Complementing the Lagrangian Density of the E . M . Field and the Surface Integral of the pv Vector Product

Considering the Lagrangian density of the electromagnetic field, a 4 × 4 transformation matrix is found which can be used to include two of the symmetrized Maxwell’s equations as one of the Euler-Lagrange equations of the complete Lagrangian density. The 4 × 4 transformation matrix introduces newly defined vector products. In a Theorem the surface integral of one of the newly defined vector products is shown to be reduced to a line integral.


Introduction
In physics there are four equations which are known as the Maxwell's equations.The two of these equations are the Gauss's law for the electric field and the Gauss's law for the magnetic field and the other two are the Ampère's law and the Faraday's law [1].In the theory for the massless electromagnetic vector field [2] the Gauss's law for the electric field and the Ampère's law are written in a succinct form as Equation (1) below and the Lagrangian density is found for this Equation (1).However, the Gauss's law for the magnetic field and Faraday's law are omitted apparently because of the fact that there are no magnetic charges detected yet.In this article the Faraday's law and the Gauss's law for the magnetic field are formulated as one tensor Equation (8) and the Lagrangian density is written for Equation (8).Even if there are no magnetic charges the Lagrangian density of the two omitted Maxwell's equations which contains the difference between the potential energy density and the kinetic energy density of the electromagnetic field [2] should not be omitted since the Ampère's law alone does not imply electromagnetic waves and it is the Ampère's law together with Faraday's law which among others implied the electromagnetic waves and there is therefore a need to complement the existing Lagrangian density [2].Reference [1] gives the Maxwell's equations in the integral form, however, using the existing mathematical theorems the Maxwell's equations can be written in differential form [3]. In this article the relation between the tensor of Equation ( 1) and the tensor of Equation ( 8) is found as a 4 × 4 matrix the components of which contain newly defined vector products which are called here as the parallel-vertical (abbreviated as the p-v) vector product and the parallel-horizontal vector product.In finding the Lagrangian from the Lagrangian density one should integrate over the volume concerned.
In this article the surface integral [4] of the parallelvertical vector product is shown to be reducible to a line integral.

The Maxwell's Equations in Tensor Forms
The four Maxwell's equations are written with tensor notations in more compact forms as the following two Equations [2] and , , , , , , The Lagrangian density which leads to Equation (1) as its Euler-Lagrange Equation of motion [5], is the following with v A being the vector potential with four components, the first component of which is the electrical po-tential, and the three other components of which are the components of the three dimensional vector potential A, the curl of which is the magnetic field B.
However, Equation ( 6) which is well known in the published literature does not lead to Equation (2) directly as its Euler-Lagrange equation of motion.
To write the complete Lagrangian density of the Maxwell's equations withterms for eventual magnetic monopoles, Equation ( 2) is written in a modified form as follows Equations ( 1) and ( 2) apparently seem to be less compact than the Maxwell's equations in the differential-form as presented in the article of the Journal PIER, mentioned here as reference [6].However, Equation ( 2) can be made to have a compacter form than its presented form of Equation ( 2) with the tensor notation, as its symmetrized form is shown as Equation (5).Equation ( 2) can be obtained from Equation ( 7) when the vector on the right hand side of Equation ( 7) is set to be equal to zero.Denoting the 4 × 4 ma-trix in Equation ( 7), as v K  , one may write Equation ( 7) in analogy with Equation (1) as follows One may presume, firstly, a transformation to exist which changes then a transformation matrix of the following form is found with two newly defined vector products in the components of the matrix of Equation (10), one of which is defined as follows One may call this product of two arbitrary vectors P and Q (having three components) in Equation (40), the "parallel-horizontal product".The second newly defined vector product is as follows this multiplication of two vectors in Equation ( 12) may be called the parallel-vertical multiplication.The singularity (division by zero) in the prefactor 1  E B of the transformation matrix (10) is holding in the free space [7] and the singularity of the mentioned prefactor is avoided in a space of charge densities where the electric and magnetic fields are not necessarily perpendicular to each other.
To write the Lagrangian density for the Equations ( 7) or (8) in a form analogous to Equation ( 6) one needs to define a new vector field Z  such that In this way one can see that the following equations hold for Z    The form of Equation ( 8) suggests easily the following Lagrangian density which would give the Equation ( 8) as its equation of motion through the following Euler-Lagrange equation with the canonical coordinate being v Z and its derivative being The total Lagrangian density of the electromagnetic field would be the sum 1 2    , the expressions of the added terms of which can be taken, respectively, from Equation (6) and Equation (18), the two equations which have analogous forms, but having their differently defined respective vectors, namely, v A and v Z in addition to the difference in the upper and lower tensor indices.In taking the derivative of the second term on the left hand side of Equation ( 19), and obtaining Equation (8), the factor 1 4 in Equation ( 18) is cancelled outduring the tensor algebra manipulations due to the Einstein summation convention that a repeated index in a multiplication of tensors implies a summation over the repeated index through all the conventional values of the concerned repeated index [8].
The electric field for a conductor or a semiconductor is proportional to the drift velocity and thus to the drift momentum as follows [1,5] When the following condition is valid Then using the approximation and the quantum me- The P vector in equation ( 12) may be replaced by the   operator.The quantity of the surface integral of the parallelvertical product can be reduced to a line integral around a curve in the counterclockwise direction analogous to Stokes' Theorem [4].
Let the parametric representation of a smooth differentiable surface be described by the following equation Let the parallel-vertical product be written as follows where Q is a continuously differentiable vector field def y z The following vector product is a vector field perpendicular to the surface And the unit vector perpendicular to the surface can be written as follows Then one can write the following mathematically stated theorem Theorem: Or: Proof: Using Green's Theorem [4] one can write