Page:Elektrische und Optische Erscheinungen (Lorentz) 011.jpg

 and its components occasionally by

$$[Rot\ \mathfrak{A}]_{l}.$$

If s is the border line of surface $$\sigma$$, then we have

Furthermore we will easily find

$$Div\ Rot\ \mathfrak{A}=0,$$

and for the components of vector $$Rot\ Rot\ \mathfrak{A}$$

$$\frac{\partial}{\partial x}Div\ \mathfrak{A}-\triangle\mathfrak{A}_{x},$$ etc.

Here, the letter $$\Delta$$ has, like in all our formulas, the meaning

$$\triangle=\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}+\frac{\partial^{2}}{\partial z^{2}}.$$

i. If m and n are scalar magnitudes, then we attribute to the expressions

$$-\mathfrak{A},\ m\ \mathfrak{A},\ m\ \mathfrak{A}\pm n\ \mathfrak{B}$$

the known meanings.

j. By $$[\mathfrak{A}.\mathfrak{B}]$$ we understand the so-called "vector product", namely a vector whose magnitude is given by the area of the parallelogram drawn over $$\mathfrak{A}$$ and $$\mathfrak{B}$$, and whose direction is perpendicular to the plane that is laid trough $$\mathfrak{A}$$ and $$\mathfrak{B}$$, and namely in a way, by that the direction of $$\mathfrak{A}$$ is transformed into the direction of $$\mathfrak{B}$$.

As regards the components it can be written $$[\mathfrak{A}.\mathfrak{B}]_{l}$$; the components into the axis-directions are:

$$\mathfrak{A}_{y}\mathfrak{B}_{z}-\mathfrak{A}_{z}\mathfrak{B}_{y},\ \mathfrak{A}_{z}\mathfrak{B}_{x}-\mathfrak{A}_{x}\mathfrak{B}_{z},\ \mathfrak{A}_{x}\mathfrak{B}_{y}-\mathfrak{A}_{y}\mathfrak{B}_{x}{,}$$

and

$$[\mathfrak{B}.\mathfrak{A}]= - [\mathfrak{A}.\mathfrak{B}].$$

k. The advantage of the previously introduced expressions mainly consists in the fact, that now three equations like

$$\mathfrak{A}_{x}=\mathfrak{B}_{x},\ \mathfrak{A}_{y}=\mathfrak{B}_{y},\ \mathfrak{A}_{z}=\mathfrak{B}_{z}$$

can be summarized in one formula

$$\mathfrak{A}=\mathfrak{B}$$

However, in the course of the investigation we will