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

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

By (1):

Introducing the supplements

where

is the discriminant of the arc on $M_2$, thus because of

it follows

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

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

it follows

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

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

By (1):

or

Template:Pagenum 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 $\frac{dx}{ds}$, $N^{\prime }$, $N$ are following each other like the coordinate axes, where $N^{\prime }$ 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:

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

Introducing the supplements

where $b$ is the discriminant of the arc-element on $M_3$, so because of

it is given

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

$\mathrm{d}\mathrm{i}\mathrm{v}\left(B_1{B}_2{B}_3{B}_4\right)=\frac{1}{\sqrt{c}}\sum _{\alpha }\frac{\partial }{\partial x^{(\alpha )}}\left(\sqrt{c}\sum _{\beta }{c}^{(\alpha \beta )}{B}_{\beta }\right),$,

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

Introducing the supplements

where $a$ or $b$ are the discriminants of the arc-element of $M_2$ or $M_3$:

Template:Pagenum where the normal plane $N_{\alpha \beta }$ is given by $[N^{\prime }N]$ and $N^{\prime }$, 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:

The system

shall be called the system dual to $A_{\alpha \beta }$ and be denoted as $A_{{\alpha }_3{\alpha }_4}^{\ast }$. So it follows

and

Template:Pagenum 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

it therefore follows

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

or

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

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, Template:Pagenum it can hardly be avoided in connection with other vectorial formations which are more combined. In the mentioned work, Template:Sc shows, based on the differential equations for the second derivative $\frac{{\partial }^2{x}^{(\alpha )}}{\partial {\overline{x}}^{(\lambda )}\partial {\overline{x}}^{(\mu )}}$, 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:

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

Template:Sc and Template:Sc 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:

Template:Pagenum Then we have, as it can be easily shown:

with the connection

with the connection

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

Template:Center Thus

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

and

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 $\sum _{\beta ,\gamma }{c}^{(\beta \gamma )}{A}_{\beta /\gamma }$ with respect to $x^{(1)}$.

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