Quaternionic Differential Equations

### Generalized field equations

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Generalized field equations hold for all basic fields. Generalized field equations fit best in a quaternionic setting.

Quaternions consist of a real number valued scalar part and a three-dimensional spatial vector that represents the imaginary part.

The multiplication rule of quaternions indicates that several independent parts constitute the product. In this comment, we use a suffix  r {\displaystyle _{r}} to indicate the scalar real part of a quaternion, and we use  a → {\displaystyle {\vec {a}}} to indicate the imaginary vector part of quaternion  a {\displaystyle a} .

c = c r + c → = a b = ( a r + a → ) ( b r + b → ) = a r b r − ⟨ a → , b → ⟩ + a r b → + a → b r ± a → × b → {\displaystyle c=c_{r}+{\vec {c}}=ab=(a_{r}+{\vec {a}})(b_{r}+{\vec {b}})=a_{r}b_{r}-\langle {\vec {a}},{\vec {b}}\rangle +a_{r}{\vec {b}}+{\vec {a}}b_{r}{\color {Red}\pm }{\vec {a}}\times {\vec {b}}}

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(1)

The  ± {\displaystyle {\color {Red}\pm }} in equation (1) indicates that quaternions exist in right-handed and left-handed versions.

The formula can be used to check the completeness of a set of equations that follow from the application of the product rule.

The quaternionic conjugate of a is  a ∗ = a r − a → {\displaystyle a^{*}=a_{r}-{\vec {a}}} From the product rule follows the formula for the norm  | a | {\displaystyle |a|} of quaternion  a {\displaystyle a} .

| a | 2 = a a ∗ = ( a r + a → ) ( a r − a → ) = a r a r − ⟨ a → , a → ⟩ {\displaystyle |a|^{2}=aa^{*}=(a_{r}+{\vec {a}})(a_{r}-{\vec {a}})=a_{r}a_{r}-\langle {\vec {a}},{\vec {a}}\rangle }

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(2)

We define the quaternionic nabla as  ∇ = { ∂ ∂ τ , ∂ ∂ x , ∂ ∂ y , ∂ ∂ z } = ∇ r + ∇ → ; ∇ r = ∂ ∂ τ ; ∇ → = { ∂ ∂ x , ∂ ∂ y , ∂ ∂ z } {\displaystyle \nabla =\left\\{{\partial \over \partial \tau },{\partial \over \partial x},{\partial \over \partial y},{\partial \over \partial z}\right\\}=\nabla _{r}+{\vec {\nabla }};\quad \nabla _{r}={\partial \over \partial \tau };\quad {\vec {\nabla }}=\left\\{{\partial \over \partial x},{\partial \over \partial y},{\partial \over \partial z}\right\\}} .

The quaternionic nabla  ∇ {\displaystyle \nabla } acts like a multiplying operator. The (partial) differential  ∇ ψ {\displaystyle \nabla \psi } represents the full first-order change of field  ψ {\displaystyle \psi } . We assume that  φ = ∇ ψ {\displaystyle \varphi =\nabla \psi } exists in an enclosed region of the domain of  ψ {\displaystyle \psi } .

#### First order partial differential equation

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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 }}}

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(3)

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.

#### Subfields

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∇ → 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 }

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(4)

(Equation (4) 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}}}

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(5)

E →   = d e f   − ∇ r ψ → − ∇ → ψ r {\displaystyle {\vec {E}}\ {\overset {\underset {\mathrm {def} }{}}{=}}\ -\nabla _{r}{\vec {\psi }}-{\vec {\nabla }}\psi _{r}}

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(6)

B →   = d e f   ∇ → × ψ → {\displaystyle {\vec {B}}\ {\overset {\underset {\mathrm {def} }{}}{=}}\ {\vec {\nabla }}\times {\vec {\psi }}}

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(7)

J →   = d e f   ∇ → × B → − ∇ r E → {\displaystyle {\vec {J}}\ {\overset {\underset {\mathrm {def} }{}}{=}}\ {\vec {\nabla }}\times {\vec {B}}-\nabla _{r}{\vec {E}}}

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(8)

#### Properties of the spatial nabla operator

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The 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}}}}

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(9)

∇ → 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 }}}}

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(10)

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 }}}}

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(11)

⟨ ∇ → , ∇ → ⟩ 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}}}}}

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(12)

∇ → × 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}}}}

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(13)

= 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 }}}}

∇ → × 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 }}}}

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(14)

##### Special formulas

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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}}}}

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(15)

⟨ ∇ → , ∇ → 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}}

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(16)

⟨ ∇ → , ∇ → ⟩ a →   = d e f   ∇ 2 a → {\displaystyle \langle {\vec {\nabla }},{\vec {\nabla }}\rangle {\vec {a}}\ {\overset {\underset {\mathrm {def} }{}}{=}}\ \nabla ^{2}{\vec {a}}}

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(17)

⟨ ∇ → × ∇ → , a → ⟩ = 0 {\displaystyle \langle {\vec {\nabla }}\times {\vec {\nabla }},{\vec {a}}\rangle =0}

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(18)

∇ → × ∇ → a r = 0 → {\displaystyle {\vec {\nabla }}\times {\vec {\nabla }}a_{r}={\vec {0}}}

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(19)

⟨ ∇ → , ∇ → × a → ⟩ = 0 {\displaystyle \langle {\vec {\nabla }},{\vec {\nabla }}\times {\vec {a}}\rangle =0}

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(20)

∇ → × ( ∇ → × 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}}}

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(21)

∇ → ∇ → a = ( ∇ → × ∇ → ) a − ⟨ ∇ → , ∇ → ⟩ a === Q u a t e r ( ∇ → × ∇ → ) a → − ⟨ ∇ → , ∇ → ⟩ a = ∇ → ⟨ ∇ → , a → ⟩ − 2 ⟨ ∇ → , ∇ → ⟩ a + ⟨ ∇ → , ∇ → ⟩ a r {\displaystyle {\vec {\nabla }}{\vec {\nabla }}a=({\vec {\nabla }}\times {\vec {\nabla }})a-\langle {\vec {\nabla }},{\vec {\nabla }}\rangle a===Quater({\vec {\nabla }}\times {\vec {\nabla }}){\vec {a}}-\langle {\vec {\nabla }},{\vec {\nabla }}\rangle a={\vec {\nabla }}\langle {\vec {\nabla }},{\vec {a}}\rangle -2\langle {\vec {\nabla }},{\vec {\nabla }}\rangle a+\langle {\vec {\nabla }},{\vec {\nabla }}\rangle a_{r}}

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(22)

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}}}

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(23)

⟨ ∇ → , q → ⟩ = 3 {\displaystyle \langle {\vec {\nabla }},{\vec {q}}\rangle =3}

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(24)

∇ → × q → = 0 → {\displaystyle {\vec {\nabla }}\times {\vec {q}}={\vec {0}}}

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(25)

∇ → | q → | = q → | q → | ;   ∇ | q | = q | q | {\displaystyle {\vec {\nabla }}\,|{\vec {q}}|={\frac {\vec {q}}{|{\vec {q}}|}};\ \nabla \,|q|={\frac {q}{|q|}}}

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(26)

∇ → 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}}}}

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(27)

⟨ ∇ → , ∇ → 1 | q → − q → ′ | ⟩ = ⟨ ∇ → , ∇ → ⟩ 1 | q → − q → ′ | = − ⟨ ∇ → , q → − q → ′ | q → − q → ′ | 3 ⟩ = 4 ϕ 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\,\phi \,\delta ({\vec {q}}-{\vec {q}}^{'})}

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(28)

∇   q = 1 − 3 ; ∇ ∗ q = 1 + 3 ; ∇ q ∗ = 1 + 3 {\displaystyle \nabla \ q=1-3;\nabla ^{*}q=1+3;\nabla q^{*}=1+3}

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(29)

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}

##### Derivation of higher order equations

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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}

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(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}}

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(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}}

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(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}}

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(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}}}

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(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 }}}

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(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}}}

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(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}}

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(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 }

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(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}}

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(39)

∇ → 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 }

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(40)

∇ 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 }

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(41)

∇ ∗ 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 }

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(42)

∇ ∗ φ = ( ⟨ ∇ → , ∇ → ⟩ + ∇ r ∇ r ) ψ → + ∇ r ∇ → ψ r − ∇ → ⟨ ∇ → , ψ → ⟩ + ∇ r ⟨ ∇ → , ψ → ⟩ + ⟨ ∇ → , ∇ → ⟩ ψ r − ∇ r ∇ → ψ r − ∇ → ⟨ ∇ → , ψ → ⟩ + ∇ r ∇ r ψ r − ∇ r ⟨ ∇ → , ψ → ⟩ {\displaystyle \nabla ^{*}\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}-\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 }

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(43)

χ = ∇ ∗ φ = ( ⟨ ∇ → , ∇ → ⟩ + ∇ r ∇ r ) ψ + ∇ r ∇ → ψ r − ∇ → ⟨ ∇ → , ψ → ⟩ + ∇ r ⟨ ∇ → , ψ → ⟩ − ∇ r ∇ → ψ r − ∇ → ⟨ ∇ → , ψ → ⟩ − ∇ r ⟨ ∇ → , ψ → ⟩ = ( ⟨ ∇ → , ∇ → ⟩ + ∇ r ∇ r ) ψ {\displaystyle \chi =\nabla ^{*}\varphi =(\langle {\vec {\nabla }},{\vec {\nabla }}\rangle +\nabla _{r}\nabla _{r})\psi +{\color {red}\nabla _{r}{\vec {\nabla }}\psi _{r}}-{\color {blue}{\vec {\nabla }}\langle {\vec {\nabla }},{\vec {\psi }}\rangle }+{\color {green}\nabla _{r}\langle {\vec {\nabla }},{\vec {\psi }}\rangle }-{\color {red}\nabla _{r}{\vec {\nabla }}\psi _{r}}-{\color {blue}{\vec {\nabla }}\langle {\vec {\nabla }},{\vec {\psi }}\rangle }-{\color {green}\nabla _{r}\langle {\vec {\nabla }},{\vec {\psi }}\rangle }=(\langle {\vec {\nabla }},{\vec {\nabla }}\rangle +\nabla _{r}\nabla _{r})\psi }

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(44)

Most of the terms vanish. 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 }

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(45)

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.

#### Second order partial differential equations

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We start with the quaternionic equivalent of the Maxwell-Faraday equation.

∇ r B → = ∇ → × ( ∇ r ψ → ) = − ∇ → × E → {\displaystyle \nabla _{r}{\vec {B}}={\vec {\nabla }}\times (\nabla _{r}{\vec {\psi }})=-{\vec {\nabla }}\times {\vec {E}}}

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(46)

Two interesting second order quaternionic partial differential equations exist.

ζ = ( ∇ r ∇ r − ⟨ ∇ → , ∇ → ⟩ ) ψ {\displaystyle \zeta =(\nabla _{r}\nabla _{r}-\langle {\vec {\nabla }},{\vec {\nabla }}\rangle )\psi }

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(47)

This is the quaternionic equivalent of the wave equation. It offers waves as part of the solutions of the homogeneous equation.

χ = ∇ ∗ φ = ∇ ∗ ∇ ψ = ∇ ∇ ∗ ψ = ( ∇ r − ∇ → ) ( φ r + φ → ) = ( ∇ r + ∇ → ) ( ∇ r − ∇ → ) ( ψ r + ψ → ) = ( ∇ r ∇ r + ⟨ ∇ → , ∇ → ⟩ ) ψ {\displaystyle \chi =\nabla ^{*}\varphi =\nabla ^{*}\nabla \psi =\nabla \nabla ^{*}\psi =(\nabla _{r}-{\vec {\nabla }})(\varphi _{r}+{\vec {\varphi }})=(\nabla _{r}+{\vec {\nabla }})(\nabla _{r}-{\vec {\nabla }})(\psi _{r}+{\vec {\psi }})=(\nabla _{r}\nabla _{r}+\langle {\vec {\nabla }},{\vec {\nabla }}\rangle )\psi }

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(48)

This equation can be split into two first order partial wave equations  χ = ∇ ∗ φ {\displaystyle \chi =\nabla ^{*}\varphi } and  φ = ∇ ψ {\displaystyle \varphi =\nabla \psi } .

This equation does not offer waves as part of the solutions of the homogeneous equation.

#### Differential operators

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D = ( ∇ r ∇ r − ⟨ ∇ → , ∇ → ⟩ ) {\displaystyle D=(\nabla _{r}\nabla _{r}-\langle {\vec {\nabla }},{\vec {\nabla }}\rangle )}

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(49)

This is the quaternionic equivalent of d'Alembert's operator.

⊡ = ( ∇ r ∇ r + ⟨ ∇ → , ∇ → ⟩ ) {\displaystyle \boxdot =(\nabla _{r}\nabla _{r}+\langle {\vec {\nabla }},{\vec {\nabla }}\rangle )}

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(50)

This operator does not yet have a known name.

Operator  ⟨ ∇ → , ∇ → ⟩ {\displaystyle \langle {\vec {\nabla }},{\vec {\nabla }}\rangle } represents the main part of the Poisson equation. Together with  ∇ r ∇ r {\displaystyle \nabla _{r}\nabla _{r}} this operator configures the above operators.

As is shown above,  ⊡ {\displaystyle \boxdot } can be derived from the nabla operator. That cannot be said from the  D {\displaystyle D} operator.