The spatial nabla and the quaternionic nabla are particular operators that play an important role in the partial differential equations that control the behavior of fields in the Hilbert Book Model.
Here we treat three kinds of nabla operators.
spatial nabla
∇
→
=
i
→
∂
∂
x
+
j
→
∂
∂
y
+
k
→
∂
∂
z
{\textstyle {\vec {\nabla }}={\vec {i}}{\frac {\partial {}}{\partial {x}}}+{\vec {j}}{\frac {\partial {}}{\partial {y}}}+{\vec {k}}{\frac {\partial {}}{\partial {z}}}}
quaternionic nabla
∇
=
∂
∂
τ
+
i
→
∂
∂
x
+
j
→
∂
∂
y
+
k
→
∂
∂
z
=
∇
r
+
∇
→
{\textstyle \nabla ={\frac {\partial {}}{\partial {\tau }}}+{\vec {i}}{\frac {\partial {}}{\partial {x}}}+{\vec {j}}{\frac {\partial {}}{\partial {y}}}+{\vec {k}}{\frac {\partial {}}{\partial {z}}}=\nabla _{r}+{\vec {\nabla }}}
Dirac nabla
∇
=
∂
∂
τ
+
I
{
i
→
∂
∂
x
+
j
→
∂
∂
y
+
k
→
∂
∂
z
}
=
∇
r
+
I
∇
→
{\textstyle \nabla ={\frac {\partial {}}{\partial {\tau }}}+\mathbb {I} \left\{{\vec {i}}{\frac {\partial {}}{\partial {x}}}+{\vec {j}}{\frac {\partial {}}{\partial {y}}}+{\vec {k}}{\frac {\partial {}}{\partial {z}}}\right\}=\nabla _{r}+\mathbb {I} \,{\vec {\nabla }}}
The Dirac plays a role in the interpretation of the Dirac equation .
Properties of the spatial nabla operator
edit
The nabla product is not necessarily associative . Thus
∇
(
∇
∗
ψ
)
=
∇
∗
(
∇
ψ
)
=
(
∇
r
∇
r
+
⟨
∇
→
,
∇
→
⟩
)
ψ
≠
(
∇
∇
∗
)
ψ
=
(
∇
∗
∇
)
ψ
=
(
∇
r
∇
r
−
∇
→
∇
→
)
ψ
=
(
∇
r
∇
r
+
⟨
∇
→
,
∇
→
⟩
∓
∇
→
×
∇
→
)
ψ
{\displaystyle \nabla (\nabla ^{*}\psi )=\nabla ^{*}(\nabla \psi )=(\nabla _{r}\nabla _{r}+\langle {\vec {\nabla }},{\vec {\nabla }}\rangle )\psi \neq (\nabla \nabla ^{*})\psi =(\nabla ^{*}\nabla )\psi =(\nabla _{r}\nabla _{r}-{\vec {\nabla }}{\vec {\nabla }})\psi =(\nabla _{r}\nabla _{r}+\langle {\vec {\nabla }},{\vec {\nabla }}\rangle \mp {\vec {\nabla }}\times {\vec {\nabla }})\psi }
(1 )
Nabla in different coordinate systems
edit
The spatial nabla exists in several coordinate systems. This section shows the representation of the quaternionic nabla for Cartesian coordinate systems and for polar coordinate systems .
∇
→
=
{
∂
∂
x
,
∂
∂
y
,
∂
∂
z
}
=
∂
∂
x
x
^
→
+
∂
∂
y
y
^
→
+
∂
∂
z
z
^
→
{\displaystyle {\vec {\nabla }}=\{{\frac {\partial {}}{\partial {x}}},\,{\frac {\partial {}}{\partial {y}}},\,{\frac {\partial {}}{\partial {z}}}\}={\frac {\partial {}}{\partial {x}}}{\vec {\hat {x}}}+{\frac {\partial {}}{\partial {y}}}{\vec {\hat {y}}}+{\frac {\partial {}}{\partial {z}}}{\vec {\hat {z}}}}
(2 )
∇
→
a
r
=
∂
∂
x
x
^
→
+
∂
∂
y
y
^
→
+
∂
∂
z
z
^
→
=
∂
a
r
∂
ρ
ρ
^
→
+
1
ρ
∂
a
r
∂
θ
θ
^
→
+
1
ρ
sin
(
θ
)
∂
a
r
∂
ϕ
ϕ
^
→
{\displaystyle {\vec {\nabla }}a_{r}={\frac {\partial {}}{\partial {x}}}{\vec {\hat {x}}}+{\frac {\partial {}}{\partial {y}}}{\vec {\hat {y}}}+{\frac {\partial {}}{\partial {z}}}{\vec {\hat {z}}}={\frac {\partial {a_{r}}}{\partial {\rho }}}{\vec {\hat {\rho }}}+{\frac {1}{\rho }}{\frac {\partial {a_{r}}}{\partial {\theta }}}{\vec {\hat {\theta }}}+{\frac {1}{\rho \sin(\theta )}}{\frac {\partial {a_{r}}}{\partial {\phi }}}{\vec {\hat {\phi }}}}
(3 )
Here
{
ρ
,
θ
,
ϕ
}
{\displaystyle \{\rho ,\theta ,\phi \}}
are the coordinates with
{
ρ
^
→
,
θ
^
→
,
ϕ
^
→
}
{\displaystyle \{{\vec {\hat {\rho }}},{\vec {\hat {\theta }}},{\vec {\hat {\phi }}}\}}
as coordinate axes.
⟨
∇
→
,
a
→
⟩
=
∂
a
x
∂
x
x
^
→
+
∂
a
y
∂
y
y
^
→
+
∂
a
z
∂
z
z
^
→
=
1
ρ
2
∂
(
ρ
2
a
ρ
)
∂
ρ
+
1
ρ
sin
(
θ
)
∂
(
a
θ
sin
(
θ
)
∂
θ
)
+
1
ρ
sin
(
θ
)
∂
a
ϕ
∂
ϕ
{\displaystyle \langle {\vec {\nabla }},{\vec {a}}\rangle ={\frac {\partial {a_{x}}}{\partial {x}}}{\vec {\hat {x}}}+{\frac {\partial {a_{y}}}{\partial {y}}}{\vec {\hat {y}}}+{\frac {\partial {a_{z}}}{\partial {z}}}{\vec {\hat {z}}}={\frac {1}{\rho ^{2}}}{\frac {\partial {(\rho ^{2}a_{\rho })}}{\partial {\rho }}}+{\frac {1}{\rho \sin(\theta )}}{\frac {\partial {(a_{\theta }\sin(\theta )}}{\partial {\theta )}}}+{\frac {1}{\rho \sin(\theta )}}{\frac {\partial {a_{\phi }}}{\partial {\phi }}}}
(4 )
⟨
∇
→
,
∇
→
⟩
f
r
=
1
ρ
2
∂
(
ρ
2
∂
f
r
∂
ρ
)
∂
ρ
+
1
ρ
2
sin
(
θ
)
∂
(
sin
(
θ
)
∂
f
r
∂
θ
∂
θ
)
+
1
ρ
2
sin
2
(
θ
)
∂
2
f
r
∂
ϕ
2
{\displaystyle \langle {\vec {\nabla }},{\vec {\nabla }}\rangle f_{r}={\frac {1}{\rho ^{2}}}{\frac {\partial {(\rho ^{2}{\frac {\partial {f_{r}}}{\partial \rho }})}}{\partial {\rho }}}+{\frac {1}{\rho ^{2}\sin(\theta )}}{\frac {\partial {(\sin(\theta ){\frac {\partial {f_{r}}}{\partial {\theta }}}}}{\partial {\theta )}}}+{\frac {1}{\rho ^{2}\sin ^{2}(\theta )}}{\frac {\partial ^{2}{f_{r}}}{\partial {\phi ^{2}}}}}
(5 )
∇
→
×
a
→
=
(
∂
a
z
∂
y
−
∂
a
y
∂
z
)
x
^
→
+
(
∂
a
x
∂
z
−
∂
a
z
∂
x
)
y
^
→
+
(
∂
a
y
∂
x
−
∂
a
x
∂
y
)
z
^
→
{\displaystyle {\vec {\nabla }}\times {\vec {a}}=({\frac {\partial {a_{z}}}{\partial {y}}}-{\frac {\partial {a_{y}}}{\partial {z}}}){\vec {\hat {x}}}+({\frac {\partial {a_{x}}}{\partial {z}}}-{\frac {\partial {a_{z}}}{\partial {x}}}){\vec {\hat {y}}}+({\frac {\partial {a_{y}}}{\partial {x}}}-{\frac {\partial {a_{x}}}{\partial {y}}}){\vec {\hat {z}}}}
(6 )
=
1
ρ
sin
(
θ
)
(
∂
a
ϕ
∂
ϕ
−
∂
(
a
θ
sin
(
θ
)
∂
θ
)
)
ρ
^
→
+
(
1
ρ
2
∂
(
ρ
2
a
ρ
)
∂
ρ
−
1
ρ
sin
(
θ
)
∂
a
ϕ
∂
ϕ
)
θ
^
→
+
(
1
ρ
sin
(
θ
)
∂
(
a
θ
sin
(
θ
)
∂
θ
)
−
1
ρ
2
∂
(
ρ
2
a
ρ
)
∂
ρ
)
ϕ
^
→
{\displaystyle \qquad ={\frac {1}{\rho \sin(\theta )}}({\frac {\partial {a_{\phi }}}{\partial {\phi }}}-{\frac {\partial {(a_{\theta }\sin(\theta )}}{\partial {\theta )}}}){\vec {\hat {\rho }}}+({\frac {1}{\rho ^{2}}}{\frac {\partial {(\rho ^{2}a_{\rho })}}{\partial {\rho }}}-{\frac {1}{\rho \sin(\theta )}}{\frac {\partial {a_{\phi }}}{\partial {\phi }}}){\vec {\hat {\theta }}}+({\frac {1}{\rho \sin(\theta )}}{\frac {\partial {(a_{\theta }\sin(\theta )}}{\partial {\theta )}}}-{\frac {1}{\rho ^{2}}}{\frac {\partial {(\rho ^{2}a_{\rho })}}{\partial {\rho }}}){\vec {\hat {\phi }}}}
(7 )
∇
→
×
a
→
=
1
ρ
sin
(
θ
)
(
∂
a
ϕ
∂
ϕ
−
∂
(
a
θ
sin
(
θ
)
∂
θ
)
)
ρ
^
→
+
(
1
ρ
2
∂
(
ρ
2
a
ρ
)
∂
ρ
−
1
ρ
sin
(
θ
)
∂
a
ϕ
∂
ϕ
)
θ
^
→
+
(
1
ρ
sin
(
θ
)
∂
(
a
θ
sin
(
θ
)
∂
θ
)
−
1
ρ
2
∂
(
ρ
2
a
ρ
)
∂
ρ
)
ϕ
^
→
{\displaystyle {\vec {\nabla }}\times {\vec {a}}={\frac {1}{\rho \sin(\theta )}}({\frac {\partial {a_{\phi }}}{\partial {\phi }}}-{\frac {\partial {(a_{\theta }\sin(\theta )}}{\partial {\theta )}}}){\vec {\hat {\rho }}}+({\frac {1}{\rho ^{2}}}{\frac {\partial {(\rho ^{2}a_{\rho })}}{\partial {\rho }}}-{\frac {1}{\rho \sin(\theta )}}{\frac {\partial {a_{\phi }}}{\partial {\phi }}}){\vec {\hat {\theta }}}+({\frac {1}{\rho \sin(\theta )}}{\frac {\partial {(a_{\theta }\sin(\theta )}}{\partial {\theta )}}}-{\frac {1}{\rho ^{2}}}{\frac {\partial {(\rho ^{2}a_{\rho })}}{\partial {\rho }}}){\vec {\hat {\phi }}}}
(8 )
The spatial nabla operator shows behavior that is valid for all quaternionic functions for which the first order partial differential equation exists.
Here the quaternionic field
a
=
a
r
+
a
→
{\displaystyle a=a_{r}+{\vec {a}}}
obeys the requirement that the first order partial differential
∇
a
{\displaystyle \nabla a}
exists.
Δ
=
d
e
f
∇
2
=
d
e
f
⟨
∇
→
,
∇
→
⟩
=
d
e
f
∂
2
∂
x
2
+
∂
2
∂
y
2
+
∂
2
∂
z
2
{\displaystyle \Delta \ {\overset {\underset {\mathrm {def} }{}}{=}}\ \nabla ^{2}\ {\overset {\underset {\mathrm {def} }{}}{=}}\ \langle {\vec {\nabla }},{\vec {\nabla }}\rangle \ {\overset {\underset {\mathrm {def} }{}}{=}}\ {\frac {\partial {}^{2}}{\partial {x}^{2}}}+{\frac {\partial {}^{2}}{\partial {y}^{2}}}+{\frac {\partial {}^{2}}{\partial {z}^{2}}}}
(9 )
⟨
∇
→
,
∇
→
a
r
⟩
=
⟨
∇
→
,
∇
→
⟩
a
r
=
∇
2
a
r
{\displaystyle \langle {\vec {\nabla }},{\vec {\nabla }}a_{r}\rangle =\langle {\vec {\nabla }},{\vec {\nabla }}\rangle a_{r}=\nabla ^{2}a_{r}}
(10 )
⟨
∇
→
,
∇
→
⟩
a
→
=
d
e
f
∇
2
a
→
{\displaystyle \langle {\vec {\nabla }},{\vec {\nabla }}\rangle {\vec {a}}\ {\overset {\underset {\mathrm {def} }{}}{=}}\ \nabla ^{2}{\vec {a}}}
(11 )
⟨
∇
→
×
∇
→
,
a
→
⟩
=
0
{\displaystyle \langle {\vec {\nabla }}\times {\vec {\nabla }},{\vec {a}}\rangle =0}
(12 )
∇
→
×
∇
→
a
r
=
0
→
{\displaystyle {\vec {\nabla }}\times {\vec {\nabla }}a_{r}={\vec {0}}}
(13 )
⟨
∇
→
,
∇
→
×
a
→
⟩
=
0
{\displaystyle \langle {\vec {\nabla }},{\vec {\nabla }}\times {\vec {a}}\rangle =0}
(14 )
∇
→
×
(
∇
→
×
a
→
)
=
∇
→
⟨
∇
→
,
a
→
⟩
−
⟨
∇
→
,
∇
→
⟩
a
→
{\displaystyle {\vec {\nabla }}\times ({\vec {\nabla }}\times {\vec {a}})={\vec {\nabla }}\langle {\vec {\nabla }},{\vec {a}}\rangle -\langle {\vec {\nabla }},{\vec {\nabla }}\rangle {\vec {a}}}
(15 )
∇
→
(
∇
→
a
)
=
∇
→
(
∇
→
×
a
−
⟨
∇
→
,
a
→
⟩
+
∇
→
a
r
)
=
∇
→
×
(
∇
→
×
a
→
)
−
∇
→
⟨
∇
→
,
a
→
⟩
−
⟨
∇
→
,
∇
→
⟩
a
r
=
∇
→
⟨
∇
→
,
a
→
⟩
−
⟨
∇
→
,
∇
→
⟩
a
→
−
∇
→
⟨
∇
→
,
a
→
⟩
−
⟨
∇
→
,
∇
→
⟩
a
r
=
−
⟨
∇
→
,
∇
→
⟩
a
{\displaystyle {\vec {\nabla }}({\vec {\nabla }}a)={\vec {\nabla }}({\vec {\nabla }}\times a-\langle {\vec {\nabla }},{\vec {a}}\rangle +{\vec {\nabla }}a_{r})={\vec {\nabla }}\times ({\vec {\nabla }}\times {\vec {a}})-{\vec {\nabla }}\langle {\vec {\nabla }},{\vec {a}}\rangle -\langle {\vec {\nabla }},{\vec {\nabla }}\rangle a_{r}={\vec {\nabla }}\langle {\vec {\nabla }},{\vec {a}}\rangle -\langle {\vec {\nabla }},{\vec {\nabla }}\rangle {\vec {a}}-{\vec {\nabla }}\langle {\vec {\nabla }},{\vec {a}}\rangle -\langle {\vec {\nabla }},{\vec {\nabla }}\rangle a_{r}=-\langle {\vec {\nabla }},{\vec {\nabla }}\rangle a}
(16 )
For constant
k
→
{\displaystyle {\vec {k}}}
and parameter
q
=
q
r
+
q
→
=
{
q
r
,
q
x
,
q
y
,
q
z
}
{\displaystyle q=q_{r}+{\vec {q}}=\{q_{r},q_{x},q_{y},q_{z}\}}
holds
∇
→
⟨
k
→
,
q
→
⟩
=
k
→
{\displaystyle {\vec {\nabla }}\langle {\vec {k}},{\vec {q}}\rangle ={\vec {k}}}
(17 )
⟨
∇
→
,
q
→
⟩
=
3
{\displaystyle \langle {\vec {\nabla }},{\vec {q}}\rangle =3}
(18 )
∇
→
×
q
→
=
0
→
{\displaystyle {\vec {\nabla }}\times {\vec {q}}={\vec {0}}}
(19 )
∇
→
|
q
→
|
=
q
→
|
q
→
|
;
∇
|
q
|
=
q
|
q
|
{\displaystyle {\vec {\nabla }}\,|{\vec {q}}|={\frac {\vec {q}}{|{\vec {q}}|}};\quad \nabla \,|q|={\frac {q}{|q|}}}
(20 )
∇
→
1
|
q
→
−
q
→
′
|
=
−
q
→
−
q
→
′
|
q
→
−
q
→
′
|
3
;
∇
1
|
q
−
q
′
|
=
−
q
−
q
′
|
q
−
q
′
|
3
{\displaystyle {\vec {\nabla }}{\frac {1}{|{\vec {q}}-{\vec {q}}^{'}|}}=-{\frac {{\vec {q}}-{\vec {q}}^{'}}{|{\vec {q}}-{\vec {q}}^{'}|^{3}}};\ \nabla {\frac {1}{|q-q^{'}|}}=-{\frac {q-q^{'}}{|q-q^{'}|^{3}}}}
(21 )
⟨
∇
→
,
∇
→
1
|
q
→
−
q
→
′
|
⟩
=
⟨
∇
→
,
∇
→
⟩
1
|
q
→
−
q
→
′
|
=
−
⟨
∇
→
,
q
→
−
q
→
′
|
q
→
−
q
→
′
|
3
⟩
=
4
π
δ
(
q
→
−
q
→
′
)
{\displaystyle \langle {\vec {\nabla }},{\vec {\nabla }}{\frac {1}{|{\vec {q}}-{\vec {q}}^{'}|}}\rangle =\langle {\vec {\nabla }},{\vec {\nabla }}\rangle {\frac {1}{|{\vec {q}}-{\vec {q}}^{'}|}}=-\langle {\vec {\nabla }},{\frac {{\vec {q}}-{\vec {q}}^{'}}{|{\vec {q}}-{\vec {q}}^{'}|^{3}}}\rangle =4\,\pi \,\delta ({\vec {q}}-{\vec {q}}^{'})}
(22 )
This indicates the relation between the Poisson equation and the Green's function.
∇
q
=
1
−
3
;
∇
∗
q
=
1
+
3
;
∇
q
∗
=
1
+
3
{\displaystyle \nabla \ q=1-3;\quad \nabla ^{*}q=1+3;\quad \nabla q^{*}=1+3}
(23 )
The term
(
∇
→
×
∇
→
)
f
{\displaystyle ({\vec {\nabla }}\times {\vec {\nabla }})f}
indicates the curvature of field
f
{\displaystyle f}
.
The term
⟨
∇
→
,
∇
→
⟩
f
{\displaystyle \langle {\vec {\nabla }},{\vec {\nabla }}\rangle f}
indicates the stress of field
f
{\displaystyle f}
First order partial differential equation
edit
φ
=
φ
r
+
φ
→
=
∇
ψ
=
(
∇
r
+
∇
→
)
(
ψ
r
+
ψ
→
)
=
∇
r
ψ
r
−
⟨
∇
→
,
ψ
→
⟩
+
∇
r
ψ
→
+
∇
→
ψ
r
±
∇
→
×
ψ
→
{\displaystyle \varphi =\varphi _{r}+{\vec {\varphi }}=\nabla \psi =(\nabla _{r}+{\vec {\nabla }})(\psi _{r}+{\vec {\psi }})=\nabla _{r}\psi _{r}-\langle {\vec {\nabla }},{\vec {\psi }}\rangle +\nabla _{r}{\vec {\psi }}+{\vec {\nabla }}\psi _{r}{\color {Red}\pm }{\vec {\nabla }}\times {\vec {\psi }}}
(24 )
The equation is a quaternionic first order partial differential equation.
The five terms on the right side show the components that constitute the full first-order change.
They represent subfields of field
φ
{\displaystyle \varphi }
, and often they get special names and symbols.
∇
→
ψ
r
{\displaystyle {\vec {\nabla }}\psi _{r}}
is the gradient of
ψ
r
{\displaystyle \psi _{r}}
⟨
∇
→
,
ψ
→
⟩
{\displaystyle \langle {\vec {\nabla }},{\vec {\psi }}\rangle }
is the divergence of
ψ
→
{\displaystyle {\vec {\psi }}}
.
∇
→
×
ψ
→
{\displaystyle {\vec {\nabla }}\times {\vec {\psi }}}
is the curl of
ψ
→
{\displaystyle {\vec {\psi }}}
φ
r
=
∇
r
ψ
r
−
⟨
∇
→
,
ψ
→
⟩
{\displaystyle \varphi _{r}=\nabla _{r}\psi _{r}-\langle {\vec {\nabla }},{\vec {\psi }}\rangle }
(25 )
(Equation (25 ) has no equivalent in Maxwell's equations!)
φ
→
=
∇
r
ψ
→
+
∇
→
ψ
r
±
∇
→
×
ψ
→
=
∓
B
→
−
E
→
{\displaystyle {\vec {\varphi }}=\nabla _{r}{\vec {\psi }}+{\vec {\nabla }}\psi _{r}\pm {\vec {\nabla }}\times {\vec {\psi }}=\mp {\vec {B}}-{\vec {E}}}
(26 )
E
→
=
d
e
f
−
∇
r
ψ
→
−
∇
→
ψ
r
{\displaystyle {\vec {E}}\ {\overset {\underset {\mathrm {def} }{}}{=}}\ -\nabla _{r}{\vec {\psi }}-{\vec {\nabla }}\psi _{r}}
(27 )
B
→
=
d
e
f
∇
→
×
ψ
→
{\displaystyle {\vec {B}}\ {\overset {\underset {\mathrm {def} }{}}{=}}\ {\vec {\nabla }}\times {\vec {\psi }}}
(28 )
J
→
=
d
e
f
∇
→
×
B
→
−
∇
r
E
→
{\displaystyle {\vec {J}}\ {\overset {\underset {\mathrm {def} }{}}{=}}\ {\vec {\nabla }}\times {\vec {B}}-\nabla _{r}{\vec {E}}}
(29 )
Derivation of higher order equations
edit
With the help of the properties of the spatial nabla operator follows an interesting second-order partial differential equation.
⟨
∇
→
,
B
→
⟩
=
⟨
∇
→
,
∇
→
×
ψ
→
⟩
=
0
{\displaystyle \langle {\vec {\nabla }},{\vec {B}}\rangle =\langle {\vec {\nabla }},{\vec {\nabla }}\times {\vec {\psi }}\rangle =0}
(30 )
⟨
∇
→
,
E
→
⟩
=
−
∇
r
⟨
∇
→
,
ψ
→
⟩
−
⟨
∇
→
,
∇
→
ψ
r
⟩
=
−
∇
r
⟨
∇
→
,
ψ
→
⟩
−
⟨
∇
→
,
∇
→
⟩
ψ
r
{\displaystyle \langle {\vec {\nabla }},{\vec {E}}\rangle =-\nabla _{r}\langle {\vec {\nabla }},{\vec {\psi }}\rangle -\langle {\vec {\nabla }},{\vec {\nabla }}\psi _{r}\rangle =-\nabla _{r}\langle {\vec {\nabla }},{\vec {\psi }}\rangle -\langle {\vec {\nabla }},{\vec {\nabla }}\rangle \psi _{r}}
(31 )
⟨
∇
→
,
B
→
−
E
→
⟩
=
∇
r
⟨
∇
→
,
ψ
→
⟩
+
⟨
∇
→
,
∇
→
⟩
ψ
r
{\displaystyle \langle {\vec {\nabla }},{\vec {B}}-{\vec {E}}\rangle =\nabla _{r}\langle {\vec {\nabla }},{\vec {\psi }}\rangle +\langle {\vec {\nabla }},{\vec {\nabla }}\rangle \psi _{r}}
(32 )
∇
r
E
→
=
−
∇
r
∇
r
ψ
→
−
∇
r
∇
→
ψ
r
{\displaystyle \nabla _{r}{\vec {E}}=-\nabla _{r}\nabla _{r}{\vec {\psi }}-\nabla _{r}{\vec {\nabla }}\psi _{r}}
(33 )
∇
→
×
E
→
=
−
∇
→
×
(
∇
r
ψ
→
)
−
∇
→
×
(
∇
→
ψ
r
)
=
−
∇
r
∇
→
×
ψ
→
=
−
∇
r
B
→
{\displaystyle {\vec {\nabla }}\times {\vec {E}}=-{\vec {\nabla }}\times (\nabla _{r}{\vec {\psi }})-{\vec {\nabla }}\times ({\vec {\nabla }}\psi _{r})=-\nabla _{r}{\vec {\nabla }}\times {\vec {\psi }}=-\nabla _{r}{\vec {B}}}
(34 )
∇
→
×
B
→
=
∇
→
×
(
∇
→
×
ψ
→
)
=
∇
→
⟨
∇
→
,
ψ
→
⟩
−
⟨
∇
→
,
∇
→
⟩
ψ
→
{\displaystyle {\vec {\nabla }}\times {\vec {B}}={\vec {\nabla }}\times ({\vec {\nabla }}\times {\vec {\psi }})={\vec {\nabla }}\langle {\vec {\nabla }},{\vec {\psi }}\rangle -\langle {\vec {\nabla }},{\vec {\nabla }}\rangle {\vec {\psi }}}
(35 )
∇
→
×
B
→
−
∇
→
×
E
→
=
∇
→
⟨
∇
→
,
ψ
→
⟩
−
⟨
∇
→
,
∇
→
⟩
ψ
→
+
∇
r
B
→
{\displaystyle {\vec {\nabla }}\times {\vec {B}}-{\vec {\nabla }}\times {\vec {E}}={\vec {\nabla }}\langle {\vec {\nabla }},{\vec {\psi }}\rangle -\langle {\vec {\nabla }},{\vec {\nabla }}\rangle {\vec {\psi }}+\nabla _{r}{\vec {B}}}
(36 )
∇
→
×
(
B
→
−
E
→
)
−
∇
r
(
B
→
−
E
→
)
=
∇
→
⟨
∇
→
,
ψ
→
⟩
−
⟨
∇
→
,
∇
→
⟩
ψ
→
−
∇
r
∇
r
ψ
→
−
∇
r
∇
→
ψ
r
{\displaystyle {\vec {\nabla }}\times ({\vec {B}}-{\vec {E}})-\nabla _{r}({\vec {B}}-{\vec {E}})={\vec {\nabla }}\langle {\vec {\nabla }},{\vec {\psi }}\rangle -\langle {\vec {\nabla }},{\vec {\nabla }}\rangle {\vec {\psi }}-\nabla _{r}\nabla _{r}{\vec {\psi }}-\nabla _{r}{\vec {\nabla }}\psi _{r}}
(37 )
∇
→
×
(
B
→
−
E
→
)
−
∇
r
(
B
→
−
E
→
)
=
−
(
⟨
∇
→
,
∇
→
⟩
+
∇
r
∇
r
)
ψ
→
−
∇
r
∇
→
ψ
r
+
∇
→
⟨
∇
→
,
ψ
→
⟩
{\displaystyle {\vec {\nabla }}\times ({\vec {B}}-{\vec {E}})-\nabla _{r}({\vec {B}}-{\vec {E}})=-(\langle {\vec {\nabla }},{\vec {\nabla }}\rangle +\nabla _{r}\nabla _{r}){\vec {\psi }}-\nabla _{r}{\vec {\nabla }}\psi _{r}+{\vec {\nabla }}\langle {\vec {\nabla }},{\vec {\psi }}\rangle }
(38 )
∇
→
×
(
B
→
−
E
→
)
−
⟨
∇
→
,
B
→
−
E
→
⟩
−
∇
r
(
B
→
−
E
→
)
=
∇
∗
φ
→
=
−
(
⟨
∇
→
,
∇
→
⟩
+
∇
r
∇
r
)
ψ
→
−
∇
r
∇
→
ψ
r
+
∇
→
⟨
∇
→
,
ψ
→
⟩
−
∇
r
⟨
∇
→
,
ψ
→
⟩
−
⟨
∇
→
,
∇
→
⟩
ψ
r
{\displaystyle {\vec {\nabla }}\times ({\vec {B}}-{\vec {E}})-\langle {\vec {\nabla }},{\vec {B}}-{\vec {E}}\rangle -\nabla _{r}({\vec {B}}-{\vec {E}})=\nabla ^{*}{\vec {\varphi }}=-(\langle {\vec {\nabla }},{\vec {\nabla }}\rangle +\nabla _{r}\nabla _{r}){\vec {\psi }}-\nabla _{r}{\vec {\nabla }}\psi _{r}+{\vec {\nabla }}\langle {\vec {\nabla }},{\vec {\psi }}\rangle -\nabla _{r}\langle {\vec {\nabla }},{\vec {\psi }}\rangle -\langle {\vec {\nabla }},{\vec {\nabla }}\rangle \psi _{r}}
(39 )
Further
J
→
=
d
e
f
∇
→
×
B
→
−
∇
r
E
→
=
∇
→
⟨
∇
→
,
ψ
→
⟩
−
⟨
∇
→
,
∇
→
⟩
ψ
→
+
∇
r
∇
r
ψ
→
+
∇
r
∇
→
ψ
r
=
(
∇
r
∇
r
−
⟨
∇
→
,
∇
→
⟩
)
ψ
→
+
∇
r
∇
→
ψ
r
+
∇
→
⟨
∇
→
,
ψ
→
⟩
{\displaystyle {\vec {J}}\ {\overset {\underset {\mathrm {def} }{}}{=}}\ {\vec {\nabla }}\times {\vec {B}}-\nabla _{r}{\vec {E}}={\vec {\nabla }}\langle {\vec {\nabla }},{\vec {\psi }}\rangle -\langle {\vec {\nabla }},{\vec {\nabla }}\rangle {\vec {\psi }}+\nabla _{r}\nabla _{r}{\vec {\psi }}+\nabla _{r}{\vec {\nabla }}\psi _{r}=(\nabla _{r}\nabla _{r}-\langle {\vec {\nabla }},{\vec {\nabla }}\rangle ){\vec {\psi }}+\nabla _{r}{\vec {\nabla }}\psi _{r}+{\vec {\nabla }}\langle {\vec {\nabla }},{\vec {\psi }}\rangle }
(40 )
From the above formulas follows that the Maxwell equations do not form a complete set.
Physicists use gauge equations to make Maxwell equations more complete.
Derivation of second order partial differential equation 1
edit
φ
=
∇
ψ
=
(
∇
r
+
∇
→
)
ψ
{\displaystyle \varphi =\nabla \psi =(\nabla _{r}+{\vec {\nabla }})\psi }
(41 )
φ
r
=
∇
r
ψ
r
−
⟨
∇
→
,
ψ
→
⟩
{\displaystyle \varphi _{r}=\nabla _{r}\psi _{r}-\langle {\vec {\nabla }},{\vec {\psi }}\rangle }
(42 )
φ
→
=
∇
r
ψ
→
+
∇
→
ψ
r
±
∇
→
×
ψ
→
{\displaystyle {\vec {\varphi }}=\nabla _{r}{\vec {\psi }}+{\vec {\nabla }}\psi _{r}\pm {\vec {\nabla }}\times {\vec {\psi }}}
(43 )
∇
→
φ
r
=
∇
→
(
∇
r
ψ
r
−
⟨
∇
→
,
ψ
→
⟩
)
=
∇
r
∇
→
ψ
r
−
∇
→
⟨
∇
→
,
ψ
→
⟩
{\displaystyle {\vec {\nabla }}\varphi _{r}={\vec {\nabla }}(\nabla _{r}\psi _{r}-\langle {\vec {\nabla }},{\vec {\psi }}\rangle )=\nabla _{r}{\vec {\nabla }}\psi _{r}-{\vec {\nabla }}\langle {\vec {\nabla }},{\vec {\psi }}\rangle }
(44 )
∇
r
φ
r
=
∇
r
(
∇
r
ψ
r
−
⟨
∇
→
,
ψ
→
⟩
)
=
∇
r
∇
r
ψ
r
−
∇
r
⟨
∇
→
,
ψ
→
⟩
{\displaystyle \nabla _{r}\varphi _{r}=\nabla _{r}(\nabla _{r}\psi _{r}-\langle {\vec {\nabla }},{\vec {\psi }}\rangle )=\nabla _{r}\nabla _{r}\psi _{r}-\nabla _{r}\langle {\vec {\nabla }},{\vec {\psi }}\rangle }
(45 )
∇
∗
φ
r
=
−
∇
r
∇
→
ψ
r
+
∇
→
⟨
∇
→
,
ψ
→
⟩
+
∇
r
∇
r
ψ
r
−
∇
r
⟨
∇
→
,
ψ
→
⟩
{\displaystyle \nabla ^{*}\varphi _{r}=-\nabla _{r}{\vec {\nabla }}\psi _{r}+{\vec {\nabla }}\langle {\vec {\nabla }},{\vec {\psi }}\rangle +\nabla _{r}\nabla _{r}\psi _{r}-\nabla _{r}\langle {\vec {\nabla }},{\vec {\psi }}\rangle }
(46 )
∇
→
φ
→
=
∇
→
(
∇
r
ψ
→
+
∇
→
ψ
r
±
∇
→
×
ψ
→
⟩
)
=
−
∇
r
⟨
∇
→
,
ψ
→
⟩
±
∇
r
∇
→
×
ψ
→
−
⟨
∇
→
,
∇
→
⟩
ψ
r
∓
⟨
∇
→
,
∇
→
×
ψ
→
⟩
+
∇
→
×
∇
→
×
ψ
→
{\displaystyle {\vec {\nabla }}{\vec {\varphi }}={\vec {\nabla }}(\nabla _{r}{\vec {\psi }}+{\vec {\nabla }}\psi _{r}\pm {\vec {\nabla }}\times {\vec {\psi }}\rangle )=-\nabla _{r}\langle {\vec {\nabla }},{\vec {\psi }}\rangle \pm \nabla _{r}{\vec {\nabla }}\times {\vec {\psi }}-\langle {\vec {\nabla }},{\vec {\nabla }}\rangle \psi _{r}\mp \langle {\vec {\nabla }},{\vec {\nabla }}\times {\vec {\psi }}\rangle +{\vec {\nabla }}\times {\vec {\nabla }}\times {\vec {\psi }}}
(47 )
∇
→
φ
→
=
−
∇
r
⟨
∇
→
,
ψ
→
⟩
±
∇
r
∇
→
×
ψ
→
−
⟨
∇
→
,
∇
→
⟩
ψ
r
+
∇
→
×
∇
→
×
ψ
→
=
−
∇
r
⟨
∇
→
,
ψ
→
⟩
±
∇
r
∇
→
×
ψ
→
−
⟨
∇
→
,
∇
→
⟩
ψ
r
+
∇
→
⟨
∇
→
,
ψ
→
⟩
−
⟨
∇
→
,
∇
→
⟩
ψ
→
{\displaystyle {\vec {\nabla }}{\vec {\varphi }}=-\nabla _{r}\langle {\vec {\nabla }},{\vec {\psi }}\rangle \pm \nabla _{r}{\vec {\nabla }}\times {\vec {\psi }}-\langle {\vec {\nabla }},{\vec {\nabla }}\rangle \psi _{r}+{\vec {\nabla }}\times {\vec {\nabla }}\times {\vec {\psi }}=-\nabla _{r}\langle {\vec {\nabla }},{\vec {\psi }}\rangle \pm \nabla _{r}{\vec {\nabla }}\times {\vec {\psi }}-\langle {\vec {\nabla }},{\vec {\nabla }}\rangle \psi _{r}+{\vec {\nabla }}\langle {\vec {\nabla }},{\vec {\psi }}\rangle -\langle {\vec {\nabla }},{\vec {\nabla }}\rangle {\vec {\psi }}}
(48 )
∇
r
φ
→
=
∇
r
(
∇
r
ψ
→
+
∇
→
ψ
r
±
∇
→
×
ψ
→
)
=
∇
r
∇
r
ψ
→
+
∇
r
∇
→
ψ
r
±
∇
r
∇
→
×
ψ
→
{\displaystyle \nabla _{r}{\vec {\varphi }}=\nabla _{r}(\nabla _{r}{\vec {\psi }}+{\vec {\nabla }}\psi _{r}\pm {\vec {\nabla }}\times {\vec {\psi }})=\nabla _{r}\nabla _{r}{\vec {\psi }}+\nabla _{r}{\vec {\nabla }}\psi _{r}\pm \nabla _{r}{\vec {\nabla }}\times {\vec {\psi }}}
(49 )
∇
∗
φ
→
=
∇
r
∇
r
ψ
→
+
∇
r
∇
→
ψ
r
±
∇
r
∇
→
×
ψ
→
+
∇
r
⟨
∇
→
,
ψ
→
⟩
∓
∇
r
∇
→
×
ψ
→
+
⟨
∇
→
,
∇
→
⟩
ψ
r
−
∇
→
⟨
∇
→
,
ψ
→
⟩
+
⟨
∇
→
,
∇
→
⟩
ψ
→
{\displaystyle \nabla ^{*}{\vec {\varphi }}=\nabla _{r}\nabla _{r}{\vec {\psi }}+\nabla _{r}{\vec {\nabla }}\psi _{r}\pm \nabla _{r}{\vec {\nabla }}\times {\vec {\psi }}+\nabla _{r}\langle {\vec {\nabla }},{\vec {\psi }}\rangle \mp \nabla _{r}{\vec {\nabla }}\times {\vec {\psi }}+\langle {\vec {\nabla }},{\vec {\nabla }}\rangle \psi _{r}-{\vec {\nabla }}\langle {\vec {\nabla }},{\vec {\psi }}\rangle +\langle {\vec {\nabla }},{\vec {\nabla }}\rangle {\vec {\psi }}}
(50 )
∇
∗
φ
→
=
∇
r
∇
r
ψ
→
+
⟨
∇
→
,
∇
→
⟩
ψ
→
+
∇
r
∇
→
ψ
r
+
∇
r
⟨
∇
→
,
ψ
→
⟩
+
⟨
∇
→
,
∇
→
⟩
ψ
r
−
∇
→
⟨
∇
→
,
ψ
→
⟩
{\displaystyle \nabla ^{*}{\vec {\varphi }}=\nabla _{r}\nabla _{r}{\vec {\psi }}+\langle {\vec {\nabla }},{\vec {\nabla }}\rangle {\vec {\psi }}+\nabla _{r}{\vec {\nabla }}\psi _{r}+\nabla _{r}\langle {\vec {\nabla }},{\vec {\psi }}\rangle +\langle {\vec {\nabla }},{\vec {\nabla }}\rangle \psi _{r}-{\vec {\nabla }}\langle {\vec {\nabla }},{\vec {\psi }}\rangle }
(51 )
∇
∗
φ
=
−
∇
r
∇
→
ψ
r
+
∇
→
⟨
∇
→
,
ψ
→
⟩
+
∇
r
∇
r
ψ
r
−
∇
r
⟨
∇
→
,
ψ
→
⟩
+
∇
r
∇
r
ψ
→
+
⟨
∇
→
,
∇
→
⟩
ψ
→
+
∇
r
∇
→
ψ
r
+
∇
r
⟨
∇
→
,
ψ
→
⟩
+
⟨
∇
→
,
∇
→
⟩
ψ
r
−
∇
→
⟨
∇
→
,
ψ
→
⟩
{\displaystyle \nabla ^{*}\varphi =-\nabla _{r}{\vec {\nabla }}\psi _{r}+{\vec {\nabla }}\langle {\vec {\nabla }},{\vec {\psi }}\rangle +\nabla _{r}\nabla _{r}\psi _{r}-\nabla _{r}\langle {\vec {\nabla }},{\vec {\psi }}\rangle +\nabla _{r}\nabla _{r}{\vec {\psi }}+\langle {\vec {\nabla }},{\vec {\nabla }}\rangle {\vec {\psi }}+\nabla _{r}{\vec {\nabla }}\psi _{r}+\nabla _{r}\langle {\vec {\nabla }},{\vec {\psi }}\rangle +\langle {\vec {\nabla }},{\vec {\nabla }}\rangle \psi _{r}-{\vec {\nabla }}\langle {\vec {\nabla }},{\vec {\psi }}\rangle }
(52 )
χ
=
∇
∗
φ
=
(
∇
r
−
∇
→
)
φ
=
(
∇
r
−
∇
→
)
(
∇
r
+
∇
→
)
ψ
=
(
∇
r
∇
r
+
⟨
∇
→
,
∇
→
⟩
)
(
ψ
r
+
ψ
→
)
+
∇
→
⟨
∇
→
,
ψ
→
⟩
−
∇
→
⟨
∇
→
,
ψ
→
⟩
+
∇
r
⟨
∇
→
,
ψ
→
⟩
−
∇
r
⟨
∇
→
,
ψ
→
⟩
−
∇
r
∇
→
ψ
r
+
∇
r
∇
→
ψ
r
{\displaystyle \chi =\nabla ^{*}\varphi =(\nabla _{r}-{\vec {\nabla }})\varphi =(\nabla _{r}-{\vec {\nabla }})(\nabla _{r}+{\vec {\nabla }})\psi =(\nabla _{r}\nabla _{r}+\langle {\vec {\nabla }},{\vec {\nabla }}\rangle )(\psi _{r}+{\vec {\psi }})+{\vec {\nabla }}\langle {\vec {\nabla }},{\vec {\psi }}\rangle -{\vec {\nabla }}\langle {\vec {\nabla }},{\vec {\psi }}\rangle +\nabla _{r}\langle {\vec {\nabla }},{\vec {\psi }}\rangle -\nabla _{r}\langle {\vec {\nabla }},{\vec {\psi }}\rangle -\nabla _{r}{\vec {\nabla }}\psi _{r}+\nabla _{r}{\vec {\nabla }}\psi _{r}}
(53 )
Most of the terms vanish.
χ
=
∇
∗
(
∇
ψ
)
=
(
∇
r
−
∇
→
)
(
(
∇
r
+
∇
→
)
(
ψ
r
+
ψ
→
)
)
=
(
∇
r
∇
r
+
⟨
∇
→
,
∇
→
⟩
)
ψ
{\displaystyle \chi =\nabla ^{*}(\nabla \psi )=(\nabla _{r}-{\vec {\nabla }})((\nabla _{r}+{\vec {\nabla }})(\psi _{r}+{\vec {\psi }}))=(\nabla _{r}\nabla _{r}+\langle {\vec {\nabla }},{\vec {\nabla }}\rangle )\psi }
(54 )
Derivation of second order partial differential equation 2
edit
We add the complex imaginary base number
I
=
−
1
{\textstyle \mathbb {I} ={\sqrt {-1}}}
to the spatial nabla operator
∇
→
{\textstyle {\vec {\nabla }}}
.
g
=
(
∇
r
+
I
∇
→
)
f
{\displaystyle g=(\nabla _{r}+\mathbb {I} {\vec {\nabla }})f}
(55 )
g
r
=
∇
r
f
r
−
⟨
I
∇
→
,
f
→
⟩
{\displaystyle g_{r}=\nabla _{r}f_{r}-\langle \mathbb {I} {\vec {\nabla }},{\vec {f}}\rangle }
(56 )
g
→
=
∇
r
f
→
+
I
∇
→
f
r
±
I
∇
→
×
f
→
{\displaystyle {\vec {g}}=\nabla _{r}{\vec {f}}+\mathbb {I} {\vec {\nabla }}f_{r}\pm \mathbb {I} {\vec {\nabla }}\times {\vec {f}}}
(57 )
I
∇
→
g
r
=
I
∇
→
(
∇
r
f
r
−
⟨
I
∇
→
,
f
→
⟩
)
=
∇
r
I
∇
→
f
r
−
I
∇
→
⟨
I
∇
→
,
f
→
⟩
{\displaystyle \mathbb {I} {\vec {\nabla }}g_{r}=\mathbb {I} {\vec {\nabla }}(\nabla _{r}f_{r}-\langle \mathbb {I} {\vec {\nabla }},{\vec {f}}\rangle )=\nabla _{r}\mathbb {I} {\vec {\nabla }}f_{r}-\mathbb {I} {\vec {\nabla }}\langle \mathbb {I} {\vec {\nabla }},{\vec {f}}\rangle }
(58 )
∇
r
g
r
=
∇
r
(
∇
r
f
r
−
⟨
I
∇
→
,
f
→
⟩
)
=
∇
r
∇
r
f
r
−
∇
r
⟨
I
∇
→
,
f
→
⟩
{\displaystyle \nabla _{r}g_{r}=\nabla _{r}(\nabla _{r}f_{r}-\langle \mathbb {I} {\vec {\nabla }},{\vec {f}}\rangle )=\nabla _{r}\nabla _{r}f_{r}-\nabla _{r}\langle \mathbb {I} {\vec {\nabla }},{\vec {f}}\rangle }
(59 )
(
∇
r
−
I
∇
→
)
g
r
=
−
∇
r
I
∇
→
f
r
+
I
∇
→
⟨
I
∇
→
,
f
→
⟩
+
∇
r
∇
r
f
r
−
∇
r
⟨
I
∇
→
,
f
→
⟩
{\displaystyle (\nabla _{r}-\mathbb {I} {\vec {\nabla }})g_{r}=-\nabla _{r}\mathbb {I} {\vec {\nabla }}f_{r}+\mathbb {I} {\vec {\nabla }}\langle \mathbb {I} {\vec {\nabla }},{\vec {f}}\rangle +\nabla _{r}\nabla _{r}f_{r}-\nabla _{r}\langle \mathbb {I} {\vec {\nabla }},{\vec {f}}\rangle }
(60 )
I
∇
→
g
→
=
I
∇
→
(
∇
r
f
→
+
I
∇
→
f
r
±
I
∇
→
×
f
→
⟩
)
=
−
∇
r
⟨
I
∇
→
,
f
→
⟩
±
∇
r
I
∇
→
×
f
→
−
⟨
I
∇
→
,
I
∇
→
⟩
f
r
∓
⟨
I
∇
→
,
I
∇
→
×
f
→
⟩
+
I
∇
→
×
I
∇
→
×
f
→
{\displaystyle \mathbb {I} {\vec {\nabla }}{\vec {g}}=\mathbb {I} {\vec {\nabla }}(\nabla _{r}{\vec {f}}+\mathbb {I} {\vec {\nabla }}f_{r}\pm \mathbb {I} {\vec {\nabla }}\times {\vec {f}}\rangle )=-\nabla _{r}\langle \mathbb {I} {\vec {\nabla }},{\vec {f}}\rangle \pm \nabla _{r}\mathbb {I} {\vec {\nabla }}\times {\vec {f}}-\langle \mathbb {I} {\vec {\nabla }},\mathbb {I} {\vec {\nabla }}\rangle f_{r}\mp \langle \mathbb {I} {\vec {\nabla }},\mathbb {I} {\vec {\nabla }}\times {\vec {f}}\rangle +\mathbb {I} {\vec {\nabla }}\times \mathbb {I} {\vec {\nabla }}\times {\vec {f}}}
(61 )
I
∇
→
g
→
=
−
∇
r
⟨
I
∇
→
,
f
→
⟩
±
∇
r
I
∇
→
×
f
→
−
⟨
I
∇
→
,
I
∇
→
⟩
f
r
+
I
∇
→
×
I
∇
→
×
f
→
=
−
∇
r
⟨
I
∇
→
,
f
→
⟩
±
∇
r
I
∇
→
×
f
→
−
⟨
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∇
→
,
I
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⟩
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r
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{\displaystyle \mathbb {I} {\vec {\nabla }}{\vec {g}}=-\nabla _{r}\langle \mathbb {I} {\vec {\nabla }},{\vec {f}}\rangle \pm \nabla _{r}\mathbb {I} {\vec {\nabla }}\times {\vec {f}}-\langle \mathbb {I} {\vec {\nabla }},\mathbb {I} {\vec {\nabla }}\rangle f_{r}+\mathbb {I} {\vec {\nabla }}\times \mathbb {I} {\vec {\nabla }}\times {\vec {f}}=-\nabla _{r}\langle \mathbb {I} {\vec {\nabla }},{\vec {f}}\rangle \pm \nabla _{r}\mathbb {I} {\vec {\nabla }}\times {\vec {f}}-\langle \mathbb {I} {\vec {\nabla }},\mathbb {I} {\vec {\nabla }}\rangle f_{r}+\mathbb {I} {\vec {\nabla }}\langle \mathbb {I} {\vec {\nabla }},{\vec {f}}\rangle -\langle \mathbb {I} {\vec {\nabla }},\mathbb {I} {\vec {\nabla }}\rangle {\vec {f}}}
(62 )
∇
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→
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∇
r
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=
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{\displaystyle \nabla _{r}{\vec {g}}=\nabla _{r}(\nabla _{r}{\vec {f}}+\mathbb {I} {\vec {\nabla }}f_{r}\pm \mathbb {I} {\vec {\nabla }}\times {\vec {f}})=\nabla _{r}\nabla _{r}{\vec {f}}+\nabla _{r}\mathbb {I} {\vec {\nabla }}f_{r}\pm \nabla _{r}\mathbb {I} {\vec {\nabla }}\times {\vec {f}}}
(63 )
(
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∓
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{\displaystyle (\nabla _{r}-\mathbb {I} {\vec {\nabla }}){\vec {g}}=\nabla _{r}\nabla _{r}{\vec {f}}+\nabla _{r}\mathbb {I} {\vec {\nabla }}f_{r}\pm \nabla _{r}\mathbb {I} {\vec {\nabla }}\times {\vec {f}}+\nabla _{r}\langle \mathbb {I} {\vec {\nabla }},{\vec {f}}\rangle \mp \nabla _{r}\mathbb {I} {\vec {\nabla }}\times {\vec {f}}+\langle \mathbb {I} {\vec {\nabla }},\mathbb {I} {\vec {\nabla }}\rangle f_{r}-\mathbb {I} {\vec {\nabla }}\langle \mathbb {I} {\vec {\nabla }},{\vec {f}}\rangle +\langle \mathbb {I} {\vec {\nabla }},\mathbb {I} {\vec {\nabla }}\rangle {\vec {f}}}
(64 )
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→
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∇
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∇
r
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∇
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{\displaystyle (\nabla _{r}-\mathbb {I} {\vec {\nabla }}){\vec {g}}=\nabla _{r}\nabla _{r}{\vec {f}}+\langle \mathbb {I} {\vec {\nabla }},\mathbb {I} {\vec {\nabla }}\rangle {\vec {f}}+\nabla _{r}\mathbb {I} {\vec {\nabla }}f_{r}+\nabla _{r}\langle \mathbb {I} {\vec {\nabla }},{\vec {f}}\rangle +\langle \mathbb {I} {\vec {\nabla }},\mathbb {I} {\vec {\nabla }}\rangle f_{r}-\mathbb {I} {\vec {\nabla }}\langle \mathbb {I} {\vec {\nabla }},{\vec {f}}\rangle }
(65 )
(
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→
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g
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−
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⟩
{\displaystyle (\nabla _{r}-\mathbb {I} {\vec {\nabla }})g=-\nabla _{r}\mathbb {I} {\vec {\nabla }}f_{r}+\mathbb {I} {\vec {\nabla }}\langle \mathbb {I} {\vec {\nabla }},{\vec {f}}\rangle +\nabla _{r}\nabla _{r}f_{r}-\nabla _{r}\langle \mathbb {I} {\vec {\nabla }},{\vec {f}}\rangle +\nabla _{r}\nabla _{r}{\vec {f}}+\langle \mathbb {I} {\vec {\nabla }},\mathbb {I} {\vec {\nabla }}\rangle {\vec {f}}+\nabla _{r}\mathbb {I} {\vec {\nabla }}f_{r}+\nabla _{r}\langle \mathbb {I} {\vec {\nabla }},{\vec {f}}\rangle +\langle \mathbb {I} {\vec {\nabla }},\mathbb {I} {\vec {\nabla }}\rangle f_{r}-\mathbb {I} {\vec {\nabla }}\langle \mathbb {I} {\vec {\nabla }},{\vec {f}}\rangle }
(66 )
ζ
=
(
∇
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→
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g
=
(
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→
)
(
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f
=
(
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∇
→
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→
⟩
)
(
f
r
+
f
→
)
+
I
∇
→
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I
∇
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,
f
→
⟩
−
I
∇
→
⟨
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,
f
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⟩
+
∇
r
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,
f
→
⟩
−
∇
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∇
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∇
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∇
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r
{\displaystyle \zeta =(\nabla _{r}-\mathbb {I} {\vec {\nabla }})g=(\nabla _{r}-\mathbb {I} {\vec {\nabla }})(\nabla _{r}+\mathbb {I} {\vec {\nabla }})f=(\nabla _{r}\nabla _{r}+\langle \mathbb {I} {\vec {\nabla }},\mathbb {I} {\vec {\nabla }}\rangle )(f_{r}+{\vec {f}})+\mathbb {I} {\vec {\nabla }}\langle \mathbb {I} {\vec {\nabla }},{\vec {f}}\rangle -\mathbb {I} {\vec {\nabla }}\langle \mathbb {I} {\vec {\nabla }},{\vec {f}}\rangle +\nabla _{r}\langle \mathbb {I} {\vec {\nabla }},{\vec {f}}\rangle -\nabla _{r}\langle \mathbb {I} {\vec {\nabla }},{\vec {f}}\rangle -\nabla _{r}\mathbb {I} {\vec {\nabla }}f_{r}+\nabla _{r}\mathbb {I} {\vec {\nabla }}f_{r}}
(67 )
ζ
=
(
∇
r
−
I
∇
→
)
g
=
(
∇
r
−
I
∇
→
)
(
∇
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+
I
∇
→
)
f
=
(
∇
r
∇
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−
⟨
∇
→
,
∇
→
⟩
)
(
f
r
+
f
→
)
−
∇
→
⟨
∇
→
,
f
→
⟩
+
∇
→
⟨
∇
→
,
f
→
⟩
+
I
(
∇
r
⟨
∇
→
,
f
→
⟩
−
∇
r
⟨
∇
→
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−
∇
r
∇
→
f
r
+
∇
r
∇
→
f
r
)
{\displaystyle \zeta =(\nabla _{r}-\mathbb {I} {\vec {\nabla }})g=(\nabla _{r}-\mathbb {I} {\vec {\nabla }})(\nabla _{r}+\mathbb {I} {\vec {\nabla }})f=(\nabla _{r}\nabla _{r}-\langle {\vec {\nabla }},{\vec {\nabla }}\rangle )(f_{r}+{\vec {f}})-{\vec {\nabla }}\langle {\vec {\nabla }},{\vec {f}}\rangle +{\vec {\nabla }}\langle {\vec {\nabla }},{\vec {f}}\rangle +\mathbb {I} (\nabla _{r}\langle {\vec {\nabla }},{\vec {f}}\rangle -\nabla _{r}\langle {\vec {\nabla }},{\vec {f}}\rangle -\nabla _{r}{\vec {\nabla }}f_{r}+\nabla _{r}{\vec {\nabla }}f_{r})}
(68 )
ζ
=
(
∇
r
∇
r
−
⟨
∇
→
,
∇
→
⟩
)
f
{\displaystyle \zeta =(\nabla _{r}\nabla _{r}-\langle {\vec {\nabla }},{\vec {\nabla }}\rangle )f}
(69 )
Thus also this quaternionic second order partial differential equation splits in two first order partial differential equations. But these are no quaternionic partial differential equations !
g
=
(
∇
r
+
I
∇
→
)
f
;
ζ
=
(
∇
r
−
I
∇
→
)
g
{\displaystyle g=(\nabla _{r}+\mathbb {I} {\vec {\nabla }})f;\qquad \zeta =(\nabla _{r}-\mathbb {I} {\vec {\nabla }})g}
(70 )