stands for what is called the vector product of the two vectors,
namely the vector
(
Y
α
′
Z
α
−
Z
α
′
Y
α
,
Z
α
′
X
α
−
X
α
′
Z
α
,
X
α
′
Y
α
−
Y
α
′
X
α
)
{\displaystyle ({{Y_{\alpha }}^{\prime }}{Z_{\alpha }}-{{Z_{\alpha }}^{\prime }}{Y_{\alpha }},{{Z_{\alpha }}^{\prime }}{X_{\alpha }}-{{X_{\alpha }}^{\prime }}{Z_{\alpha }},{{X_{\alpha }}^{\prime }}{Y_{\alpha }}-{{Y_{\alpha }}^{\prime }}{X_{\alpha }})}
It is evident that
c
u
r
l
α
(
X
α
,
Y
α
,
Z
α
)
{\displaystyle {curl_{\alpha }}(X_{\alpha },Y_{\alpha },Z_{\alpha })}
can be expressed in the
symbolic form
[
(
∂
∂
x
α
,
∂
∂
y
α
,
∂
∂
z
α
)
.
(
X
α
,
Y
α
,
Z
α
)
]
{\displaystyle {\bigg [}{\bigg (}{\frac {\partial }{\partial x_{\alpha }}},{\frac {\partial }{\partial y_{\alpha }}},{\frac {\partial }{\partial z_{\alpha }}}{\bigg )}.(X_{\alpha },Y_{\alpha },Z_{\alpha }){\bigg ]}}
The vector equation
(
X
α
,
Y
α
,
Z
α
)
=
(
X
α
′
,
Y
α
′
,
Z
α
′
)
{\displaystyle (X_{\alpha },Y_{\alpha },Z_{\alpha })=({X_{\alpha }}^{\prime },{Y_{\alpha }}^{\prime },{Z_{\alpha }}^{\prime })}
is an abbreviation of the three equations
X
α
=
X
α
′
,
Y
α
=
Y
α
′
,
Z
α
=
Z
α
′
{\displaystyle X_{\alpha }={X_{\alpha }}^{\prime },Y_{\alpha }={Y_{\alpha }}^{\prime },Z_{\alpha }={Z_{\alpha }}^{\prime }}
Let
(
F
α
,
G
α
,
H
α
)
{\displaystyle (F_{\alpha },G_{\alpha },H_{\alpha })}
be the electric force at
(
x
α
,
y
α
,
z
α
,
t
α
)
{\displaystyle (x_{\alpha },y_{\alpha },z_{\alpha },t_{\alpha })}
, and let
(
L
α
,
M
α
,
N
α
)
{\displaystyle (L_{\alpha },M_{\alpha },N_{\alpha })}
be the magnetic force at the same point and time.
Also let
ρ
α
{\displaystyle \rho _{\alpha }}
a be the volume density of the electric charge and
(
u
α
,
v
α
,
w
α
)
{\displaystyle (u_{\alpha },v_{\alpha },w_{\alpha })}
its velocity; and let
(
P
α
,
Q
α
,
R
α
)
{\displaystyle (P_{\alpha },Q_{\alpha },R_{\alpha })}
be the ponderomotive
force: all equally at
(
x
α
,
y
α
,
z
α
,
t
α
)
{\displaystyle (x_{\alpha },y_{\alpha },z_{\alpha },t_{\alpha })}
. Finally let
c
{\displaystyle c}
be the velocity
of light in vacuo .
Then Lorentz’s form of Maxwell’s equations is
(1)
div
α
(
F
α
,
G
α
,
H
α
)
=
ρ
α
{\displaystyle {{\text{div}}_{\alpha }}(F_{\alpha },G_{\alpha },H_{\alpha })=\rho _{\alpha }}
(2)
div
α
(
L
α
,
M
α
,
N
α
)
=
0
{\displaystyle {{\text{div}}_{\alpha }}(L_{\alpha },M_{\alpha },N_{\alpha })=0}
(3)
c
u
r
l
α
(
L
α
,
M
α
,
N
α
)
=
−
1
c
{
∂
∂
t
α
(
F
α
,
G
α
,
H
α
)
+
ρ
α
(
u
α
,
v
α
,
w
α
)
}
{\displaystyle {curl_{\alpha }}(L_{\alpha },M_{\alpha },N_{\alpha })=-{\frac {1}{c}}{\bigg \{}{\frac {\partial }{\partial t_{\alpha }}}(F_{\alpha },G_{\alpha },H_{\alpha })+{\rho _{\alpha }}(u_{\alpha },v_{\alpha },w_{\alpha }){\bigg \}}}
(4)
c
u
r
l
α
(
F
α
,
G
α
,
H
α
)
=
−
1
c
∂
∂
t
α
(
L
α
,
M
α
,
N
α
)
{\displaystyle {curl_{\alpha }}(F_{\alpha },G_{\alpha },H_{\alpha })=-{\frac {1}{c}}{\frac {\partial }{\partial t_{\alpha }}}(L_{\alpha },M_{\alpha },N_{\alpha })}
(5)
(
P
α
,
Q
α
,
R
α
)
=
(
F
α
,
G
α
,
H
α
)
+
1
c
[
(
u
α
,
v
α
,
w
α
)
.
(
L
α
,
M
α
,
N
α
)
]
{\displaystyle (P_{\alpha },Q_{\alpha },R_{\alpha })=(F_{\alpha },G_{\alpha },H_{\alpha })+{\frac {1}{c}}[(u_{\alpha },v_{\alpha },w_{\alpha }).(L_{\alpha },M_{\alpha },N_{\alpha })]}
It will be noted that each of the vector equations (3), (4),
(5) stands for three ordinary equations, so that there are eleven
equations in the five formulae.