Page:AbrahamMinkowski1.djvu/4

§ 2. Mathematical auxiliary formulas.

The time differentiation for fixed space points, is represented by ${\displaystyle {\tfrac {\partial }{\partial t}}}$. The temporal change of a surface integral, extended over a surface whose points are moving with velocity ${\displaystyle {\mathfrak {w}}}$, namely

${\displaystyle {\frac {d}{dt}}\int df\ {\mathfrak {A}}_{n}=\int df\left\{{\frac {\partial '{\mathfrak {A}}}{\partial t}}\right\}_{n}}$

defines another kind of time differentiation of a vector

 (1) ${\displaystyle {\frac {\partial '{\mathfrak {A}}}{\partial t}}={\frac {\partial {\mathfrak {A}}}{\partial t}}+{\mathfrak {w}}\ \mathrm {div} {\mathfrak {A}}+\mathrm {curl} [{\mathfrak {Aw}}]}$

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

 (2) ${\displaystyle {\dot {\mathfrak {A}}}={\frac {\partial {\mathfrak {A}}}{\partial t}}+({\mathfrak {w}}\nabla ){\mathfrak {A}}}$

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

 (2a) ${\displaystyle {\begin{array}{c}{\frac {d}{dt}}\int dv\ {\mathfrak {A}}=\int dv{\frac {\delta {\mathfrak {A}}}{\delta t}}\\\\{\frac {\delta {\mathfrak {A}}}{\delta t}}={\dot {\mathfrak {A}}}+{\mathfrak {A}}\ \mathrm {div} {\mathfrak {w}}\end{array}}}$

From (2) and (2a) it follows

 (3) ${\displaystyle {\frac {\delta {\mathfrak {A}}}{\delta t}}={\frac {\partial {\mathfrak {A}}}{\partial t}}+({\mathfrak {w}}\nabla ){\mathfrak {A}}+{\mathfrak {A}}\ \mathrm {div} {\mathfrak {w}}}$

Accordingly it is given for the scalars:

 (3a) ${\displaystyle {\frac {\delta \psi }{\delta t}}={\frac {\partial \psi }{\partial t}}+\mathrm {div} \psi {\mathfrak {w}}}$

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

${\displaystyle \mathrm {curl} [{\mathfrak {Aw}}]=({\mathfrak {w}}\nabla ){\mathfrak {A}}-({\mathfrak {A}}\nabla ){\mathfrak {w}}+{\mathfrak {A}}\ \mathrm {div} {\mathfrak {w}}-{\mathfrak {w}}\ \mathrm {div} {\mathfrak {A}}}$,

the relation

 (4) ${\displaystyle {\frac {\partial '{\mathfrak {A}}}{\partial t}}={\frac {\delta {\mathfrak {A}}}{\delta t}}-({\mathfrak {A}}\nabla ){\mathfrak {w}}}$.

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

${\displaystyle [{\mathfrak {{\dot {A}}B}}]+[{\mathfrak {A{\dot {B}}}}]={\frac {\delta }{\delta t}}[{\mathfrak {AB}}]-[{\mathfrak {AB}}]\mathrm {div} {\mathfrak {w}}}$

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

${\displaystyle {\begin{array}{l}{\frac {\partial '{\mathfrak {A}}}{\partial t}}={\mathfrak {\dot {A}}}+{\mathfrak {A}}\ \mathrm {div} {\mathfrak {w}}-({\mathfrak {A}}\nabla ){\mathfrak {w}},\\\\{\frac {\partial '{\mathfrak {B}}}{\partial t}}={\mathfrak {\dot {B}}}+{\mathfrak {B}}\ \mathrm {div} {\mathfrak {w}}-({\mathfrak {B}}\nabla ){\mathfrak {w}},\end{array}}}$