Page:AbrahamMinkowski1.djvu/5

 one obtains

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

Due to the identity which is easily to be verified

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

the relation is obtained

§ 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:

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:

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

Here, $$Q$$ means the -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),