Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/269

We have also the following scalar functions:—

The electric potential Ψ.

The magnetic potential (where it exists) Ω.

The electric density e.

The density of magnetic 'matter' m.

Besides these we have the following quantities, indicating physical properties of the medium at each point:—

C, the conductivity for electric currents.

K, the dielectric inductive capacity.

μ, the magnetic inductive capacity.

These quantities are, in isotropic media, mere scalar functions of ρ, but in general they are linear and vector operators on the vector functions to which they are applied. K and μ are certainly always self-conjugate, and C is probably so also.

619.] The equations (A) of magnetic induction, of which the first is,

$a={\frac {dH}{dy}}-{\frac {dG}{dz}},$

may now be written

${\mathfrak {B}}=V\nabla {\mathfrak {A}},$

where $\nabla$is the operator

$i{\frac {d}{dx}}+j{\frac {d}{dy}}+k{\frac {d}{dz}},$

and V indicates that the vector part of the result of this operation is to be taken.

Since ${\mathfrak {A}}$ is subject to the condition $S\nabla {\mathfrak {A}}=0,\nabla {\mathfrak {A}}$ is a pure vector, and the symbol V is unnecessary.

The equations (B) of electromotive force, of which the first is

$P=c{\dot {y}}-b{\dot {z}}-{\frac {dF}{dt}}-{\frac {d\Psi }{dx}},$

become

${\mathfrak {E}}=V{\mathfrak {GB}}-{\dot {\mathfrak {A}}}-\nabla \Psi .$

The equations (C) of mechanical force, of which the first is

$X=cv-bw-e{\frac {d\Psi }{dx}}-m{\frac {d\Omega }{dx}},$

become

${\mathfrak {F}}=V{\mathfrak {CB}}-e\nabla \Psi -m\nabla \Omega .$