Page:AbrahamMinkowski2.djvu/3

 three-dimensional space $$x, y, z$$, the three first components of vector $$V_{I}^{4}$$ constitute a three-dimensional vector $$V^{3}$$, $$\mathfrak{r}$$, the fourth ($$u$$) a three-dimensional scalar ($$S^{3}$$).

A four-dimensional vector of the second kind $$\left(V_{II}^{4}\right)$$ denotes a system of six magnitudes, which are transformed like the following expressions, formed by the components $$x_{1},y_{1},z_{1},u_{1}$$ and $$x_{2},y_{2},z_{2},u_{2}$$ of two $$V_{I}^{4}$$:

{{MathForm1|(1)|$$\left\{ \begin{array}{ccccc} \mathfrak{a}_{x}=\left|\begin{array}{cc} y_{1} & z_{1}\\ y_{2} & z_{2}\end{array}\right|, & & \mathfrak{a}_{y}=\left|\begin{array}{cc} z_{1} & x_{1}\\ z_{2} & x_{2}\end{array}\right|, & & \mathfrak{a}_{z}=\left|\begin{array}{cc} x_{1} & y_{1}\\ x_{2} & y_{2}\end{array}\right|;\\ \\\mathfrak{b}_{x}=\left|\begin{array}{cc} x_{1} & u_{1}\\ x_{2} & u_{2}\end{array}\right|, & & \mathfrak{b}_{y}=\left|\begin{array}{cc} y_{1} & u_{1}\\ y_{2} & u_{2}\end{array}\right|, & & \mathfrak{b}_{z}=\left|\begin{array}{cc} z_{1} & u_{1}\\ z_{2} & u_{2}\end{array}\right|.\end{array}\right.$$}}

Obviously, when projecting into three-dimensional space, $$V_{II}^{4}$$ is composed of two $$V^{3}$$, which, in the symbolism of ordinary vector analysis, we can write:

From two $$V_{I}^{4}$$:

$\mathfrak{r},\ u$ and $\mathfrak{r}_{1},\ u_{1}$

we can compose a four-dimensional scalar ($$S^4$$) as follows:

Conversely, from any four-dimensional scalar $$\varphi(x,y,z,u)$$, we obtain (derived with respect to their coordinates) a $$V_{I}^{4}$$:

So the operators

$\frac{\partial}{\partial x},\ \frac{\partial}{\partial y},\ \frac{\partial}{\partial z},\ \frac{\partial}{\partial u}$

transform as the components of a $$V_{I}^{4}$$, and these operators were denoted by as the components of the operator "lor".

We can compose a $$S^4$$ from four $$V_{I}^{4}$$, which determines the space of the parallelepiped of the four vectors:

If we apply scheme (3) to $$S^{4}$$, we obtain a $$V_{I}^{4}$$, which is composed of three other $$V_{I}^{4},\ \mathfrak{r}_{1}u_{1},\ \mathfrak{r}_{2}u_{2},\ \mathfrak{r}_{3}u_{3}$$, whose components are:

{{MathForm1|(5)|$$\left\{ \begin{array}{ccc} X=\frac{\partial\varphi}{\partial x}=\left|\begin{array}{ccccc} y_{1} & & z_{1} &  & u_{1}\\ y_{2} & & z_{2} &  & u_{2}\\ y_{3} & & z_{3} &  & u_{3}\end{array}\right|, &  & Y=\frac{\partial\varphi}{\partial y}=\left|\begin{array}{ccccc} z_{1} & & x_{1} &  & u_{1}\\ z_{2} & & x_{2} &  & u_{2}\\ z_{3} & & x_{3} &  & u_{3}\end{array}\right|,\\ \\Z=\frac{\partial\varphi}{\partial z}=\left|\begin{array}{ccccc} x_{1} & & y_{1} &  & u_{1}\\ x_{2} & & y_{2} &  & u_{2}\\ x_{3} & & y_{3} &  & u_{3}\end{array}\right|, &  & U=\frac{\partial\varphi}{\partial u}=-\left|\begin{array}{ccccc} x_{1} & & y_{1} &  & z_{1}\\ x_{2} & & y_{2} &  & z_{2}\\ x_{3} & & y_{3} &  & z_{3}\end{array}\right|;\end{array}\right.$$}}