the Jacobian of Lorentz Transform is:

The Maxwell’s equations can be viewed like this for matching the order and sign:

……………..(1)

……..(2)

and

……………….(3)

………..(4)

Lets simplify (2). the cross product can be changed to Einstein notation.

where . Set a matrix F be :

If we let the index i be zero. ,

the equation (2) can be written like :

where and . each column represent 1 equation. notices that the minus sign before the rot in equation (2) is automatically included.

If we extend the j to be zero, define , . the minus sign perseveres the matrix F to be anti-symmetric. Then, the equation 1 will be absorbed. if we define a 4-current.

the equation (1) and (2) can be combined into:

where

Using similar method, by defining a matrix G, such that:

The transformation of field is simply use the Jacobian. Since the 4-current also need to transform, and the Zero 4-vector in equation G also. Thus

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