Page:Zur Elektrodynamik bewegter Systeme II.djvu/3

 where $$\tfrac{d}{dt}\left(\tfrac{\partial}{\partial t}\right)$$ denotes the differentiation with respect to a fixed material point (space point). Furthermore P is rotation, $$\Gamma$$ is divergence, $$\nabla$$ is gradient,

$(A\cdot\nabla)=A_{x}\cdot\frac{\partial}{\partial x}+A_{y}\cdot\frac{\partial}{\partial y}+A_{z}\cdot\frac{\partial}{\partial z}$.

§ 2. Transformation to a moving coordinate system and local time.

We decompose the velocities $$u$$ into a common translational velocity $$p$$ (which is constant with respect to time) of the whole system, and the "relative" velocity $$v$$:

and we denote a differentiation with respect to time, in relation to a relatively stationary point, by $$\tfrac{\delta}{\delta t}$$:

Then it is given

Simultaneously, instead of the "general time" $$t$$ we introduce the "local time" $$t'$$. It is defined at a point whose radius vector is $$r$$, by:

Differentiations with respect to relative coordinates, in which local time is assumed as the fourth independent variable, shall be denoted by an upper index prime. Then it is

Eventually, we decompose $$\mathfrak{E}$$ and $$\mathfrak{M}$$: