Explicitly covariant form of the integral Maxwell equations
DOI:
https://doi.org/10.31349/RevMexFisE.18.76Keywords:
Maxwell integral equations in explicitly covariant form, Covariant Maxwell integral equations, Classical integral electrodynamics in covariant form.Abstract
It is analyzed in detail the covariance of the integral forms of the Maxwell Equations. Different forms of writing the integral Maxwell equations in explicitly covariant form are analyzed. First we show how one of these ways can be obtained from the differential Maxwell equations, both from their usual three vector form as from their covariant four vector form. Then we discuss how this covariant integral Maxwell equations can be obtained from usual integral Maxwell equations. It is emphasized the necessity to write the usual integral equations without time derivatives. The integrations regions in the three-dimensional space and time are identified with parts of the same hyper-surface in four-dimensional space-time. This point is carefully analyzed. Later we discuss other forms of the integral Maxwell equations. We show how these new versions can be expressed in an explicitly covariant form.
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Perhaps the most familiar form of the divergence theorem is
where is an arbitrary volume, the closed surface that surrounds it, and a vector valued function. If, for example, this theorem is applied to the function one obtains where is the projection of on the plane When this surface integral is evaluated using the parameters the projection is equal to (see Ref. [19]).
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