Page:Grundgleichungen (Minkowski).djvu/40



§ 13. The Product of the Field-vectors fF.
Finally let us enquire about the laws which lead to the determination of the vector w as a function of x, y, z, t. In these investigations, the expressions which are obtained by the multiplication of two alternating matrices

$$f=\left|\begin{array}{cccc} 0, & f_{12}, & f_{13}, & f_{14}\\ f_{21}, & 0, & f_{23}, & f_{24}\\ f_{31}, & f_{32}, & 0, & f_{34}\\ f_{41}, & f_{42}, & f_{43}, & 0\end{array}\right|,\ F=\left|\begin{array}{cccc} 0, & F_{12}, & F_{13}, & F_{14}\\ F_{21}, & 0, & F_{23}, & F_{24}\\ F_{31}, & F_{32}, & 0, & F_{34}\\ F_{41}, & F_{42}, & F_{43}, & 0\end{array}\right|$$

are of much importance. Let us write.

Then (71)

Let L now denote the symmetrical combination of the indices 1, 2, 3, 4, given by

Then we shall have

In order to express in a real form, we write

Now