(1786) |

It is easily demonstrated that

(1787) |

in a right-handed coordinate system.

Consider the *dual electromagnetic field tensor*,
,
which is defined

(1788) |

This tensor is clearly an antisymmetric pseudo-4-tensor. We have

(1789) |

plus

(1790) |

where use has been made of . The previous expression yields

(1791) |

It follows that

(1792) | ||

(1793) |

or

(1794) |

The previous expression is, again, slightly misleading, because stands for the component of the pseudo-3-tensor , and not for an element of the proper-3-vector . Of course, in this case, really does represent the first element of the pseudo-3-vector . Note that the elements of are obtained from those of by making the transformation and .

The covariant elements of the dual electromagnetic field tensor are given by

(1795) | ||

(1796) |

or

(1797) |

The elements of are obtained from those of by making the transformation and .

Let us now consider the two Maxwell equations

(1798) | ||

(1799) |

The first of these equations can be written

because . The second equation takes the form

(1801) |

or

Equations (1802) and (1804) can be combined to give

(1803) |

Thus, we conclude that Maxwell's equations for the electromagnetic fields are equivalent to the following pair of 4-tensor equations:

(1804) | ||

(1805) |

It is obvious from the form of these equations that the laws of electromagnetism are invariant under translations, rotations, special Lorentz transformations, parity inversions, or any combination of these transformations.