Page:AbrahamMinkowski1.djvu/4

From Wikisource
Jump to: navigation, search
This page has been proofread, but needs to be validated.


§ 2. Mathematical auxiliary formulas.

The time differentiation for fixed space points, is represented by . The temporal change of a surface integral, extended over a surface whose points are moving with velocity , namely

defines another kind of time differentiation of a vector

(1)

Furthermore, the derivative (with respect to time) which is related to moving points, is

(2)

This is connected with the temporal change of the volume integral of a vector, by the relations

(2a)

From (2) and (2a) it follows

(3)

Accordingly it is given for the scalars:

(3a)

From (1) and (3) it finally follows, with respect to the general rule

,

the relation

(4) .

Since the time differentiation introduced in (2) follows the ordinary calculation rules, we have with respect to (2a)

From this equation, together with the ones following from (4) and (2a)