# Page:AbrahamMinkowski1.djvu/5

one obtains

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

Due to the identity which is easily to be verified

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

the relation is obtained

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

§ 3. The energy equation and the momentum equations.

We understand under ${\displaystyle xyzt}$ 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) ${\displaystyle {\begin{cases}{\mathfrak {K}}_{x}={\frac {\partial X_{x}}{\partial x}}+{\frac {\partial X_{y}}{\partial y}}+{\frac {\partial X_{z}}{\partial z}}-{\frac {\partial {\mathfrak {g}}_{x}}{\partial t}},\\\\{\mathfrak {K}}_{y}={\frac {\partial Y_{x}}{\partial x}}+{\frac {\partial Y_{y}}{\partial y}}+{\frac {\partial Y_{z}}{\partial z}}-{\frac {\partial {\mathfrak {g}}_{y}}{\partial t}},\\\\{\mathfrak {K}}_{z}={\frac {\partial Z_{x}}{\partial x}}+{\frac {\partial Z_{y}}{\partial y}}+{\frac {\partial Z_{z}}{\partial z}}-{\frac {\partial {\mathfrak {g}}_{z}}{\partial t}}.\end{cases}}}$

The vector ${\displaystyle {\mathfrak {g}}}$ 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 ${\displaystyle X_{x},\ Y_{y},\ Z_{z}}$, and the pairwise shear-stresses which are mutually equal:

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

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

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

Here, ${\displaystyle Q}$ means the Joule-head, ${\displaystyle \psi }$ the electromagnetic energy density, ${\displaystyle {\mathfrak {S}}}$ 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),