§ 2. Mathematical auxiliary formulas.
The time differentiation for fixed space points, is represented by
∂
∂
t
{\displaystyle {\tfrac {\partial }{\partial t}}}
. The temporal change of a surface integral, extended over a surface whose points are moving with velocity
w
{\displaystyle {\mathfrak {w}}}
, namely
d
d
t
∫
d
f
A
n
=
∫
d
f
{
∂
′
A
∂
t
}
n
{\displaystyle {\frac {d}{dt}}\int df\ {\mathfrak {A}}_{n}=\int df\left\{{\frac {\partial '{\mathfrak {A}}}{\partial t}}\right\}_{n}}
defines another kind of time differentiation of a vector
(1)
∂
′
A
∂
t
=
∂
A
∂
t
+
w
d
i
v
A
+
c
u
r
l
[
A
w
]
{\displaystyle {\frac {\partial '{\mathfrak {A}}}{\partial t}}={\frac {\partial {\mathfrak {A}}}{\partial t}}+{\mathfrak {w}}\ \mathrm {div} {\mathfrak {A}}+\mathrm {curl} [{\mathfrak {Aw}}]}
Furthermore, the derivative (with respect to time) which is related to moving points, is
(2)
A
˙
=
∂
A
∂
t
+
(
w
∇
)
A
{\displaystyle {\dot {\mathfrak {A}}}={\frac {\partial {\mathfrak {A}}}{\partial t}}+({\mathfrak {w}}\nabla ){\mathfrak {A}}}
This is connected with the temporal change of the volume integral of a vector, by the relations
(2a)
d
d
t
∫
d
v
A
=
∫
d
v
δ
A
δ
t
δ
A
δ
t
=
A
˙
+
A
d
i
v
w
{\displaystyle {\begin{array}{c}{\frac {d}{dt}}\int dv\ {\mathfrak {A}}=\int dv{\frac {\delta {\mathfrak {A}}}{\delta t}}\\\\{\frac {\delta {\mathfrak {A}}}{\delta t}}={\dot {\mathfrak {A}}}+{\mathfrak {A}}\ \mathrm {div} {\mathfrak {w}}\end{array}}}
From (2) and (2a) it follows
(3)
δ
A
δ
t
=
∂
A
∂
t
+
(
w
∇
)
A
+
A
d
i
v
w
{\displaystyle {\frac {\delta {\mathfrak {A}}}{\delta t}}={\frac {\partial {\mathfrak {A}}}{\partial t}}+({\mathfrak {w}}\nabla ){\mathfrak {A}}+{\mathfrak {A}}\ \mathrm {div} {\mathfrak {w}}}
Accordingly it is given for the scalars:
(3a)
δ
ψ
δ
t
=
∂
ψ
∂
t
+
d
i
v
ψ
w
{\displaystyle {\frac {\delta \psi }{\delta t}}={\frac {\partial \psi }{\partial t}}+\mathrm {div} \psi {\mathfrak {w}}}
From (1) and (3) it finally follows, with respect to the general rule
c
u
r
l
[
A
w
]
=
(
w
∇
)
A
−
(
A
∇
)
w
+
A
d
i
v
w
−
w
d
i
v
A
{\displaystyle \mathrm {curl} [{\mathfrak {Aw}}]=({\mathfrak {w}}\nabla ){\mathfrak {A}}-({\mathfrak {A}}\nabla ){\mathfrak {w}}+{\mathfrak {A}}\ \mathrm {div} {\mathfrak {w}}-{\mathfrak {w}}\ \mathrm {div} {\mathfrak {A}}}
,
the relation
(4)
∂
′
A
∂
t
=
δ
A
δ
t
−
(
A
∇
)
w
{\displaystyle {\frac {\partial '{\mathfrak {A}}}{\partial t}}={\frac {\delta {\mathfrak {A}}}{\delta t}}-({\mathfrak {A}}\nabla ){\mathfrak {w}}}
.
Since the time differentiation introduced in (2) follows the ordinary calculation rules, we have with respect to (2a)
[
A
˙
B
]
+
[
A
B
˙
]
=
δ
δ
t
[
A
B
]
−
[
A
B
]
d
i
v
w
{\displaystyle [{\mathfrak {{\dot {A}}B}}]+[{\mathfrak {A{\dot {B}}}}]={\frac {\delta }{\delta t}}[{\mathfrak {AB}}]-[{\mathfrak {AB}}]\mathrm {div} {\mathfrak {w}}}
From this equation, together with the ones following from (4) and (2a)
∂
′
A
∂
t
=
A
˙
+
A
d
i
v
w
−
(
A
∇
)
w
,
∂
′
B
∂
t
=
B
˙
+
B
d
i
v
w
−
(
B
∇
)
w
,
{\displaystyle {\begin{array}{l}{\frac {\partial '{\mathfrak {A}}}{\partial t}}={\mathfrak {\dot {A}}}+{\mathfrak {A}}\ \mathrm {div} {\mathfrak {w}}-({\mathfrak {A}}\nabla ){\mathfrak {w}},\\\\{\frac {\partial '{\mathfrak {B}}}{\partial t}}={\mathfrak {\dot {B}}}+{\mathfrak {B}}\ \mathrm {div} {\mathfrak {w}}-({\mathfrak {B}}\nabla ){\mathfrak {w}},\end{array}}}