User:Peeter.joot

fixup and add back to Geometric_algebra

One of the most striking applications of the geometric product is the ability to formulate the eight Maxwell's equations in a coherent fashion as a single equation.

This isn't a new idea, and this has been done historically using formulations based on quaterions (~1910. dig up citation). A formulation in terms of antisymetric second rank tensors $F_{\mu \nu }$ and $G_{\mu \nu }$ (See: Formulation of Maxwell's equations in special relativity) reduces the eight equations to two, but also introduces complexity and obfuscates the connection to the physically measurable quantities.

A formulation in terms of differential forms Maxwell's_equations is also possible. This doesn't have the complexity of the tensor formulation, but requires the electromagnetic field to be expressed as a differential form. This is arguably strange given a traditional vector calculus education.

To explore the ideas, the starting point is the traditional set of Maxwell's equations

$\nabla \cdot \mathbf {E} ={\frac {\rho }{\epsilon _{0}}}$

$\nabla \cdot \mathbf {B} =0$

$\nabla \times \mathbf {E} +{\frac {\partial \mathbf {B} }{\partial t}}=0$

Failed to parse (Conversion error. Server ("/media/api/rest_") reported: "Cannot get mml. upstream connect error or disconnect/reset before headers. reset reason: connection termination"): {\displaystyle c^{2}\nabla \times \mathbf {B} -{\frac {\partial \mathbf {E} }{\partial t}}=\mathbf {J} /\epsilon _{0}}

It is customary in relativistic treatments of electrodynamics to introduce a four vector $(x,y,z,ict)$. Using this as a hint, one can write the time partials in terms of $ict$ and regrouping slightly

$\nabla \cdot \mathbf {E} ={\frac {\rho }{\epsilon _{0}}}$

$\nabla \cdot (ic\mathbf {B} )=0$

$\nabla \times \mathbf {E} +{\frac {\partial (ic\mathbf {B} )}{\partial (ict)}}=0$

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \nabla \times (ic\mathbf{B}) + \frac{\partial \mathbf{E}} {\partial (ict)} = i\mathbf{J}/{\epsilon_0 c} }

There is no use of geometric or wedge products here, but the opposing signs in the two sets of curl and time partial equations is removed. The pairs of equations can be added together without loss of information since the original equations can be recovered by taking real and imaginary parts.

$\nabla \cdot (\mathbf {E} +ic\mathbf {B} )={\frac {\rho }{\epsilon _{0}}}$

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \nabla \times (\mathbf E + ic \mathbf B) + \frac{\partial (\mathbf E + ic \mathbf B)} {\partial (ict)} = i\mathbf{J}/{\epsilon_0 c} }

It is thus natural to define a combined electrodynamic field as a complex vector, expressing the natural orthogality of the electric and magnetic fields

$\mathbf {F} =\mathbf {E} +ic\mathbf {B}$.

The electric and magnetic fields can be recovered from this composite field by taking real and imaginary parts respectively. With multiplication o of the second by a $-i$ factor to convert to a wedge product representation the remaining pair of equations can be written

$\nabla \cdot \mathbf {F} ={\frac {\rho }{\epsilon _{0}}}$

$-i\nabla \times \mathbf {F} -{\frac {\partial \mathbf {F} }{\partial (ct)}}=\mathbf {J} /{\epsilon _{0}c}=\nabla \wedge \mathbf {F} -{\frac {\partial \mathbf {F} }{\partial (ct)}}$

These equations can be added without loss

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \nabla \cdot \mathbf F + \nabla \wedge \mathbf F - \frac{\partial \mathbf F} {\partial (ct)} = \frac {\rho} {\epsilon_0} + \mathbf{J}/{\epsilon_0 c} }

Leading to the end result

Failed to parse (Conversion error. Server ("/media/api/rest_") reported: "Cannot get mml. upstream connect error or disconnect/reset before headers. reset reason: connection termination"): {\displaystyle (-{\frac {\partial }{\partial (ct)}}+\nabla )\mathbf {F} ={\frac {\rho }{\epsilon _{0}}}+\mathbf {J} /{\epsilon _{0}c}}