Abstract
Maxwell’s differential equations (503), presented in Sect. 3, Chap. 9, or Minskowski’s equations (580), are indispensable to solve problems in electrodynamics, the scattering and bending of electromagnetic waves, the derivation of the field of moving charges and currents and so on. On the other hand, we have seen that important results, following in general from tedious calculations, can be obtained relatively simple from an investigation of the underlying algebraic structure. An example is given by the discussion of the angular momentum operators in quantum mechanics, s. Sect. 5.1.1, Chap. 10. In this chapter, we shall present a formulation of electrodynamics, from which the algebraic structure of these equations becomes especially obvious. Such a formulation of the theory that makes especially visible its structural peculiarities is especially important for theoretical investigations, for global problems and for extensions of the theory.
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Notes
- 1.
To the covariant differentials defined by \(dx_i := g_{ik}dx^k\) there exist in general no global covariant coordinates \(x_i\). However, they exist in the case that the metric \(g_{ik}\) originates from the unit matrix \(\delta _{ik}\) ( \(\delta _{ik} = 1\quad \text{ for }\quad i = k\) and zero else) by a coordinate transformation, i.e. if \(g_{ik} = (\partial X^r/\partial x^i)(\partial X^s/\partial x^k)\delta _{rs}\) with Cartesian coordinates \(X^r \equiv X_r\) in an Euclidean space with metric \(\delta _{ik}\).
- 2.
In the older algebraic language, a form represents any homogeneous polynom. Now it is used only for linear differential forms. Forms of higher order are defined by the condition of antisymmetry.
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Günther, H., Müller, V. (2019). Electrodynamics in Exterior Calculus. In: The Special Theory of Relativity. Springer, Singapore. https://doi.org/10.1007/978-981-13-7783-9_11
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DOI: https://doi.org/10.1007/978-981-13-7783-9_11
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