Page:AbrahamMinkowski1.djvu/4

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§ 2. Mathematical auxiliary formulas.

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

\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) \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) \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) \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) \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) \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

\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) \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)

[\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)

\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}