Page:AbrahamMinkowski1.djvu/5

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one obtains

\left[\frac{\partial'\mathfrak{A}}{\partial t}\mathfrak{B}\right]+\left[\mathfrak{A}\frac{\partial'\mathfrak{B}}{\partial t}\right]=\frac{\delta}{\delta t}[\mathfrak{AB}]+[\mathfrak{AB}]\mathrm{div}\mathfrak{w}-\left[\mathfrak{A},\ (\mathfrak{B}\nabla)\mathfrak{w}\right]+\left[\mathfrak{B},\ (\mathfrak{A}\nabla)\mathfrak{w}\right]

Due to the identity which is easily to be verified

\left[\mathfrak{A},\ (\mathfrak{B}\nabla)\mathfrak{w}\right]-\left[\mathfrak{B},\ (\mathfrak{A}\nabla)\mathfrak{w}\right]=[\mathfrak{AB}]\mathrm{div}\mathfrak{w}-([\mathfrak{AB}]\nabla)\mathfrak{w}-\left[[\mathfrak{AB}]\mathrm{curl}\mathfrak{w}\right]

the relation is obtained

(5) \left[\frac{\partial'\mathfrak{A}}{\partial t}\mathfrak{B}\right]+\left[\mathfrak{A}\frac{\partial'\mathfrak{B}}{\partial t}\right]=\frac{\delta}{\delta t}[\mathfrak{AB}]+([\mathfrak{AB}]\nabla)\mathfrak{w}-\left[[\mathfrak{AB}]\mathrm{curl}\mathfrak{w}\right]


§ 3. The energy equation and the momentum equations.


We understand under xyzt coordinates and the time, measured in a reference system in which the observer has a fixed location. The ponderomotive force measured by him, which is acting (due to the electromagnetic process) on the unit volume of moving matter, shall have the components:

(6) \begin{cases}
\mathfrak{K}_{x}=\frac{\partial X_{x}}{\partial x}+\frac{\partial X_{y}}{\partial y}+\frac{\partial X_{z}}{\partial z}-\frac{\partial\mathfrak{g}_{x}}{\partial t},\\
\\\mathfrak{K}_{y}=\frac{\partial Y_{x}}{\partial x}+\frac{\partial Y_{y}}{\partial y}+\frac{\partial Y_{z}}{\partial z}-\frac{\partial\mathfrak{g}_{y}}{\partial t},\\
\\\mathfrak{K}_{z}=\frac{\partial Z_{x}}{\partial x}+\frac{\partial Z_{y}}{\partial y}+\frac{\partial Z_{z}}{\partial z}-\frac{\partial\mathfrak{g}_{z}}{\partial t}.\end{cases}

The vector \mathfrak{g} which arises here, is denoted by us as "electromagnetic momentum density" or shortly as "momentum density". The system of "fictitious electromagnetic stresses" consists of six quantities, namely the normal stresses X_{x},\ Y_{y},\ Z_{z}, and the pairwise shear-stresses which are mutually equal:

(6a) X_{y}=Y_{x},\ Y_{z}=Z_{y},\ Z_{x}=X_{z}

To the "momentum equations" (6), the energy equation is added:

(7) \mathfrak{wK}+Q=-\mathrm{div}\mathfrak{S}-\frac{\partial\psi}{\partial t}

Here, Q means the Joule-head, \psi the electromagnetic energy density, \mathfrak{S} the energy current.

While the momentum equations determine the momentum exerted by the electromagnetic field, the energy equation determines which energy-quantity per unit space and time is converted into a non-electromagnetic form (work and heat).

If one introduces into (6) and (7) the temporal derivative defined by (3) and (3a),