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and its components occasionally by

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

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

$\int {\mathfrak {A}}_{s}\ d\ s=\int {\mathfrak {B}}_{n}\ d\ \sigma .$

(1)

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