(3)}B^{(4)}\right)={\sqrt {c}}{\underset {\alpha ,\beta }{\sum }}c_{\alpha \beta }B^{(\alpha /\beta )}={\underset {\alpha }{\sum }}{\frac {\partial }{\partial x^{(\alpha )}}}\left({\sqrt {c}}B^{(\alpha )}\right),\\&=\mathrm {div} \left(B_{1}B_{2}B_{3}B_{4}\right)={\sqrt {c}}{\underset {\alpha ,\beta }{\sum }}c^{(\alpha \beta )}B_{\alpha /\beta }={\underset {\alpha }{\sum }}{\frac {\partial }{\partial x^{(\alpha )}}}\left({\sqrt {c}}{\underset {\beta }{\sum }}c^{(\alpha \beta )}B_{\beta }\right)\end{aligned}}$

with the connection

$-{\frac {1}{\sqrt {c}}}A_{234}=B^{(1)}={\underset {\beta }{\sum }}c^{(1\beta )}B_{\beta }$,

${\frac {1}{\sqrt {c}}}A_{134}=B^{(2)}={\underset {\beta }{\sum }}c^{(2\beta )}B_{\beta }$ etc.

${\begin{aligned}\mathrm {A} _{234}&=A_{23/4}-A_{24/3}+A_{34/2}\\&={\sqrt {c}}{\mathfrak {Div}}^{(1)}\left(B^{(12)}B^{(13)}B^{(14)}B^{(23)}B^{(24)}B^{(34)}\right)={\sqrt {c}}{\underset {\alpha ,\beta }{\sum }}c_{\alpha \beta }B^{(1\alpha /\beta )}={\underset {\alpha }{\sum }}{\frac {\partial }{\partial x^{(\alpha )}}}\left({\sqrt {c}}B^{(1\alpha )}\right),\\&={\sqrt {c}}{\mathfrak {Div}}^{(1)}\left(B_{12}B_{13}B_{14}B_{23}B_{24}B_{34}\right)={\sqrt {c}}{\underset {\alpha ,\beta ,\gamma }{\sum }}c^{(1\alpha )}c^{(\beta \gamma )}B_{\alpha \beta /\gamma }={\underset {\alpha }{\sum }}{\frac {\partial }{\partial x^{(\alpha )}}}\left({\sqrt {c}}{\underset {\beta ,\gamma }{\sum }}c^{(1\beta )}B_{\beta \gamma }\right)\end{aligned}}$

with the connection

${\frac {1}{\sqrt {c}}}A_{12}=B^{(34)}={\underset {\alpha ,\beta }{\sum }}c^{(3\alpha )}c^{(4\beta )}B_{\alpha \beta }$ etc.,

where we could write, following the things stated above, also $A_{\alpha \beta }^{\ast }$ instead of $B_{\alpha \beta }$.

1685

Thus

${\frac {1}{\sqrt {c}}}\mathrm {A} _{234}={\mathfrak {Div}}^{(1)}\left(A_{\alpha \beta }^{\ast }\right)={\underset {\alpha }{\sum }}c^{(1\alpha )}{\underset {\beta ,\gamma }{\sum }}c^{(\beta \gamma )}A_{\alpha \beta /\gamma }^{\ast }$ etc.

${\frac {1}{\sqrt {c}}}\mathrm {A} _{234}^{\ast }={\mathfrak {Div}}^{(1)}\left(A_{\alpha \beta }\right)={\underset {\alpha }{\sum }}c^{(1\alpha )}{\underset {\beta ,\gamma }{\sum }}c^{(\beta \gamma )}A_{\alpha \beta /\gamma }$

For ${\mathfrak {Div}}\left(A_{\alpha \beta }^{\ast }\right)\equiv 0$ we have, as already mentioned above,

$A_{\alpha \beta }=\mathrm {A} _{\alpha \beta }=A_{\alpha /\beta }-A_{\beta /\alpha }$

and

${\begin{aligned}{\mathfrak {Div}}\left(A_{\alpha \beta }\right)&={\underset {\alpha }{\sum }}c^{(1\alpha )}{\underset {\beta ,\gamma }{\sum }}c^{(\beta \gamma )}A_{\alpha /\beta \gamma }-{\underset {\alpha }{\sum }}c^{(1\alpha )}{\underset {\beta ,\gamma }{\sum }}c^{(\beta \gamma )}A_{\beta /\alpha \gamma }=\\&={\underset {\alpha }{\sum }}c^{(1\alpha )}\left[{\underset {\beta ,\gamma }{\sum }}c^{(\beta \gamma )}A_{\alpha /\beta \gamma }-{\frac {\partial }{\partial x^{(\alpha )}}}\left({\underset {\beta ,\gamma }{\sum }}c^{(\beta \gamma )}A_{\beta /\gamma }\right)\right].\end{aligned}}$

since in Euclidean space (vanishing of the Riemann symbols) the permutation of the differentiation order $A_{\beta /\alpha \gamma }=A_{\beta /\gamma \alpha }$ is allowed, and $\sum c^{(1\alpha )}\sum c^{(\beta \gamma )}A_{\beta /\gamma \alpha }$ represents the contravariant differential quotient of ${\underset {\beta ,\gamma }{\sum }}c^{(\beta \gamma )}A_{\beta /\gamma }$ with respect to $x^{(1)}$.^[8]

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1. The factor ${\sqrt {\frac {1}{b}}}$ makes $N$ invariant in $u$.
2. Herein one recognizes Beltrami's second differential operator. Wright, l.c., p. 56, for $n=2$.
3. Sommerfeld, Ann. d. Physik, 33, p. 650 (1910).
4. Vector product of two four-vectors $[AB]^{(\alpha \beta )}=A^{(\alpha )}B^{(\beta )}-A^{(\beta )}B^{(\alpha )}$. Sommerfeld, Ann. d. Phys., 32, p. 765 (1910).
5. Sommerfeld, l. c., 33, p. 651; as with Stokes' theorem, the minus sign is not included in the definition.
6. Sommerfeld, l. c., 33, p. 653 und 654.
7. See Wright, l. c., p. 13 und 22.
8. Wright, l. c., p. 23.