# Direct derivation of Liénard–Wiechert potentials, Maxwell’s equations and
Lorentz force from Coulomb’s law

Hrvoje Dodig  University of Split, Ruđera Boškovića 37, 21000 Split, Croatia
<EMAIL_ADDRESS>
###### Abstract

In 19th century Maxwell derived famous Maxwell equations from the knowledge of
three experimental physical laws: the electrostatic Coulomb’s law, Ampere’s
force law and Faraday’s law of induction. However, theoretical basis for
Ampere’s force law and Faraday’s law remains unknown to this day. Furthermore,
the Lorentz force is considered as summary of experimental phenomena, the
theoretical foundation that explains generation of this force is still
unknown.

To answer these fundamental theoretical questions, in this paper we derive
relativistically correct Liénard – Wiechert potentials, Maxwell’s equations
and Lorentz force from two simple postulates: (a) when all charges are at rest
the Coulomb’s force acts between the charges, and (b) that disturbances caused
by charge in motion propagate away from the source with finite velocity. The
special relativity was not used in our derivations nor the Lorentz
transformation. In effect, it was shown in this paper that all the
electrodynamic laws, including the Lorentz force, can be derived from
Coulomb’s law and time retardation.

This was accomplished by analysis of hypothetical experiment where test charge
is at rest and where previously moving source charge stops at some time in the
past. Then the generalized Helmholtz decomposition theorem, also derived in
this paper, was applied to reformulate Coulomb’s force acting at present time
as the function of positions of source charge at previous time when the source
charge was moving. From this reformulation of Coulomb’s law the
Liénard–Wiechert potentials and Maxwell’s equations were derived by careful
mathematical manipulation.

In the second part of this paper, the energy conservation principle valid for
moving charges is derived from the knowledge of electrostatic energy
conservation principle valid for stationary charges. This again was
accomplished by using generalized Helmholtz decomposition theorem. From this
dynamic energy conservation principle the Lorentz force is finally derived.

###### keywords:

Coulomb’s law , Liénard–Wiechert potentials , Maxwell equations , Lorentz
force

††journal: arXiv

##  1 Introduction

In his famous _Treatise_[1, 2] Maxwell derived equations of electrodynamics
based on the knowledge about the three experimental laws known at the time:
the Coulomb’s law describing the electric force between charges at rest;
Ampere’s law describing the force between current carrying wires, and the
Faraday’s law of induction. Prior to Maxwell, magnetism and electricity were
regarded to as separate phenomena. It was James Clerk Maxwell who unified
these seemingly disparate phenomena into the set of equations collectively
known today as Maxwell’s equations. In modern vector notation, the four
Maxwell’s equations that govern the behavior of electromagnetic fields are
written as:

$\displaystyle\nabla\cdot\boldsymbol{\mathbf{D}}$ $\displaystyle=\rho$ (1)
$\displaystyle\nabla\cdot\boldsymbol{\mathbf{B}}$ $\displaystyle=0$ (2)
$\displaystyle\nabla\times\boldsymbol{\mathbf{E}}$
$\displaystyle=-\frac{\partial\boldsymbol{\mathbf{B}}}{\partial t}$ (3)
$\displaystyle\nabla\times\boldsymbol{\mathbf{B}}$
$\displaystyle=\mu\boldsymbol{\mathbf{J}}+\frac{1}{c^{2}}\frac{\partial\boldsymbol{\mathbf{E}}}{\partial
t}$ (4)

where symbol $\boldsymbol{\mathbf{D}}$ denotes electric displacement vector,
$\boldsymbol{\mathbf{E}}$ is electric field vector, $\boldsymbol{\mathbf{B}}$
is a vector called magnetic flux density, vector $\boldsymbol{\mathbf{J}}$ is
called current density and scalar $\rho$ is the charge density. Furthermore,
there are two more important equations in electrodynamics that relate magnetic
vector potential $\boldsymbol{\mathbf{A}}$ and scalar potential $\phi$ to
electromagnetic fields $\boldsymbol{\mathbf{B}}$ and
$\boldsymbol{\mathbf{E}}$:

$\displaystyle\boldsymbol{\mathbf{B}}$
$\displaystyle=\nabla\times\boldsymbol{\mathbf{A}}$ (5)
$\displaystyle\boldsymbol{\mathbf{E}}$
$\displaystyle=-\nabla\phi-\frac{\partial\boldsymbol{\mathbf{A}}}{\partial t}$
(6)

In standard electromagnetic theory, if a point charge $q_{s}$ is moving with
velocity $\boldsymbol{\mathbf{v}}_{s}(t)$ along arbitrary path
$\boldsymbol{\mathbf{r}}_{s}(t)$ the scalar potential $\phi$ and vector
potential $\boldsymbol{\mathbf{A}}$ caused by moving charge $q_{s}$ are
described by well known, relativistically correct, Liénard–Wiechert potentials
[3, 4]:

$\displaystyle\phi$
$\displaystyle=\phi(\boldsymbol{\mathbf{r}},t)=\frac{1}{4\pi\epsilon}\left(\frac{q_{s}}{\left(1-\boldsymbol{\mathbf{n}}_{s}(t_{r})\cdot\boldsymbol{\mathbf{\beta}}_{s}(t_{r})\right)\left|\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}_{s}(t_{r})\right|}\right)$
(7) $\displaystyle\boldsymbol{\mathbf{A}}$
$\displaystyle=\boldsymbol{\mathbf{A}}(\boldsymbol{\mathbf{r}},t)=\frac{\mu
c}{4\pi}\left(\frac{q_{s}\boldsymbol{\mathbf{\beta}}_{s}(t_{r})}{\left(1-\boldsymbol{\mathbf{n}}_{s}(t_{r})\cdot\boldsymbol{\mathbf{\beta}}_{s}(t_{r})\right)\left|\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}_{s}(t_{r})\right|}\right)$
(8)

where $t_{r}$ is retarded time, $\boldsymbol{\mathbf{r}}$ is the position
vector of observer and vectors $\boldsymbol{\mathbf{n}}_{s}(t_{r})$ and
$\boldsymbol{\mathbf{\beta}}_{s}(t_{r})$ are:

$\displaystyle\boldsymbol{\mathbf{\beta}}(t_{r})$
$\displaystyle=\frac{\boldsymbol{\mathbf{v}}_{s}(t_{r})}{c}$ (9)
$\displaystyle\boldsymbol{\mathbf{n}}(t_{r})$
$\displaystyle=\frac{\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}_{s}(t_{r})}{\left|\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}_{s}(t_{r})\right|}$
(10)

These equations were almost simultaneously discovered by Liénard and Wiechert
around 1900’s and they represent explicit expressions for time-varying
electromagnetic fields caused by charge in arbitrary motion. Nevertheless,
Liénard-Wiechert potentials were derived from retarded potentials, which in
turn, are derived from Maxwell equations.

Maxwell’s electrodynamic equations provide the complete description of
electromagnetic fields, however, these equations say nothing about mechanical
forces experienced by the charge moving in electromagnetic field. If the
charge $q$ is moving in electromagnetic field with velocity
$\boldsymbol{\mathbf{v}}$ then the force $\boldsymbol{\mathbf{F}}$ experienced
by the charge $q$ is:

$\boldsymbol{\mathbf{F}}=q\left(\boldsymbol{\mathbf{E}}+\boldsymbol{\mathbf{v}}\times\boldsymbol{\mathbf{B}}\right)$
(11)

The force described by equation (11) is well known Lorentz force. Discovery of
this electrodynamic force is historically credited to H.A. Lorentz [5],
however, the similar expression for electromagnetic force can be found in
Maxwell’s _Treatise_ , article 598 [2]. The difference between the two is that
Maxwell’s electromotive force acts on moving circuits and Lorentz force acts
on moving charges.

However, it is not yet explained what causes the Lorentz force, Ampere’s force
law and Faraday’s law. Maxwell derived his expression for electromotive force
along moving circuit from the knowledge of experimental Faraday’s law. Later,
Lorentz extended Maxwell’s reasoning to discover the force acting on charges
moving in electromagnetic field [5]. Nevertheless, it would be impossible for
Lorentz to derive his force law without the prior knowledge of Maxwell
equations [6].

Nowadays, the Lorentz force
($q\boldsymbol{\mathbf{v}}\times\boldsymbol{\mathbf{B}}$ term) is commonly
viewed as an effect of Einstein’s special relativity. For example, an observer
co-moving with source charge would not measure any magnetic field, while on
the other hand, the stationary observer would measure the magnetic field
caused by moving source charge. However, in this work, we demonstrate that the
special relativity is not needed to derive the Lorentz force and Maxwell
equations. In fact, we derive Maxwell’s equations and Lorentz force from more
fundamental principles: the Coulomb’s law and time retardation.

There is another reason why the idea to derive Maxwell’s equations and Lorentz
force from Coulomb’s law may seem plausible. Because of mathematical
similarity between Coulomb’s law and Newton’s law of gravity many researchers
thought that if Maxwell’s equations and Lorentz force could be derived from
Coulomb’s law that this would be helpful in understanding of gravity. These
two inverse-square physical laws are written:

$\displaystyle\boldsymbol{\mathbf{F}}_{C}$
$\displaystyle=\frac{q_{1}q_{2}}{4\pi\epsilon}\frac{\boldsymbol{\mathbf{r}}_{1}-\boldsymbol{\mathbf{r}}_{2}}{\left|\boldsymbol{\mathbf{r}}_{1}-\boldsymbol{\mathbf{r}}_{2}\right|^{3}}$
(12) $\displaystyle\boldsymbol{\mathbf{F}}_{G}$
$\displaystyle=-Gm_{1}m_{2}\frac{\boldsymbol{\mathbf{r}}_{1}-\boldsymbol{\mathbf{r}}_{2}}{\left|\boldsymbol{\mathbf{r}}_{1}-\boldsymbol{\mathbf{r}}_{2}\right|^{3}}$
(13)

The expressions for Coulomb’s force and Newton’s gravitational force are
indeed similar, however, these two forces significantly differ in physical
nature. The latter force is always attractive while the former can be either
attractive or repulsive. Nevertheless, a number of researchers attempted to
derive Maxwell’s equations from Coulomb’s law, and most of these attempts rely
on Lorentz transformation of space-time coordinates between the rest frame of
the moving charge and laboratory frame.

The first hint that Maxwell’s equations could be derived from Coulomb’s law
and Lorentz transformation can be found in Einstein’s original 1905 paper on
special relativity [7]. Einstein suggested that the Lorentz force term
($\boldsymbol{\mathbf{v}}\times\boldsymbol{\mathbf{B}}$) is to be attributed
to Lorentz transformation of the electrostatic field from the rest frame of
moving charge to the laboratory frame where the charge has constant velocity.
Later, in 1912, Leigh Page derived Faraday’s law and Ampere’s law from
Coulomb’s law using Lorentz transformation [8]. Frisch and Willets discussed
the derivation of Lorentz force from Coulomb’s law using relativistic
transformation of force [9]. Similar route to derivation of Maxwell’s
equations and Lorentz force from Coulomb’s law was taken by Elliott in 1966
[10]. Kobe in 1986 derives Maxwell’s equations as the generalization of
Coulomb’s law using special relativity [11]. Lorrain and Corson derive Lorentz
force from Coulomb’s law, again, by using Lorentz transformation and special
relativity[12]. Field in 2006 derives Lorentz force and magnetic field
starting from Coulomb’s law by relating the electric field to electrostatic
potential in a manner consistent with special relativity [13]. The most recent
attempt comes from Singal [14] who attempted to derive electromagnetic fields
of accelerated charge from Coulomb’s law and Lorentz transformation.

All of the mentioned attempts have in common that they attempt to derive
Maxwell equations from Coulomb’s law by exploiting Lorentz transformation or
Einstein’s special theory of relativity. However, historically the Lorentz
transformation was derived from Maxwell’s equations [15], thus, the attempt to
to derive Maxwell’s equations using Lorentz transformation seems to involve
circular reasoning [16]. The strongest criticism came from Jackson who pointed
out that it should be immediately obvious that, without additional
assumptions, it is impossible to derive Maxwell’s equations from Coulomb’s law
using theory of special relativitiy [17]. Schwartz addresses these additional
assumptions and starting from Gauss’ law of electrostatics and by exploiting
the Lorentz invariance and properties of Lorentz transformation he derives the
Maxwell’s equations [18].

In addition to the criticism above, we point out that the derivations of
Maxwell’s equations from Coulomb’s law using Lorentz transformation should
only be considered valid for the special case of the charge moving along the
straight line with constant velocity. This is because the Lorentz
transformation is derived under the assumption that electron moves with
constant velocity along straight line [15]. For example, if the particle moves
with uniform acceleration along straight line the transformation of
coordinates between the rest frame of the particle and the laboratory frame
takes the different mathematical form than that of the Lorentz transformation
[19]. If the particle is in uniform circular motion yet another coordinate
transformation from the rest frame to laboratory frame, called Franklin
transform, is valid [20]. None of the above cited papers consider the fact
that Lorentz transformation is no longer valid when the charge is not moving
along straight line with constant velocity.

Figure 1: In (a) the source charge $q_{s}$ is at rest at present time $t_{p}$.
Each point on closed contour $C$ is affected by Coulomb’s electrostatic field
$\boldsymbol{\mathbf{E}}_{c}$. Energy conservation principle at present time
is $\oint_{C}{\boldsymbol{\mathbf{E}}_{c}\cdot d\boldsymbol{\mathbf{r}}}=0$.
In (b) the source charge $q_{s}$ is moving along arbitrary path and it stops
at past time $t_{s}<t_{p}$. Dynamic energy conservation principle valid in the
past is assumed to be unknown when source charge was moving.

To circumvent problems with special relativity and Lorentz transformation we
take entirely different approach to derive Liénard–Wiechert potentials and
Maxwell’s equations from Coulomb law. We start our derivation from the
analysis of the following hypothetical experiment: consider two charges at
rest at present time, one called the test charge, and the other called the
source charge. The source charge was moving in the past but it is at rest at
present time. Because both charges are at rest at present the force acting on
test charge at present time is the Coulomb’s force.

However, in the past when the source charge was moving, we assume that the
force acting on test charge was not the Coulomb’s force. To discover the
mathematical form of this ”unknown” electrodynamic force acting in the past
from the knowledge of known electrostatic force (Coulomb’s law) acting at
present time the generalized Helmholtz decomposition theorem was applied. This
theorem, derived in A, allowed us to relate Coulomb’s force acting at present
time to the positions of source charge at past time. From here,
Liénard–Wiechert potentials and Maxwell’s equations were derived by careful
mathematical manipulation.

It should be emphasized that we did not resort to theory of special relativity
nor to Lorentz transformation in our derivation of Maxwell’s equations. Not
less importantly, the presented derivation of Maxwell’s equations from
Coulomb’s law is valid for charges in arbitrary motion. In effect, we may say
that more general physical law (Maxwell’s equations) acting at past time is
derived from the knowledge of limited physical law (Coulomb’s law) acting at
present time.

However, from Maxwell’s equations, it is very difficult, if not entirely
impossible, to derive the Lorentz force without resorting to some form of
energy conservation law. As shown in Fig. 1a, at present time, the single
stationary charge creates Coulomb’s electrostatic field. Known energy
conservation law valid at present time states that contour integral of
Coulomb’s field along closed contour $C$ is equal to zero.

But, this electrostatic energy conservation law valid at present is not
necessarily valid in the past when the source charge was moving. Thus, in the
second part of this paper we derive this ”unknown” dynamic energy conservation
principle valid in the past from the knowledge of electrostatic energy
conservation principle valid at present time. This was again achieved by the
careful application of generalized Helmholtz decomposition theorem which
allowed us to transform electrostatic energy conservation law valid at present
to dynamic energy conservation law valid in the past.

This dynamic energy conservation law states that the work of non-conservative
force along closed contour is equal to the time derivative of the flux of
certain vector field through the surface bounded by this closed contour. From
this dynamic energy conservation law the Lorentz force was finally derived.

##  2 Generalized Helmholtz decomposition theorem

Because generalized Helmholtz decomposition theorem is central for deriving
Maxwell equations and Lorentz force from Coulomb’s law, in this section, we
briefly present this important theorem while the derivation itself is moved to
A. There have been several previous attempts in the literature to generalize
classical Helmholtz decomposition theorem to time dependent vector fields [21,
22, 23]. However, in none of the cited articles the Helmholtz theorem for
functions of space and time is presented in the mathematical form usable for
the mathematical developments described in this paper. This is probably caused
by difficulties in stating such a theorem and this was clearly stated in [22]:
”There does not exist any simple generalization of this theorem for time-
dependent vector fields”.

However, we show that there indeed exists the simple generalization of
Helmholtz decomposition theorem for time-dependent vector fields and that it
can derived from time-dependent inhomogeneous wave equation. To improve the
clarity of this paper, the complete derivation of Helmholtz decomposition
theorem for functions of space and time is moved to A, subsection A.2. As it
was shown in A, the generalization of Helmholtz decomposition theorem for the
vector function of space and time
$\boldsymbol{\mathbf{F}}(\boldsymbol{\mathbf{r}},t)$ can be written as:

$\displaystyle\boldsymbol{\mathbf{F}}(\boldsymbol{\mathbf{r}},t)=$
$\displaystyle-\nabla\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\bigg{(}\nabla^{\prime}\cdot\boldsymbol{\mathbf{F}}(\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})\bigg{)}G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}$
(14) $\displaystyle+\frac{1}{c^{2}}\frac{\partial}{\partial
t}\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\left(\frac{\partial}{\partial
t^{\prime}}\boldsymbol{\mathbf{F}}(\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})\right)G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}$
$\displaystyle+\nabla\times\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\bigg{(}\nabla^{\prime}\times\boldsymbol{\mathbf{F}}(\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})\bigg{)}G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}$

where scalar function
$G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})$ is
the fundamental solution of time-dependent inhomogeneous wave equation given
as:

$\nabla^{2}G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})-\frac{1}{c^{2}}\frac{\partial^{2}}{\partial
t^{2}}G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})=-\delta(\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}^{\prime})\delta(t-t^{\prime})$
(15)

In the equation above,
$\delta(\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}^{\prime})=\delta(x-x^{\prime})\delta(y-y^{\prime})\delta(z-z^{\prime})$
is 3D Dirac delta function, and $\delta(t-t^{\prime})$ is Dirac delta function
in one dimension. Fundamental solution
$G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})$,
sometimes called Green’s function, represents the retarded in time solution of
the inhomogeneous time dependent wave equation and it can be written as:

$G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})=\frac{\delta\left(t^{\prime}-t+\frac{\left|\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}^{\prime}\right|}{c}\right)}{4\pi\left|\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}^{\prime}\right|}$
(16)

where position vector $\boldsymbol{\mathbf{r}}^{\prime}$ is the location of
the source at time $t^{\prime}$.

From equation (14) it is evident that the Helmholtz decomposition theorem for
functions of space and time can be regarded to as a mathematical tool that
allows us to rewrite any vector function that is function of present time $t$
and of present position $\boldsymbol{\mathbf{r}}$ as vector function of
previous time $t^{\prime}$ and of previous position
$\boldsymbol{\mathbf{r^{\prime}}}$. Furthermore, the generalized Helmholtz
decomposition theorem (14) comes with additional limitation that it is valid
if vector function
$\boldsymbol{\mathbf{F}}(\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})$
approaches zero faster than
$1/\left|\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}^{\prime}\right|$ as
$\left|\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}^{\prime}\right|\rightarrow\infty$.

Very similar theorem was presented in article written by Heras [24]; the
difference is that in Heras’ article the time integrals in equation (14) were
a priori evaluated at retarded time
$t^{\prime}=t-\left|\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}^{\prime}\right|/c$.
As such, the generalized Helmholtz theorem presented in [24] is not suitable
for the derivation of Maxwell equations and Lorentz force from the Coulomb’s
law. Reason for this, as it will become evident later in this paper, is that
if we immediately evaluate the time integrals in equation (14) the important
information is lost from the equation.

##  3 Derivation of Maxwell Equations from Coulomb’s law

In this section we derive Maxwell equations from Coulomb’s law using
generalized Helmholtz decomposition theorem represented by equation (14). To
begin the discussion, we consider hypothetical experiment shown in Fig. 2,
where source charge $q_{s}$ is moving along trajectory
$\boldsymbol{\mathbf{r}}_{s}(t)$ and it stops at some past time $t_{s}$. The
test charge $q$ is stationary at all times. We assume that the disturbances
caused by moving source charge propagate outwardly from the source charge with
finite velocity $c$. These disturbances originating from the source charge at
past time manifest itself as the force acting on stationary test charge at
present time. This means that there is a time delay $\Delta t$ between the
past time $t_{s}$ when the source charge has stopped and the present time
$t_{p}$ when this disturbance has propagated to the test charge:

$\Delta
t=t_{p}-t_{s}=\frac{\left|\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}_{s}(t_{s})\right|}{c}$
(17)

Figure 2: Source charge $q_{s}$ is moving along arbitrary trajectory
$\boldsymbol{\mathbf{r}}_{s}(t)$ and stops at time $t_{s}$. Because $q_{s}$
stops moving at past time $t_{s}$, at present time $t_{p}$, the stationary
test charge $q$ experiences electrostatic Coulomb force.

At precise moment in time $t_{p}$, that we call the present time, the force
acting on stationary test charge $q$ is the Coulomb’s force because source
charge and test charge are both at rest, and because the effect of source
charge stopping at past time $t_{s}$ had enough time to propagate to test
charge. The Coulomb’s force $\boldsymbol{\mathbf{F}}_{c}$ experienced by the
test charge $q$ at present time $t_{p}$ can be expressed by the following
equation:

$\boldsymbol{\mathbf{F}}_{c}(\boldsymbol{\mathbf{r}},t_{p})=\frac{qq_{s}}{4\pi\epsilon}\frac{\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}_{s}(t_{s})}{\left|\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}_{s}(t_{s})\right|^{3}}\hskip
30.00005ptt_{p}=t_{s}+\Delta t$ (18)

Let us now consider the time $t$ just one brief moment before the stopping
time $t_{s}$:

$t=t_{s}-\delta t$ (19)

where $\delta t\rightarrow 0$ is very small time interval. This time interval
$\delta t$ is so small that we might even call it infinitesimally small. Then
at the moment in time infinitesimally before the present time $t_{p}$ the
force felt by test charge $q$ is still the Coulomb’s force if $\delta
t\rightarrow 0$. Using these considerations, we can now rewrite equation (18)
as:

$\boldsymbol{\mathbf{F}}_{c}(\boldsymbol{\mathbf{r}},t_{p}-\delta
t)=\frac{qq_{s}}{4\pi\epsilon}\frac{\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}_{s}(t)}{\left|\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}_{s}(t)\right|^{3}}\hskip
30.00005ptt=t_{s}-\delta t;\hskip 6.00006pt\delta t\rightarrow 0$ (20)

Note that equation (20) is equivalent to equation (18) when $\delta
t\rightarrow 0$. The reason why we have written the Coulomb’s law this way is
to permit slight variation of time before stopping time $t_{s}$ so that we can
exploit generalized Helmholtz decomposition theorem in order to derive
Maxwell’s equations from Coulomb’s law. Had we not done this then the source
charge position vector $\boldsymbol{\mathbf{r}}_{s}(t_{s})$ would simply be
the constant vector and generalized Helmholtz decomposition could not be used.

Because the right hand side of equation (20) is now the function of time $t$
and position $\boldsymbol{\mathbf{r}}$ we are allowed to use the generalized
Helmholtz decomposition theorem to rewrite the right hand side of equation
(20). This is because generalized Helmholtz decomposition theorem states that
_any_ vector function of time $t$ and position $\boldsymbol{\mathbf{r}}$ can
be decomposed as described by this theorem if that function meets certain
criteria. Thus, using generalized Helmholtz decomposition theorem we can
rewrite the right hand side of equation (20) as:

$\displaystyle\boldsymbol{\mathbf{F}}_{c}(\boldsymbol{\mathbf{r}},t_{p}-\delta
t)=\frac{qq_{s}}{4\pi\epsilon}\frac{\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}_{s}(t)}{\left|\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}_{s}(t)\right|^{3}}=$
$\displaystyle\hskip 50.00008pt=$
$\displaystyle-\nabla\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\bigg{(}\nabla^{\prime}\cdot\frac{qq_{s}}{4\pi\epsilon}\frac{\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|^{3}}\bigg{)}G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}$
$\displaystyle+\frac{1}{c^{2}}\frac{\partial}{\partial
t}\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\left(\frac{\partial}{\partial
t^{\prime}}\frac{qq_{s}}{4\pi\epsilon}\frac{\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|^{3}}\right)G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}$
$\displaystyle+\nabla\times\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\bigg{(}\nabla^{\prime}\times\frac{qq_{s}}{4\pi\epsilon}\frac{\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|^{3}}\bigg{)}G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}$

To clarify the notation in equation above note that
$dV^{\prime}=dx^{\prime}dy^{\prime}dz^{\prime}$ represents the differential
volume element of an infinite volume $\mathbb{R}^{3}$. As defined in A,
subsection A.1, the primed position vector $\boldsymbol{\mathbf{r}}^{\prime}$
is written in Cartesian coordinate system as:

$\boldsymbol{\mathbf{r}}^{\prime}=x^{\prime}\boldsymbol{\mathbf{\hat{x}}}+y^{\prime}\boldsymbol{\mathbf{\hat{y}}}+z^{\prime}\boldsymbol{\mathbf{\hat{z}}}$
(22)

where variables $x^{\prime},y^{\prime},z^{\prime}\in\mathbb{R}$. Vectors
$\boldsymbol{\mathbf{\hat{x}}}$, $\boldsymbol{\mathbf{\hat{y}}}$ and
$\boldsymbol{\mathbf{\hat{z}}}$ are orthogonal Cartesian unit basis vectors.
Furthermore, in Cartesian coordinates, the primed del operator
$\nabla^{\prime}$ that appears in equation (3) is defined as:

$\nabla^{\prime}=\boldsymbol{\mathbf{\hat{x}}}\frac{\partial}{\partial
x^{\prime}}+\boldsymbol{\mathbf{\hat{y}}}\frac{\partial}{\partial
y^{\prime}}+\boldsymbol{\mathbf{\hat{z}}}\frac{\partial}{\partial z^{\prime}}$
(23)

From the definition above, it follows that primed del operator
$\nabla^{\prime}$ acts only on functions of variables
$x^{\prime},y^{\prime},z^{\prime}$, and consequently, on functions of primed
position vector
$\boldsymbol{\mathbf{r}}^{\prime}=x^{\prime}\boldsymbol{\mathbf{\hat{x}}}+y^{\prime}\boldsymbol{\mathbf{\hat{y}}}+z^{\prime}\boldsymbol{\mathbf{\hat{z}}}$.
It does not act on functions of position vector of source charge
$\boldsymbol{\mathbf{r}}_{s}(t^{\prime})$ because this position vector is
function of variable $t^{\prime}$. Using these definitions we can write the
following simple relations:

$\displaystyle\frac{\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|^{3}}$
$\displaystyle=-\nabla^{\prime}\frac{1}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|}$
(24)
$\displaystyle\nabla^{\prime}\cdot\frac{1}{4\pi}\frac{\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|^{3}}$
$\displaystyle=-\nabla^{\prime
2}\frac{1}{4\pi}\frac{1}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|}=\delta\left(\boldsymbol{\mathbf{r^{\prime}}}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right)$
(25)
$\displaystyle\nabla^{\prime}\times\frac{\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|^{3}}$
$\displaystyle=-\nabla^{\prime}\times\nabla^{\prime}\frac{1}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|}=0$
(26)

where
$\delta\left(\boldsymbol{\mathbf{r^{\prime}}}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right)$
is 3D Dirac’s delta function. Inserting equations (25) and (26) into equation
(3), and eliminating charge $q$ from the equation, yields the following
relation:

$\displaystyle\frac{q_{s}}{4\pi\epsilon}\frac{\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}_{s}(t)}{\left|\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}_{s}(t)\right|^{3}}=$
$\displaystyle-\nabla\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\frac{q_{s}}{\epsilon}\delta\left(\boldsymbol{\mathbf{r^{\prime}}}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right)G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}$
(27) $\displaystyle+\frac{1}{c^{2}}\frac{\partial}{\partial
t}\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\left(\frac{\partial}{\partial
t^{\prime}}\frac{q_{s}}{4\pi\epsilon}\frac{\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|^{3}}\right)G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}$

In C, subsection C.1, we have shown that the time derivative that appears in
the second right hand side integral of equation (27) can be written as:

$\frac{\partial}{\partial
t^{\prime}}\frac{q_{s}}{4\pi\epsilon}\frac{\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|^{3}}=\frac{q_{s}}{4\pi\epsilon}\nabla^{\prime}\times\nabla^{\prime}\times\frac{\boldsymbol{\mathbf{v}}_{s}(t^{\prime})}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|}-\frac{q_{s}}{\epsilon}\boldsymbol{\mathbf{v}}_{s}(t^{\prime})\delta\left(\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right)$
(28)

where $\boldsymbol{\mathbf{v}}_{s}(t^{\prime})$ is the velocity of the source
charge $q_{s}$ at time $t^{\prime}$:

$\boldsymbol{\mathbf{v}}_{s}(t^{\prime})=\frac{\partial\boldsymbol{\mathbf{r}}_{s}(t^{\prime})}{\partial
t^{\prime}}$ (29)

By inserting equation (28) into equation (27) it is obtained that:

$\displaystyle\frac{q_{s}}{4\pi\epsilon}\frac{\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}_{s}(t)}{\left|\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}_{s}(t)\right|^{3}}=$
$\displaystyle-\nabla\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\frac{q_{s}}{\epsilon}\delta\left(\boldsymbol{\mathbf{r^{\prime}}}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right)G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}$
(30) $\displaystyle-\frac{1}{c^{2}}\frac{\partial}{\partial
t}\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\frac{q_{s}}{\epsilon}\boldsymbol{\mathbf{v}}_{s}(t^{\prime})\delta\left(\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right)G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}$
$\displaystyle+\frac{1}{c^{2}}\frac{\partial}{\partial
t}\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\frac{q_{s}}{4\pi\epsilon}\left[\nabla^{\prime}\times\nabla^{\prime}\times\frac{\boldsymbol{\mathbf{v}}_{s}(t^{\prime})}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|}\right]G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}$

We now make use of the following identity, also derived in Appendix C,
subsection C.2, to rewrite the last right hand side term of equation (30) as:

$\displaystyle\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\frac{q_{s}}{4\pi\epsilon}\left[\nabla^{\prime}\times\nabla^{\prime}\times\frac{\boldsymbol{\mathbf{v}}_{s}(t^{\prime})}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|}\right]G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}=$
$\displaystyle=$
$\displaystyle\nabla\times\nabla\times\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\frac{q_{s}}{4\pi\epsilon}\frac{\boldsymbol{\mathbf{v}}_{s}(t^{\prime})}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|}G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}$

Replacing the last right hand side integral in equation (30) with equation (3)
and differentiating the resulting equation with respect to time $t$ yields:

$\displaystyle\frac{q_{s}}{4\pi\epsilon}\frac{\partial}{\partial
t}\frac{\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}_{s}(t)}{\left|\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}_{s}(t)\right|^{3}}=$
$\displaystyle-\frac{\partial}{\partial
t}\nabla\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\frac{q_{s}}{\epsilon}\delta\left(\boldsymbol{\mathbf{r^{\prime}}}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right)G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}$
(32) $\displaystyle-\frac{1}{c^{2}}\frac{\partial^{2}}{\partial
t^{2}}\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\frac{q_{s}}{\epsilon}\boldsymbol{\mathbf{v}}_{s}(t^{\prime})\delta\left(\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right)G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}$
$\displaystyle+\frac{1}{c^{2}}\frac{\partial^{2}}{\partial
t^{2}}\nabla\times\nabla\times\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\frac{q_{s}}{4\pi\epsilon}\frac{\boldsymbol{\mathbf{v}}_{s}(t^{\prime})}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|}G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}$

In the physical setting shown in Fig. 2 the coordinates of the test charge $q$
are fixed, hence, order in which we apply operator $\nabla\times\nabla\times$
and second order time derivative $\frac{\partial}{\partial t^{2}}$ can be
swapped (because operator $\nabla$ does not affect variable $t$). Furthermore,
because variables $t^{\prime}$, $x^{\prime}$, $y^{\prime}$ and $z^{\prime}$
are independent of time $t$ we can move the double time derivative under the
integral sign in the last right hand side integral of above equation:

$\displaystyle\frac{q_{s}}{4\pi\epsilon}\frac{\partial}{\partial
t}\frac{\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}_{s}(t)}{\left|\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}_{s}(t)\right|^{3}}=$
$\displaystyle-\frac{\partial}{\partial
t}\nabla\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\frac{q_{s}}{\epsilon}\delta\left(\boldsymbol{\mathbf{r^{\prime}}}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right)G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}$
(33) $\displaystyle-\frac{1}{c^{2}}\frac{\partial^{2}}{\partial
t^{2}}\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\frac{q_{s}}{\epsilon}\boldsymbol{\mathbf{v}}_{s}(t^{\prime})\delta\left(\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right)G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}$
$\displaystyle+\nabla\times\nabla\times\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\frac{q_{s}}{4\pi\epsilon}\frac{\boldsymbol{\mathbf{v}}_{s}(t^{\prime})}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|}\frac{1}{c^{2}}\frac{\partial^{2}}{\partial
t^{2}}G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}$

The second order time derivative of
$G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})$ in
the last term of equation (33) can be replaced with equation (15) to obtain:

$\displaystyle\frac{q_{s}}{4\pi\epsilon}\frac{\partial}{\partial
t}\frac{\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}_{s}(t)}{\left|\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}_{s}(t)\right|^{3}}=$
$\displaystyle-\frac{\partial}{\partial
t}\nabla\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\frac{q_{s}}{\epsilon}\delta\left(\boldsymbol{\mathbf{r^{\prime}}}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right)G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}$
(34) $\displaystyle-\frac{1}{c^{2}}\frac{\partial^{2}}{\partial
t^{2}}\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\frac{q_{s}}{\epsilon}\boldsymbol{\mathbf{v}}_{s}(t^{\prime})\delta\left(\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right)G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}$
$\displaystyle+\nabla\times\nabla\times\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\frac{q_{s}}{4\pi\epsilon}\frac{\boldsymbol{\mathbf{v}}_{s}(t^{\prime})}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|}\delta\left(\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}^{\prime}\right)\delta\left(t-t^{\prime}\right)dV^{\prime}}}$
$\displaystyle+\nabla\times\nabla\times\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\frac{q_{s}}{4\pi\epsilon}\frac{\boldsymbol{\mathbf{v}}_{s}(t^{\prime})}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|}\nabla^{2}G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}$

Using sifting property of Dirac’s delta function allows us to rewrite the
third right hand side term of equation (34) as:

$\displaystyle\nabla\times\nabla\times\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\frac{q_{s}}{4\pi\epsilon}\frac{\boldsymbol{\mathbf{v}}_{s}(t^{\prime})}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|}\delta\left(\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}^{\prime}\right)\delta\left(t-t^{\prime}\right)dV^{\prime}}}=$
$\displaystyle=$
$\displaystyle\nabla\times\nabla\times\int_{\mathbb{R}}{dt^{\prime}\frac{q_{s}}{4\pi\epsilon}\frac{\boldsymbol{\mathbf{v}}_{s}(t^{\prime})}{\left|\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|}\delta\left(t-t^{\prime}\right)}=$
$\displaystyle=$
$\displaystyle\nabla\times\nabla\times\frac{q_{s}}{4\pi\epsilon}\frac{\boldsymbol{\mathbf{v}}_{s}(t)}{\left|\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}_{s}(t)\right|}$

To continue the derivation of Maxwell’s equations from Coulomb’s law we should
note that operator $\nabla$ does not affect vector
$\boldsymbol{\mathbf{v}}_{s}(t)$ because $\boldsymbol{\mathbf{v}}_{s}(t)$ is a
function of variable $t$. Hence, the application of standard vector calculus
identity
$\nabla\times\nabla\times\boldsymbol{\mathbf{P}}=\nabla\left(\nabla\cdot\boldsymbol{\mathbf{P}}\right)-\nabla^{2}\boldsymbol{\mathbf{P}}$
yields:

$\displaystyle\nabla\times\nabla\times\frac{q_{s}}{4\pi\epsilon}\frac{\boldsymbol{\mathbf{v}}_{s}(t)}{\left|\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}_{s}(t)\right|}$
$\displaystyle=\frac{q_{s}}{4\pi\epsilon}\nabla\left(\nabla\cdot\frac{\boldsymbol{\mathbf{v}}_{s}(t)}{\left|\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}_{s}(t)\right|}\right)-\frac{q_{s}}{4\pi\epsilon}\nabla^{2}\frac{\boldsymbol{\mathbf{v}}_{s}(t)}{\left|\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}_{s}(t)\right|}=$
(36)
$\displaystyle=\frac{q_{s}}{4\pi\epsilon}\nabla\left(\boldsymbol{\mathbf{v}}_{s}(t)\cdot\nabla\frac{1}{\left|\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}_{s}(t)\right|}\right)-\frac{q_{s}}{4\pi\epsilon}\boldsymbol{\mathbf{v}}_{s}(t)\nabla^{2}\frac{1}{\left|\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}_{s}(t)\right|}=$
$\displaystyle=-\frac{q_{s}}{4\pi\epsilon}\nabla\frac{\partial}{\partial
t}\frac{1}{\left|\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}_{s}(t)\right|}+\frac{q_{s}}{\epsilon}\boldsymbol{\mathbf{v}}_{s}(t)\delta\left(\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}_{s}(t)\right)=$
$\displaystyle=\frac{q_{s}}{4\pi\epsilon}\frac{\partial}{\partial
t}\frac{\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}_{s}(t)}{\left|\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}_{s}(t)\right|^{3}}+\frac{q_{s}}{\epsilon}\boldsymbol{\mathbf{v}}_{s}(t)\delta\left(\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}_{s}(t)\right)$

Combining equations (34), (3) and (36), after cancellation of appropriate
terms, yields:

$\displaystyle 0=$ $\displaystyle-\frac{\partial}{\partial
t}\nabla\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\frac{q_{s}}{\epsilon}\delta\left(\boldsymbol{\mathbf{r^{\prime}}}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right)G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}$
(37) $\displaystyle-\frac{1}{c^{2}}\frac{\partial^{2}}{\partial
t^{2}}\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\frac{q_{s}}{\epsilon}\boldsymbol{\mathbf{v}}_{s}(t^{\prime})\delta\left(\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right)G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}$
$\displaystyle+\frac{q_{s}}{\epsilon}\boldsymbol{\mathbf{v}}_{s}(t)\delta\left(\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}_{s}(t)\right)$
$\displaystyle+\nabla\times\nabla\times\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\frac{q_{s}}{4\pi\epsilon}\frac{\boldsymbol{\mathbf{v}}_{s}(t^{\prime})}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|}\nabla^{2}G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}$

In Appendix C, subsection C.3, we have derived the following mathematical
identity:

$\displaystyle\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\frac{q_{s}}{4\pi\epsilon}\frac{\boldsymbol{\mathbf{v}}_{s}(t^{\prime})}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|}\nabla^{2}G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}=$
$\displaystyle=$
$\displaystyle-\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\frac{q_{s}}{\epsilon}\boldsymbol{\mathbf{v}}_{s}(t^{\prime})\delta\left(\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right)G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}$

By inserting equation (3) into equation (37) it is obtained that:

$\displaystyle 0=$ $\displaystyle-\frac{\partial}{\partial
t}\nabla\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\frac{q_{s}}{\epsilon}\delta\left(\boldsymbol{\mathbf{r^{\prime}}}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right)G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}$
(39) $\displaystyle-\frac{1}{c^{2}}\frac{\partial^{2}}{\partial
t^{2}}\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\frac{q_{s}}{\epsilon}\boldsymbol{\mathbf{v}}_{s}(t^{\prime})\delta\left(\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right)G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}$
$\displaystyle+\frac{q_{s}}{\epsilon}\boldsymbol{\mathbf{v}}_{s}(t)\delta\left(\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}_{s}(t)\right)$
$\displaystyle-\nabla\times\nabla\times\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\frac{q_{s}}{\epsilon}\boldsymbol{\mathbf{v}}_{s}(t^{\prime})\delta\left(\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right)G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}$

If we now introduce new constant $\mu=\frac{1}{c^{2}\epsilon}$ and divide
whole equation (39) by $c^{2}$ we obtain:

$\displaystyle 0=$ $\displaystyle-\frac{1}{c^{2}}\frac{\partial}{\partial
t}\nabla\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\frac{q_{s}}{\epsilon}\delta\left(\boldsymbol{\mathbf{r^{\prime}}}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right)G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}$
(40) $\displaystyle-\frac{1}{c^{2}}\frac{\partial^{2}}{\partial
t^{2}}\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{q_{s}\mu
c\frac{\boldsymbol{\mathbf{v}}_{s}(t^{\prime})}{c}\delta\left(\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right)G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}$
$\displaystyle+q_{s}\mu\boldsymbol{\mathbf{v}}_{s}(t)\delta\left(\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}_{s}(t)\right)$
$\displaystyle-\nabla\times\nabla\times\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{q_{s}\mu
c\frac{\boldsymbol{\mathbf{v}}_{s}(t^{\prime})}{c}\delta\left(\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right)G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}$

Although it is perhaps not yet apparent, equation (39) is Maxwell-Ampere
equation given in introductory part of this paper as equation (4). To evaluate
right hand side integrals in equation (40) we use sifting property of Dirac’s
delta function
$\int_{\mathbb{R}^{3}}{\delta\left(\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right)f(\boldsymbol{\mathbf{r}}^{\prime})dV^{\prime}}=f(\boldsymbol{\mathbf{r}}_{s}(t^{\prime}))$,
where $f(\boldsymbol{\mathbf{r}}^{\prime})$ is function of position vector
$\boldsymbol{\mathbf{r}}^{\prime}$, to rewrite the right hand side integrals
in equation (40) as:

$\displaystyle\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\frac{q_{s}}{\epsilon}\delta\left(\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right)G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}$
$\displaystyle=\int_{\mathbb{R}}{\frac{q_{s}}{\epsilon}G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}_{s}(t^{\prime}),t^{\prime})dt^{\prime}}$
(41) $\displaystyle\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{q_{s}\mu
c\frac{\boldsymbol{\mathbf{v}}_{s}(t^{\prime})}{c}\delta\left(\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right)G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}$
$\displaystyle=\int_{\mathbb{R}}{q_{s}\mu
c\frac{\boldsymbol{\mathbf{v}}_{s}(t^{\prime})}{c}G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}_{s}(t^{\prime}),t^{\prime})dt^{\prime}}$
(42)

To evaluate right hand side integrals in equations above we now replace
Green’s function
$G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}_{s}(t^{\prime}),t^{\prime})$
in these equations with equation (16) to obtain:

$\displaystyle\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\frac{q_{s}}{\epsilon}\delta\left(\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right)G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}$
$\displaystyle=$ (43)
$\displaystyle=\int_{\mathbb{R}}{\frac{q_{s}}{\epsilon}\frac{\delta\left(t^{\prime}-t+\frac{\left|\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|}{c}\right)}{4\pi\left|\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|}dt^{\prime}}$
$\displaystyle\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{q_{s}\mu
c\frac{\boldsymbol{\mathbf{v}}_{s}(t^{\prime})}{c}\delta\left(\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right)G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}$
$\displaystyle=$ (44) $\displaystyle=\int_{\mathbb{R}}{q_{s}\mu
c\frac{\boldsymbol{\mathbf{v}}_{s}(t^{\prime})}{c}\frac{\delta\left(t^{\prime}-t+\frac{\left|\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|}{c}\right)}{4\pi\left|\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|}dt^{\prime}}$

The right hand side integrals in equations (43) and (44) can be evaluated by
making use of the following standard mathematical identity involving Dirac’s
delta function:

$\delta\left(f(u)\right)=\frac{\delta(u-u_{0})}{\left|\frac{\partial}{\partial
u}f(u)\right|_{u=u_{0}}}$ (45)

where $f(u)$ is real function of real argument $u$, and $u_{0}$ is the
solution of equation $f(u_{0})=0$. Using identity (45), the Dirac’s delta
function in equations (43) and (44) can be written as as:

$\displaystyle\delta\left(t^{\prime}-t+\frac{\left|\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|}{c}\right)$
$\displaystyle=\frac{\delta(t^{\prime}-t_{r})}{\left|1-\frac{1}{c}\frac{\boldsymbol{\mathbf{v_{s}}}(t^{\prime})\cdot\left(\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right)}{\left|\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|}\right|}=\frac{\delta(t^{\prime}-t_{r})}{1-\frac{1}{c}\frac{\boldsymbol{\mathbf{v_{s}}}(t^{\prime})\cdot\left(\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right)}{\left|\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|}}$
(46)
$\displaystyle=\frac{\delta(t^{\prime}-t_{r})}{1-\boldsymbol{\mathbf{\beta}}(t_{r})\cdot\boldsymbol{\mathbf{n}}(t_{r})}$

where $\boldsymbol{\mathbf{\beta}}(t_{r})$ and
$\boldsymbol{\mathbf{n}}(t_{r})$ are given by equations (9) and (10),
respectively. From equation (45) it follows that the time $t_{r}$ is the
solution to the following equation:

$t-t_{r}-\frac{\left|\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}_{s}(t_{r})\right|}{c}=0$
(47)

Evidently, the time $t_{r}$ is the time when the disturbance created by moving
source charge at position in space $\boldsymbol{\mathbf{r}}_{s}(t_{r})$ was
created. This disturbance moves through the space with finite velocity $c$ and
reaches the position $\boldsymbol{\mathbf{r}}$ of the test charge at time $t$.
In the electromagnetic literature this time $t_{r}$ is commonly known as
retarded time.

To proceed with derivation of Maxwell equations, we now insert equation (46)
into equations (43) and (44), and evaluate the integrals over $t^{\prime}$
using the sifting property of Dirac’s delta function to obtain:

$\displaystyle\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\frac{q_{s}}{\epsilon}\delta\left(\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right)G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}$
$\displaystyle=\frac{1}{4\pi\epsilon}\frac{q_{s}}{\left|\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}_{s}(t_{r})\right|\left(1-\boldsymbol{\mathbf{\beta}}(t_{r})\cdot\boldsymbol{\mathbf{n}}(t_{r})\right)}$
(48) $\displaystyle\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{q_{s}\mu
c\frac{\boldsymbol{\mathbf{v}}_{s}(t^{\prime})}{c}\delta\left(\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right)G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}$
$\displaystyle=\frac{\mu
c}{4\pi}\frac{q_{s}\boldsymbol{\mathbf{\beta}}_{s}(t_{r})}{\left|\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}_{s}(t_{r})\right|\left(1-\boldsymbol{\mathbf{\beta}}(t_{r})\cdot\boldsymbol{\mathbf{n}}(t_{r})\right)}$
(49)

By inserting equations (48) and (49) into equation (40), and rearranging, it
is obtained:

$\displaystyle\nabla\times\nabla\times\frac{\mu
c}{4\pi}\frac{q_{s}\boldsymbol{\mathbf{\beta}}_{s}(t_{r})}{\left|\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}_{s}(t_{r})\right|\left(1-\boldsymbol{\mathbf{\beta}}(t_{r})\cdot\boldsymbol{\mathbf{n}}(t_{r})\right)}=$
$\displaystyle
q_{s}\mu\boldsymbol{\mathbf{v}}_{s}(t)\delta\left(\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}_{s}(t)\right)$
(50) $\displaystyle-\frac{1}{c^{2}}\frac{\partial}{\partial
t}\nabla\frac{1}{4\pi\epsilon}\frac{q_{s}}{\left|\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}_{s}(t_{r})\right|\left(1-\boldsymbol{\mathbf{\beta}}(t_{r})\cdot\boldsymbol{\mathbf{n}}(t_{r})\right)}$
$\displaystyle-\frac{1}{c^{2}}\frac{\partial^{2}}{\partial t^{2}}\frac{\mu
c}{4\pi}\frac{q_{s}\boldsymbol{\mathbf{\beta}}_{s}(t_{r})}{\left|\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}_{s}(t_{r})\right|\left(1-\boldsymbol{\mathbf{\beta}}(t_{r})\cdot\boldsymbol{\mathbf{n}}(t_{r})\right)}$

The first right hand side term of the equation above can be identified as the
current $\boldsymbol{\mathbf{J}}$ of the point charge distribution moving with
velocity $\boldsymbol{\mathbf{v}}_{s}(t)$ multiplied by constant $\mu$:

$\mu\boldsymbol{\mathbf{J}}=\mu
q_{s}\boldsymbol{\mathbf{v}}_{s}(t)\delta\left(\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}_{s}(t)\right)$
(51)

We now define scalar function $\theta(\boldsymbol{\mathbf{r}},t)$ and vector
function $\boldsymbol{\mathbf{Q}}(\boldsymbol{\mathbf{r}},t)$ as:

$\displaystyle\theta(\boldsymbol{\mathbf{r}},t)$
$\displaystyle=\frac{1}{4\pi\epsilon}\frac{q_{s}}{\left|\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}_{s}(t_{r})\right|\left(1-\boldsymbol{\mathbf{\beta}}(t_{r})\cdot\boldsymbol{\mathbf{n}}(t_{r})\right)}$
(52) $\displaystyle\boldsymbol{\mathbf{Q}}(\boldsymbol{\mathbf{r}},t)$
$\displaystyle=\frac{\mu
c}{4\pi}\frac{q_{s}\boldsymbol{\mathbf{\beta}}_{s}(t_{r})}{\left|\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}_{s}(t_{r})\right|\left(1-\boldsymbol{\mathbf{\beta}}(t_{r})\cdot\boldsymbol{\mathbf{n}}(t_{r})\right)}$
(53)

With the aid of scalar function $\theta(\boldsymbol{\mathbf{r}},t)$, vector
function $\boldsymbol{\mathbf{Q}}(\boldsymbol{\mathbf{r}},t)$, and expression
$\mu\boldsymbol{\mathbf{J}}$ given by equation (51) the equation (50) can be
written as:

$\nabla\times\nabla\times\boldsymbol{\mathbf{Q}}(\boldsymbol{\mathbf{r}},t)=\mu\boldsymbol{\mathbf{J}}-\frac{1}{c^{2}}\frac{\partial}{\partial
t}\left(-\nabla\theta(\boldsymbol{\mathbf{r}},t)-\frac{\partial}{\partial
t}\boldsymbol{\mathbf{Q}}(\boldsymbol{\mathbf{r}},t)\right)$ (54)

Furthermore, we now define two vector functions $\boldsymbol{\mathbf{M}}$ and
$\boldsymbol{\mathbf{N}}$ as:

$\displaystyle\boldsymbol{\mathbf{M}}$
$\displaystyle=\nabla\times\boldsymbol{\mathbf{Q}}(\boldsymbol{\mathbf{r}},t)$
(55) $\displaystyle\boldsymbol{\mathbf{N}}$
$\displaystyle=-\nabla\theta(\boldsymbol{\mathbf{r}},t)-\frac{\partial}{\partial
t}\boldsymbol{\mathbf{Q}}(\boldsymbol{\mathbf{r}},t)$ (56)

Using definitions of vector functions $\boldsymbol{\mathbf{M}}$ and
$\boldsymbol{\mathbf{N}}$ given by equations (55) and (56) we can rewrite
equation (54) as:

$\nabla\times\boldsymbol{\mathbf{M}}=\boldsymbol{\mathbf{J}}+\frac{1}{c^{2}}\frac{\partial\boldsymbol{\mathbf{N}}}{\partial
t}$ (57)

We shall now investigate the mathematical properties of vector fields
$\boldsymbol{\mathbf{M}}$ and $\boldsymbol{\mathbf{N}}$. Note that because for
any differentiable vector field $\boldsymbol{\mathbf{P}}$ we can write
$\nabla\cdot\nabla\times\boldsymbol{\mathbf{P}}=0$, from equation (55) it
follows that:

$\nabla\cdot\boldsymbol{\mathbf{M}}=0$ (58)

The curl of the gradient of any differentiable scalar function $\psi$ is equal
to zero, i.e. $\nabla\times\nabla\psi=0$. Thus, taking the curl of equation
(56) yields:

$\nabla\times\boldsymbol{\mathbf{N}}=-\nabla\times\frac{\partial}{\partial
t}\boldsymbol{\mathbf{Q}}(\boldsymbol{\mathbf{r}},t)=-\frac{\partial}{\partial
t}\nabla\times\boldsymbol{\mathbf{Q}}(\boldsymbol{\mathbf{r}},t)=-\frac{\partial\boldsymbol{\mathbf{M}}}{\partial
t}$ (59)

Finally, in Appendix C, subsection C.4, we have shown that the divergence of
vector field $\boldsymbol{\mathbf{N}}$ is:

$\nabla\cdot\boldsymbol{\mathbf{N}}=\frac{q_{s}}{\epsilon}\delta\left(\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}_{s}(t)\right)=\frac{\rho(\boldsymbol{\mathbf{r}},t)}{\epsilon}$
(60)

which completes the derivation of electrodynamic equations from Coulomb’s law.

To compare these equations to Maxwell’s equations, in Table 1 we have
summarized governing equations for scalar potential
$\theta(\boldsymbol{\mathbf{r}},t)$, vector potential
$\boldsymbol{\mathbf{Q}}(\boldsymbol{\mathbf{r}},t)$, vector field
$\boldsymbol{\mathbf{M}}$ and vector field $\boldsymbol{\mathbf{N}}$ which are
all derived from Coulomb’s law. By comparison with Liénard–Wiechert potentials
given in Table 2, we see that scalar potential
$\theta(\boldsymbol{\mathbf{r}},t)$ is identical to Liénard–Wiechert scalar
potential $\phi(\boldsymbol{\mathbf{r}},t)$ and vector potential
$\boldsymbol{\mathbf{Q}}(\boldsymbol{\mathbf{r}},t)$ is identical to
Liénard–Wiechert magnetic vector potential
$\boldsymbol{\mathbf{A}}(\boldsymbol{\mathbf{r}},t)$. Furthermore, by
comparing Table 1 and Table 2 we find that vector field
$\boldsymbol{\mathbf{M}}$ is identical to magnetic flux density
$\boldsymbol{\mathbf{B}}$ and that vector field $\boldsymbol{\mathbf{N}}$ is
identical to electric field $\boldsymbol{\mathbf{E}}$.

In Table 3 we have compared Maxwell’s equations governing fields
$\boldsymbol{\mathbf{B}}$ and $\boldsymbol{\mathbf{E}}$ with differential
equations governing vector fields $\boldsymbol{\mathbf{M}}$ and
$\boldsymbol{\mathbf{N}}$. Clearly, left hand side of Table 3 is identical in
the mathematical form to the right hand side of the same table, hence,
differential equations governing vector fields $\boldsymbol{\mathbf{M}}$ and
$\boldsymbol{\mathbf{N}}$ are identical to those governing vector fields
$\boldsymbol{\mathbf{B}}$ and $\boldsymbol{\mathbf{E}}$. This is expected,
because we already know that vector field
$\boldsymbol{\mathbf{N}}=\boldsymbol{\mathbf{E}}$ and vector field
$\boldsymbol{\mathbf{M}}=\boldsymbol{\mathbf{B}}$.

Thus, it should be evident by now that we have derived Maxwell equations and
Liénard–Wiechert potentials directly from Coulomb’s law. This was achieved by
mathematically relating known electrostatic Coulomb’s law acting on test
charge at present time to ”unknown” electrodynamic fields acting at past. The
mathematical link between the static case in the present and dynamic case in
the past was provided by generalized Helmholtz theorem. The derived equations
are valid for arbitrarily moving source charge and these equations are not
confined to motions along straight line. Furthermore, it should be noted that
we have derived the Maxwell equations and Liénard–Wiechert potentials directly
from Coulomb’s law without resorting to special relativity or Lorentz
transformation.

Table 1: Potentials and vector fields derived from Coulomb’s law. symbol |  equation |  description [.5]()
---|---|---
[.5]() $\theta(\boldsymbol{\mathbf{r}},t)$ |  ${\displaystyle\frac{1}{4\pi\epsilon}\frac{q_{s}}{\left|\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}_{s}(t_{r})\right|\left(1-\boldsymbol{\mathbf{\beta}}(t_{r})\cdot\boldsymbol{\mathbf{n}}(t_{r})\right)}}$ |  scalar potential derived from Coulomb’s law
1-3[.4pt/1pt] |  |
[.5]() $\boldsymbol{\mathbf{Q}}(\boldsymbol{\mathbf{r}},t)$ |  ${\displaystyle\frac{\mu c}{4\pi}\frac{q_{s}\boldsymbol{\mathbf{\beta}}_{s}(t_{r})}{\left|\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}_{s}(t_{r})\right|\left(1-\boldsymbol{\mathbf{\beta}}(t_{r})\cdot\boldsymbol{\mathbf{n}}(t_{r})\right)}}$ |  vector potential derived from Coulomb’s law
1-3[.4pt/1pt] |  |
[.5]() $\boldsymbol{\mathbf{M}}$ |  $\nabla\times\boldsymbol{\mathbf{Q}}(\boldsymbol{\mathbf{r}},t)$ |  vector field $\boldsymbol{\mathbf{M}}$ derived from Coulomb’s law
1-3[.4pt/1pt] |  |
[.5]() $\boldsymbol{\mathbf{N}}$ |  $-\nabla\phi(\boldsymbol{\mathbf{r}},t)-\frac{\partial}{\partial t}\boldsymbol{\mathbf{Q}}(\boldsymbol{\mathbf{r}},t)$ |  vector field $\boldsymbol{\mathbf{N}}$ derived from Coulomb’s law
1-3[.4pt/1pt] |  |
Table 2: Standard electromagnetic theory expressions for Liénard–Wiechert scalar potential $\phi(\boldsymbol{\mathbf{r}},t)$, Liénard–Wiechert vector potential $\boldsymbol{\mathbf{A}}(\boldsymbol{\mathbf{r}},t)$, magnetic flux density $\boldsymbol{\mathbf{B}}$ and electric field $\boldsymbol{\mathbf{E}}$. symbol |  equation |  description [.5]()
---|---|---
[.5]() $\phi(\boldsymbol{\mathbf{r}},t)$ |  ${\displaystyle\frac{1}{4\pi\epsilon}\frac{q_{s}}{\left|\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}_{s}(t_{r})\right|\left(1-\boldsymbol{\mathbf{\beta}}(t_{r})\cdot\boldsymbol{\mathbf{n}}(t_{r})\right)}}$ |  Liénard–Wiechert scalar potential
1-3[.4pt/1pt] |  |
[.5]() $\boldsymbol{\mathbf{A}}(\boldsymbol{\mathbf{r}},t)$ |  ${\displaystyle\frac{\mu c}{4\pi}\frac{q_{s}\boldsymbol{\mathbf{\beta}}_{s}(t_{r})}{\left|\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}_{s}(t_{r})\right|\left(1-\boldsymbol{\mathbf{\beta}}(t_{r})\cdot\boldsymbol{\mathbf{n}}(t_{r})\right)}}$ |  Liénard–Wiechert vector potential
1-3[.4pt/1pt] |  |
[.5]() $\boldsymbol{\mathbf{B}}$ |  $\nabla\times\boldsymbol{\mathbf{A}}(\boldsymbol{\mathbf{r}},t)$ |  magnetic flux density
1-3[.4pt/1pt] |  |
[.5]() $\boldsymbol{\mathbf{E}}$ |  $-\nabla\phi(\boldsymbol{\mathbf{r}},t)-\frac{\partial}{\partial t}\boldsymbol{\mathbf{A}}(\boldsymbol{\mathbf{r}},t)$ |  electric field
1-3[.4pt/1pt] |  |
Table 3: Maxwell equations for electromagnetic fields $\boldsymbol{\mathbf{E}}$ and $\boldsymbol{\mathbf{B}}$ compared with differential equations for vector fields $\boldsymbol{\mathbf{M}}$ and $\boldsymbol{\mathbf{N}}$ derived from Coulomb’s law. Maxwell equation |  description |  equations derived from Coulomb’s law [.5]()
---|---|---
1-3[.4pt/1pt] |  |
[.5]() $\nabla\times\boldsymbol{\mathbf{B}}=\boldsymbol{\mathbf{J}}+\frac{1}{c^{2}}\frac{\partial\boldsymbol{\mathbf{E}}}{\partial t}$ |  Maxwell-Ampere equation |  $\nabla\times\boldsymbol{\mathbf{M}}=\boldsymbol{\mathbf{J}}+\frac{1}{c^{2}}\frac{\partial\boldsymbol{\mathbf{N}}}{\partial t}$
1-3[.4pt/1pt] |  |
[.5]() $\nabla\times\boldsymbol{\mathbf{E}}=-\frac{\partial\boldsymbol{\mathbf{B}}}{\partial t}$ |  Faraday’s law |  $\nabla\times\boldsymbol{\mathbf{N}}=-\frac{\partial\boldsymbol{\mathbf{M}}}{\partial t}$
1-3[.4pt/1pt] |  |
[.5]() $\nabla\cdot\boldsymbol{\mathbf{E}}=\frac{\rho}{\epsilon}$ |  Gauss’ law for electric field |  $\nabla\cdot\boldsymbol{\mathbf{N}}=\frac{\rho}{\epsilon}$
1-3[.4pt/1pt] |  |
[.5]() $\nabla\cdot\boldsymbol{\mathbf{B}}=0$ |  Gauss’ law for magnetic field |  $\nabla\cdot\boldsymbol{\mathbf{M}}=0$
1-3[.4pt/1pt] |  |

##  4 Derivation of Electrodynamic Energy Conservation Law and Lorentz Force

To derive the electrodynamic energy conservation law from Coulomb’s law we
first consider hypothetical physical setting shown in Fig. 3 where the source
charge $q_{s}$ is moving along arbitrary trajectory
$\boldsymbol{\mathbf{r}}_{s}(t)$. Then the source charge $q_{s}$ stops at some
time in the past $t_{s}$. In this physical setting, closed contour $C$ is at
rest at all times. At present time $t_{p}>t_{s}$ all the points inside the
sphere of radius $R=c(t_{p}-t_{s})$ are affected only by electrostatic
Coulomb’s field. The known energy conservation law valid at present dictates
that contour integral of electrostatic field along any closed contour immersed
inside the sphere of radius $R$ equals to zero:

Figure 3: Source charge $q_{s}$ is moving along arbitrary trajectory
$\boldsymbol{\mathbf{r}}_{s}(t)$ and it stops at past time $t_{s}$. Closed
contour $C$ is at rest at all times. All the points $\boldsymbol{\mathbf{r}}$
on contour $C$ are inside the sphere of radius $R=c(t_{p}-t_{s})$. At present
time $t_{p}>t_{s}$ all the points on contour $C$ are affected only by
Coulomb’s electrostatic field.

$\oint_{C}{\boldsymbol{\mathbf{E}}_{c}(\boldsymbol{\mathbf{r}},t_{p})\cdot
d\boldsymbol{\mathbf{r}}}=\oint_{C}{\frac{q_{s}}{4\pi\epsilon}\frac{\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}_{s}(t_{s})}{\left|\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}_{s}(t_{s})\right|^{3}}\cdot
d\boldsymbol{\mathbf{r}}}=0$ (61)

where $\boldsymbol{\mathbf{E}}_{c}$ is Coulomb’s electrostatic field,
$\boldsymbol{\mathbf{r}}_{s}(t_{s})$ is the position vector of source charge
when it stopped moving, and vector $\boldsymbol{\mathbf{r}}$ is the position
vector of the point on contour $C$. This electrostatic energy conservation
law, valid at present time $t_{p}$, states that no net work is done in
transporting the unit charge along any closed contour immersed in
electrostatic field.

To proceed, we assume that in the past, when the source charge was moving, the
energy conservation law is unknown. However, generalized Helmholtz
decomposition theorem allows us to derive this ”unknown” electrodynamic energy
conservation law valid in the past from the knowledge of electrostatic energy
conservation law valid at present. To derive this unknown electrodynamic
electrodynamic conservation law we consider the contour integral (61) at the
moment $t$ infinitesimally before the time when the source charge stopped:

$t=t_{s}-\delta t$ (62)

where $\delta t$ is infinitesimally small time interval. If time interval
$\delta t$ approaches zero ($\delta t\rightarrow 0$) we can rewrite the
contour integral (61) as the function of time $t$:

$\oint_{C}{\boldsymbol{\mathbf{E}}_{c}(\boldsymbol{\mathbf{r}},t_{p}-\delta
t)\cdot
d\boldsymbol{\mathbf{r}}}=\oint_{C}{\frac{q_{s}}{4\pi\epsilon}\frac{\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}_{s}(t)}{\left|\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}_{s}(t)\right|^{3}}\cdot
d\boldsymbol{\mathbf{r}}}=0$ (63)

Because the integrand on the right hand side of equation (63) is the function
of varying time $t$ and position vector $\boldsymbol{\mathbf{r}}$ the
generalized Helmholtz decomposition theorem can be applied to rewrite this
integrand as the function of past positions and velocities of the source
charge. In fact, such expression is already derived in previous section as
equation (33), repeated here for clarity:

$\displaystyle\frac{q_{s}}{4\pi\epsilon}\frac{\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}_{s}(t)}{\left|\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}_{s}(t)\right|^{3}}=$
$\displaystyle-\nabla\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\frac{q_{s}}{\epsilon}\delta\left(\boldsymbol{\mathbf{r^{\prime}}}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right)G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}$
(64) $\displaystyle-\frac{1}{c^{2}}\frac{\partial}{\partial
t}\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\frac{q_{s}}{\epsilon}\boldsymbol{\mathbf{v}}_{s}(t^{\prime})\delta\left(\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right)G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}$
$\displaystyle+\frac{1}{c^{2}}\frac{\partial}{\partial
t}\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\frac{q_{s}}{4\pi\epsilon}\left[\nabla^{\prime}\times\nabla^{\prime}\times\frac{\boldsymbol{\mathbf{v}}_{s}(t^{\prime})}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|}\right]G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}$

Substituting the first two right hand side terms of equation (64) with
equations (48) and (49) and combining the result with equations (52) and (53),
and using $c^{2}=1/\mu\epsilon$ yields:

$\frac{q_{s}}{4\pi\epsilon}\frac{\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}_{s}(t)}{\left|\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}_{s}(t)\right|^{3}}=-\nabla\theta(\boldsymbol{\mathbf{r}},t)-\frac{\partial}{\partial
t}\boldsymbol{\mathbf{Q}}(\boldsymbol{\mathbf{r}},t)+\boldsymbol{\mathbf{K}}(\boldsymbol{\mathbf{r}},t)$
(65)

where vector function $\boldsymbol{\mathbf{K}}(\boldsymbol{\mathbf{r}},t)$ is
equal to the last right hand side term of equation (64):

$\boldsymbol{\mathbf{K}}(\boldsymbol{\mathbf{r}},t)=\frac{1}{c^{2}}\frac{\partial}{\partial
t}\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\frac{q_{s}}{4\pi\epsilon}\left[\nabla^{\prime}\times\nabla^{\prime}\times\frac{\boldsymbol{\mathbf{v}}_{s}(t^{\prime})}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|}\right]G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}$
(66)

Replacing the first two terms on the right hand side of equation (65) with
equation (56) yields:

$\frac{q_{s}}{4\pi\epsilon}\frac{\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}_{s}(t)}{\left|\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}_{s}(t)\right|^{3}}=\boldsymbol{\mathbf{N}}(\boldsymbol{\mathbf{r}},t)+\boldsymbol{\mathbf{K}}(\boldsymbol{\mathbf{r}},t)$
(67)

Then, by inserting equation (67) into right hand side of equation (63) it is
obtained that:

$0=\oint_{C}{\boldsymbol{\mathbf{E}}_{c}(\boldsymbol{\mathbf{r}},t_{p}-\delta
t)\cdot
d\boldsymbol{\mathbf{r}}}=\oint_{C}{\left(\boldsymbol{\mathbf{N}}(\boldsymbol{\mathbf{r}},t)+\boldsymbol{\mathbf{K}}(\boldsymbol{\mathbf{r}},t)\right)\cdot
d\boldsymbol{\mathbf{r}}}$ (68)

The space-time integral on the right hand side of equation (66) is very
difficult to evaluate. However, we can eliminate vector field
$\boldsymbol{\mathbf{K}}(\boldsymbol{\mathbf{r}},t)$ from the right hand side
of equation (68) by the application of Stokes’ theorem:

$0=\oint_{C}{\boldsymbol{\mathbf{E}}_{c}(\boldsymbol{\mathbf{r}},t_{p}-\delta
t)\cdot
d\boldsymbol{\mathbf{r}}}=\oint_{C}{\boldsymbol{\mathbf{N}}(\boldsymbol{\mathbf{r}},t)\cdot
d\boldsymbol{\mathbf{r}}}+\int_{S}{\nabla\times\boldsymbol{\mathbf{K}}(\boldsymbol{\mathbf{r}},t)\cdot
d\boldsymbol{\mathbf{S}}}$ (69)

From here, we take the curl of both sides of equation (67) and by combining
with equation (59) it is obtained that:

$\nabla\times\boldsymbol{\mathbf{K}}(\boldsymbol{\mathbf{r}},t)=-\nabla\times\boldsymbol{\mathbf{N}}(\boldsymbol{\mathbf{r}},t)=\frac{\partial}{\partial
t}\boldsymbol{\mathbf{M}}(\boldsymbol{\mathbf{r}},t)$ (70)

Because surface $S$ and contour $C$ are stationary we can write that
$\frac{\partial}{\partial
t}\boldsymbol{\mathbf{M}}(\boldsymbol{\mathbf{r}},t)=\frac{d}{dt}\boldsymbol{\mathbf{M}}(\boldsymbol{\mathbf{r}},t)$.
Inserting equation (70) into equation (69) and taking into account that
surface $S$ and contour $C$ are not moving yields:

$0=\oint_{C}{\boldsymbol{\mathbf{E}}_{c}(\boldsymbol{\mathbf{r}},t_{p}-\delta
t)\cdot
d\boldsymbol{\mathbf{r}}}=\oint_{C}{\boldsymbol{\mathbf{N}}(\boldsymbol{\mathbf{r}},t)\cdot
d\boldsymbol{\mathbf{r}}}+\frac{d}{dt}\int_{S}{\boldsymbol{\mathbf{M}}(\boldsymbol{\mathbf{r}},t)\cdot
d\boldsymbol{\mathbf{S}}}$ (71)

The right hand side of equation (71) is unknown energy conservation principle
valid for varying in time dynamic fields
$\boldsymbol{\mathbf{N}}(\boldsymbol{\mathbf{r}},t)$ and
$\boldsymbol{\mathbf{M}}(\boldsymbol{\mathbf{r}},t)$ and it is derived from
electrostatic energy conservation principle valid at present time. If
$\boldsymbol{\mathbf{N}}$ is replaced by $\boldsymbol{\mathbf{E}}$ and if
$\boldsymbol{\mathbf{M}}$ is replaced by $\boldsymbol{\mathbf{B}}$ it can be
seen that we have just obtained the physical law known in electrodynamics as
Faraday’s law.

From equation (71) the conclusion can be drawn about the nature of Faraday’s
law. It represents the energy conservation principle valid for non-
conservative dynamic fields and it is dynamic equivalent of electrostatic
energy conservation principle valid for Coulomb’s electrostatic field.

However, even the Faraday’s law itself can be considered as consequence of
something else. To see this, consider simply connected volume $V$ bounded by
surface $\partial V$ as shown in Fig. 4. The surface $\partial V$ is union of
two surfaces $S$ and $S_{1}$ bounded by respective contours $C$ and $C_{1}$.
Contours $C$ and $C_{1}$ consist of exactly the same spatial points, however,
the Stokes’ orientation of these contours is opposite $C=-C_{1}$. Then, using
$\nabla\times\boldsymbol{\mathbf{N}}(\boldsymbol{\mathbf{r}},t)=-\frac{\partial}{\partial
t}\boldsymbol{\mathbf{M}}(\boldsymbol{\mathbf{r}},t)$ the first right hand
side contour integral of equation (71) can be written as:

$\oint_{C}{\boldsymbol{\mathbf{N}}(\boldsymbol{\mathbf{r}},t)\cdot
d\boldsymbol{\mathbf{r}}}=-\oint_{C_{1}}{\boldsymbol{\mathbf{N}}(\boldsymbol{\mathbf{r}},t)\cdot
d\boldsymbol{\mathbf{r}}}=-\int_{S_{1}}{\nabla\times\boldsymbol{\mathbf{N}}(\boldsymbol{\mathbf{r}},t)\cdot
d\boldsymbol{\mathbf{S}}}=\frac{d}{dt}\int_{S_{1}}{\boldsymbol{\mathbf{M}}(\boldsymbol{\mathbf{r}},t)\cdot
d\boldsymbol{\mathbf{S}}}$ (72)

Replacing the first right hand side term of equation (71) with equation (72)
yields different form of dynamic energy conservation law:

$0=\oint_{C}{\boldsymbol{\mathbf{E}}_{c}(\boldsymbol{\mathbf{r}},t_{p}-\delta
t)\cdot d\boldsymbol{\mathbf{r}}}=\frac{d}{dt}\oint_{\partial
V}{\boldsymbol{\mathbf{M}}(\boldsymbol{\mathbf{r}},t)\cdot
d\boldsymbol{\mathbf{S}}}$ (73)

If we replace $\boldsymbol{\mathbf{M}}(\boldsymbol{\mathbf{r}},t)$ with
$\boldsymbol{\mathbf{B}}$ we see that right hand side of equation (73) is time
derivative of Gauss’ law for magnetic fields. The standard interpretation of
Gauss’ law for magnetic fields is that magnetic monopoles do not exist.
However, from equation (73) we conclude that alternative interpretation of
this law is that its time derivative represents the dynamic energy
conservation law. From the derivations presented, we might even say that
Faraday’s law is the consequence of Gauss’ law for magnetic fields. It should
be noted that these energy-conservation equations were all derived from simple
electrostatic Coulomb’s law.

Figure 4: Closed surface $\partial V$ that bounds volume $V$ is union of two
surfaces $S$ and $S_{1}$. Contour $C$ bounds surface $S$ and contour $C_{1}$
bounds surface $S_{1}$. Contours $C$ and $C_{1}$ are identical, however they
have different Stokes’ orientation.

From dynamic energy conservation law the derivation of Lorentz force is
straightforward: we now assume that all the points on surface $\partial V$
shown in Fig. 4 have some definite velocity $\boldsymbol{\mathbf{v}}$ such
that $|\boldsymbol{\mathbf{v}}|<<c$. Then the surface $\partial V$ is the
function of time, hence, $C=C(t)$ and $S=S(t)$. Hence, we can rewrite equation
(73) as the sum of two surface integrals:

$\frac{d}{dt}\oint_{\partial
V}{\boldsymbol{\mathbf{M}}(\boldsymbol{\mathbf{r}},t)\cdot
d\boldsymbol{\mathbf{S}}}=\frac{d}{dt}\oint_{S(t)}{\boldsymbol{\mathbf{M}}(\boldsymbol{\mathbf{r}},t)\cdot
d\boldsymbol{\mathbf{S}}}+\frac{d}{dt}\oint_{S_{1}(t)}{\boldsymbol{\mathbf{M}}(\boldsymbol{\mathbf{r}},t)\cdot
d\boldsymbol{\mathbf{S}}}=0$ (74)

where $\partial V=S(t)\cup S_{1}(t)$. The Leibniz identity [25] for moving
surfaces states that for any differentiable vector field
$\boldsymbol{\mathbf{P}}$ we can write:

$\frac{d}{dt}\int_{S(t)}{\boldsymbol{\mathbf{P}}\cdot
d\boldsymbol{\mathbf{S}}}=\int_{S(t)}{\left[\frac{\partial}{\partial
t}\boldsymbol{\mathbf{P}}+\left(\nabla\cdot\boldsymbol{\mathbf{P}}\right)\boldsymbol{\mathbf{v}}\right]\cdot
d\boldsymbol{\mathbf{S}}}-\oint_{C(t)}{\boldsymbol{\mathbf{v}}\times\boldsymbol{\mathbf{P}}\cdot
d\boldsymbol{\mathbf{r}}}$ (75)

Applying the Leibniz identity to the surface integral over surface $S_{1}(t)$
in equation (74) and using
$\nabla\cdot\boldsymbol{\mathbf{M}}(\boldsymbol{\mathbf{r}},t)=0$ yields:

$\frac{d}{dt}\oint_{S(t)}{\boldsymbol{\mathbf{M}}(\boldsymbol{\mathbf{r}},t)\cdot
d\boldsymbol{\mathbf{S}}}+\oint_{S_{1}(t)}{\frac{\partial}{\partial
t}\boldsymbol{\mathbf{M}}(\boldsymbol{\mathbf{r}},t)\cdot
d\boldsymbol{\mathbf{S}}}-\oint_{C_{1}(t)}{\boldsymbol{\mathbf{v}}\times\boldsymbol{\mathbf{M}}(\boldsymbol{\mathbf{r}},t)\cdot
d\boldsymbol{\mathbf{r}}}=0$ (76)

Using the result from previous section, i.e.
$\nabla\times\boldsymbol{\mathbf{N}}(\boldsymbol{\mathbf{r}},t)=-\frac{\partial}{\partial
t}\boldsymbol{\mathbf{M}}(\boldsymbol{\mathbf{r}},t)$, and applying the
Stokes’ theorem yields:

$\frac{d}{dt}\oint_{S(t)}{\boldsymbol{\mathbf{M}}(\boldsymbol{\mathbf{r}},t)\cdot
d\boldsymbol{\mathbf{S}}}-\oint_{C_{1}(t)}{\boldsymbol{\mathbf{N}}(\boldsymbol{\mathbf{r}},t)\cdot
d\boldsymbol{\mathbf{r}}}-\oint_{C_{1}(t)}{\boldsymbol{\mathbf{v}}\times\boldsymbol{\mathbf{M}}(\boldsymbol{\mathbf{r}},t)\cdot
d\boldsymbol{\mathbf{r}}}=0$ (77)

Because curves $C_{1}(t)$ and $C(t)$ comprise of same points, however, Stokes’
orientation of curves $C_{1}(t)$ and $C(t)$ is opposite, i.e.
$C_{1}(t)=-C(t)$, we can rewrite equation (77) as:

$\oint_{C(t)}{\left[\boldsymbol{\mathbf{N}}(\boldsymbol{\mathbf{r}},t)+\boldsymbol{\mathbf{v}}\times\boldsymbol{\mathbf{M}}(\boldsymbol{\mathbf{r}},t)\right]\cdot
d\boldsymbol{\mathbf{r}}}+\frac{d}{dt}\int_{S(t)}{\boldsymbol{\mathbf{M}}(\boldsymbol{\mathbf{r}},t)\cdot
d\boldsymbol{\mathbf{S}}}=0$ (78)

Note that equation (78) could not be derived from the right hand side of
equation (71), i.e. from Faraday’s law, even with Leibniz rule. For that
reason, we might take that the energy conservation law on the right hand side
of equation (73) is perhaps more general than the one given by equation (71).

Furthermore, note that the time derivative of the surface integral in equation
(78) does not represent the work of any force. However, from equation (73) we
know that the terms in equation (78) have dimensions of the work done by
electrodynamic force in moving the unit charge along contour $C(t)$. Because
the first term in equation (78) is contour integral of vector field we can
conclude that this term represents the non-zero work done by non-conservative
electrodynamic force in transporting the unit charge along contour $C(t)$.

Hence, just as the left hand side of equation (73) represents the work done by
conservative electrostatic force in transporting the unit charge along contour
$C$, the contour integral on the left of equation (78) represents the work
done by non-conservative electrodynamic force in transporting the unit charge
along the same contour. The purpose of surface integral on the left hand side
of equation (78) is to balance non-zero work of non-conservative
electrodynamic force along contour $C$. Thus, it can be concluded that the
electrodynamic force $\boldsymbol{\mathbf{F}}_{D}$ on charge $q$ moving with
velocity $\boldsymbol{\mathbf{v}}$ along contour $C$ is:

$\boldsymbol{\mathbf{F}}_{D}=q\boldsymbol{\mathbf{N}}(\boldsymbol{\mathbf{r}},t)+q\boldsymbol{\mathbf{v}}\times\boldsymbol{\mathbf{M}}(\boldsymbol{\mathbf{r}},t)$
(79)

Finally, in previous section we have shown that
$\boldsymbol{\mathbf{N}}=\boldsymbol{\mathbf{E}}$ and that
$\boldsymbol{\mathbf{M}}=\boldsymbol{\mathbf{B}}$. Thus, by replacing
$\boldsymbol{\mathbf{N}}$ with $\boldsymbol{\mathbf{E}}$ and
$\boldsymbol{\mathbf{M}}$ with $\boldsymbol{\mathbf{B}}$ it is obtained that:

$\boldsymbol{\mathbf{F}}_{D}=q\left(\boldsymbol{\mathbf{E}}+\boldsymbol{\mathbf{v}}\times\boldsymbol{\mathbf{B}}\right)$
(80)

which is expression for well known Lorentz force. It was derived theoretically
from the knowledge of electrostatic energy conservation law which, in turn,
can be derived from Coulomb’s law. Thus, we may say that we have just derived
the Lorentz force from simple electrostatic Coulomb’s law.

##  5 Conclusion

In this paper we have presented the theoretical framework that explains
Maxwell equations and the Lorentz force on more fundamental level than it was
previously done. Maxwell derived Maxwell equations from experimental Ampere’s
force law and experimental Faraday’s law, and Lorentz continued work on
Maxwell’s theory to discover the Lorentz force. In last 150 years, no
successful theory was presented that would explain Maxwell’s equations and
Lorentz force on more fundamental level.

To accomplish this, relativistically correct Liénard–Wiechert potentials,
Maxwell equations and the Lorentz force were derived directly from
electrostatic Coulomb’s law. In contrast to frequently criticized previous
attempts to derive Maxwell’s equations from Coulomb’s law using special
relativity and Lorentz transformation, the Lorentz transformation was not used
in our derivations nor the theory of special relativity. In fact, in this
work, dynamic Liénard–Wiechert potentials, Maxwell equations and Lorentz force
were derived from Coulomb’s law using the following two simple postulates:

  1. (a)

when charges are at rest the Coulomb’s law describes the force acting between
charges

  2. (b)

disturbances caused by moving charges propagate outwardly from moving charge
with finite velocity

The derivation of these dynamic physical laws from electrostatic Coulomb’s law
would not be possible without generalized Helmholtz decomposition theorem also
derived in this paper. This theorem allows the vector function of present
position and present time to be written as space-time integral of positions
and velocities at previous time. In contrast, standard Helmholtz decomposition
theorem is valid for functions of space only and it ignores time.

To derive the Lorentz force from Coulomb’s law, in section 4, the ”unknown”
dynamic energy conservation law valid in the past was derived from the
knowledge of electrostatic energy conservation law valid at present. The link
between the present and the past was again provided by generalized Helmholtz
decomposition theorem. This ”unknown” dynamic energy conservation principle
turned out to be Faraday’s law of induction. Additionally, it was shown that
Faraday’s law of induction can be considered equivalent to time derivative of
Gauss’ law for magnetic field. From these energy conservation considerations
the Lorentz force was derived.

From the presented analysis one important question naturally arises: are
Maxwell’s equations and Lorentz force the consequence of electrostatic
Coulomb’s law? They are most probably not. It is rather the opposite,
Coulomb’s law is the limiting case of Lorentz force when the source charge
becomes stationary. However, as it was shown in this paper, it is entirely
possible to deduce dynamic Maxwell equations and Lorentz force from the
knowledge of simple electrostatic Coulomb’s law.

Finally, this paper attempts to answer another important question: how can we
deduce more general dynamic physical laws from the limited knowledge provided
by static physical law? The significance of answering this question is that in
the future it will perhaps become possible that similar reasoning could deepen
the understanding of physical laws other than Maxwell equations and Lorentz
force.

##  Appendix A Derivation of generalized Helmholtz decomposition theorem

In this appendix, we derive the generalized Helmholtz decomposition theorem
for vector functions of space and time. However, in effort to enhance the
readability of this work, we first start by considering some basic identities
given in section A.1 of this appendix.

###  A.1 Preliminary considerations

To clarify notation used throughout this paper we first define position
vectors $\boldsymbol{\mathbf{r}}$ and $\boldsymbol{\mathbf{r}}^{\prime}$ as:

$\displaystyle\boldsymbol{\mathbf{r}}$
$\displaystyle=x\boldsymbol{\mathbf{\hat{x}}}+y\boldsymbol{\mathbf{\hat{y}}}+z\boldsymbol{\mathbf{\hat{z}}}$
(A.1.81) $\displaystyle\boldsymbol{\mathbf{r}}^{\prime}$
$\displaystyle=x^{\prime}\boldsymbol{\mathbf{\hat{x}}}+y^{\prime}\boldsymbol{\mathbf{\hat{y}}}+z^{\prime}\boldsymbol{\mathbf{\hat{z}}}$
(A.1.82)

where $\boldsymbol{\mathbf{\hat{x}}}$, $\boldsymbol{\mathbf{\hat{y}}}$ and
$\boldsymbol{\mathbf{\hat{z}}}$ are Cartesian, mutually orthogonal, unit basis
vectors. Variables $x,y,z\in\mathbb{R}$ and
$x^{\prime},y^{\prime},z^{\prime}\in\mathbb{R}$ are linearly independent
variables. Furthermore, throughout this paper we use position vector
$\boldsymbol{\mathbf{r}}_{s}(t^{\prime})$ to indicate the position of the
source charge. This position vector $\boldsymbol{\mathbf{r}}_{s}(t^{\prime})$
is defined as:

$\boldsymbol{\mathbf{r}}_{s}(t^{\prime})=x_{s}(t^{\prime})\boldsymbol{\mathbf{\hat{x}}}+y_{s}(t^{\prime})\boldsymbol{\mathbf{\hat{y}}}+z_{s}(t^{\prime})\boldsymbol{\mathbf{\hat{z}}}$
(A.1.83)

where $x_{s}(t^{\prime})$, $y_{s}(t^{\prime})$ and $z_{s}(t^{\prime})$ are all
functions of real variable $t^{\prime}\in\mathbb{R}$ which is independent of
variables $x,y,z\in\mathbb{R}$ and
$x^{\prime},y^{\prime},z^{\prime}\in\mathbb{R}$. The time derivative of
position vector $\boldsymbol{\mathbf{r}}_{s}(t^{\prime})$ is velocity
$\boldsymbol{\mathbf{v}}_{s}(t^{\prime})$ of the source charge:

$\boldsymbol{\mathbf{v}}_{s}(t^{\prime})=\frac{\partial\boldsymbol{\mathbf{r}}_{s}(t^{\prime})}{\partial
t^{\prime}}$ (A.1.84)

On many occasions in this paper we have used differential operators $\nabla$
and $\nabla^{\prime}$ defined as:

$\displaystyle\nabla$
$\displaystyle=\boldsymbol{\mathbf{\hat{x}}}\frac{\partial}{\partial
x}+\boldsymbol{\mathbf{\hat{y}}}\frac{\partial}{\partial
x}+\boldsymbol{\mathbf{\hat{z}}}\frac{\partial}{\partial x}$ (A.1.85)
$\displaystyle\nabla^{\prime}$
$\displaystyle=\boldsymbol{\mathbf{\hat{x}}}\frac{\partial}{\partial
x^{\prime}}+\boldsymbol{\mathbf{\hat{y}}}\frac{\partial}{\partial
y^{\prime}}+\boldsymbol{\mathbf{\hat{z}}}\frac{\partial}{\partial z^{\prime}}$
(A.1.86)

Operator $\nabla$ acts only on functions of variables $x,y,z$, hence, on
functions of position vector $\boldsymbol{\mathbf{r}}$. On the other hand,
operator $\nabla^{\prime}$ acts only on functions of variables
$x^{\prime},y^{\prime},z^{\prime}$, thus, it acts on functions of position
vector $\boldsymbol{\mathbf{r}}^{\prime}$. For example, if function $f$ is the
function of position vector $\boldsymbol{\mathbf{r}}$, that is
$f=f(\boldsymbol{\mathbf{r}})$ we can generally write:

$\nabla f(\boldsymbol{\mathbf{r}})\neq 0\hskip
20.00003pt\nabla^{\prime}f(\boldsymbol{\mathbf{r}})=0$ (A.1.87)

On the other hand, if function $f$ is the function of position vector
$\boldsymbol{\mathbf{r}}^{\prime}$, that is if
$f=f(\boldsymbol{\mathbf{r}}^{\prime})$ we can write:

$\nabla f(\boldsymbol{\mathbf{r}}^{\prime})=0\hskip
20.00003pt\nabla^{\prime}f(\boldsymbol{\mathbf{r}}^{\prime})\neq 0$ (A.1.88)

Furthermore, because variable $t^{\prime}$ is independent of variables $x,y,z$
and $x^{\prime},y^{\prime},z^{\prime}$ neither operator $\nabla$ nor
$\nabla^{\prime}$ acts on position vector
$\boldsymbol{\mathbf{r}}_{s}(t^{\prime})$ and velocity vector
$\boldsymbol{\mathbf{v}}_{s}(t^{\prime})$. Using these considerations we see
that the following equations are correct:

$\begin{array}[]{rcr}\nabla\cdot\boldsymbol{\mathbf{r}}_{s}(t^{\prime})=0&&\nabla\cdot\boldsymbol{\mathbf{v}}_{s}(t^{\prime})=0\\\
\nabla^{\prime}\cdot\boldsymbol{\mathbf{r}}_{s}(t^{\prime})=0&&\nabla^{\prime}\cdot\boldsymbol{\mathbf{v}}_{s}(t^{\prime})=0\end{array}$
(A.1.89)

However, both operators $\nabla$ and $\nabla^{\prime}$ act on Green’s function
$G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})$
given by equation (16). In fact, one can easily verify that the following
equations hold:

$\begin{array}[]{rcr}\nabla
G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})=-\nabla^{\prime}G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})&&\nabla^{2}G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})=\nabla^{\prime
2}G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})\\\
\displaystyle\frac{\partial}{\partial
t}G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})=-\frac{\partial}{\partial
t^{\prime}}G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})&&\displaystyle\frac{\partial^{2}}{\partial
t^{2}}G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})=\frac{\partial^{2}}{\partial
t^{\prime
2}}G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})\end{array}$
(A.1.90)

###  A.2 Generalized Helmholtz decomposition theorem

To start deriving generalized Helmholtz decomposition theorem for vector
functions of space and time we first consider inhomogeneous transient wave
equation:

$\nabla^{2}G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})-\frac{1}{c^{2}}\frac{\partial^{2}}{\partial
t^{2}}G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})=-\delta(\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}^{\prime})\delta(t-t^{\prime})$
(A.2.91)

where
$G(\boldsymbol{\mathbf{r}},\boldsymbol{\mathbf{t}};\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})$
is the function called fundamental solution or Green’s function and $\delta$
is Dirac’s delta function. The Green’s function for inhomogeneous wave
equation is well known and it represents an outgoing diverging spherical wave:

$G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})=\frac{\delta\left(t-t^{\prime}-\frac{\left|\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}^{\prime}\right|}{c}\right)}{4\pi\left|\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}^{\prime}\right|}$
(A.2.92)

Let us now suppose that vector field $\boldsymbol{\mathbf{F}}$ is the function
of both space $\boldsymbol{\mathbf{r}}$ and time $t$, i.e.
$\boldsymbol{\mathbf{F}}=\boldsymbol{\mathbf{F}}(\boldsymbol{\mathbf{r}},t)$.
Using sifting property of Dirac delta function we can write vector function
$\boldsymbol{\mathbf{F}}(\boldsymbol{\mathbf{r}},t)$ as the volume integral
over infinite volume $\mathbb{R}^{3}$ and over all the time $\mathbb{R}$ as:

$\boldsymbol{\mathbf{F}}(\boldsymbol{\mathbf{r}},t)=\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\boldsymbol{\mathbf{F}}(\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})\delta(\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}^{\prime})\delta(t-t^{\prime})dV^{\prime}}}$
(A.2.93)

where differential volume element $dV^{\prime}$ is
$dV^{\prime}=dx^{\prime}dy^{\prime}dz^{\prime}$. We now replace
$\delta(\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}^{\prime})\delta(t-t^{\prime})$
in equation above with left hand side of equation (A.2.91) to obtain:

$\boldsymbol{\mathbf{F}}(\boldsymbol{\mathbf{r}},t)=-\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\boldsymbol{\mathbf{F}}(\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})\left[\nabla^{2}G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})-\frac{1}{c^{2}}\frac{\partial^{2}}{\partial
t^{2}}G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})\right]dV^{\prime}}}$
(A.2.94)

From the discussion presented in section A.1 of this appendix, we know that
D’Alambert operator $\nabla^{2}-\frac{1}{c^{2}}\frac{\partial^{2}}{\partial
t^{2}}$ does not act on variables $x^{\prime}$, $y^{\prime}$,$z^{\prime}$ and
$t^{\prime}$ nor does it act on vector function
$\boldsymbol{\mathbf{F}}(\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})$. Hence,
we can write the D’Alambert operator
$\nabla^{2}-\frac{1}{c^{2}}\frac{\partial^{2}}{\partial t^{2}}$ in front of
the integral:

$\boldsymbol{\mathbf{F}}(\boldsymbol{\mathbf{r}},t)=-\left(\nabla^{2}-\frac{1}{c^{2}}\frac{\partial^{2}}{\partial
t^{2}}\right)\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\boldsymbol{\mathbf{F}}(\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}$
(A.2.95)

Using standard vector calculus identity
$\nabla\times\nabla\times\boldsymbol{\mathbf{P}}=\nabla(\nabla\cdot\boldsymbol{\mathbf{P}})-\nabla^{2}\boldsymbol{\mathbf{P}}$
we can rewrite equation (A.2.95) as:

$\displaystyle\boldsymbol{\mathbf{F}}(\boldsymbol{\mathbf{r}},t)=$
$\displaystyle\nabla\times\nabla\times\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\boldsymbol{\mathbf{F}}(\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}$
(A.2.96)
$\displaystyle-\nabla\left(\nabla\cdot\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\boldsymbol{\mathbf{F}}(\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}\right)$
$\displaystyle+\frac{1}{c^{2}}\frac{\partial^{2}}{\partial
t^{2}}\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\boldsymbol{\mathbf{F}}(\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}$

Because operators $\nabla$ and $\frac{\partial}{\partial t}$ do not act on
variables $x^{\prime}$, $y^{\prime}$, $z^{\prime}$ and $t^{\prime}$ we can
move operator $\nabla$ and partial derivative $\frac{\partial}{\partial t}$
under right hand side integrals in equation (A.2.96). Then using standard
vector calculus identities
$\nabla\times(\psi\boldsymbol{\mathbf{P}})=\nabla\psi\times\boldsymbol{\mathbf{P}}+\psi\nabla\times\boldsymbol{\mathbf{P}}$
and
$\nabla\cdot(\psi\boldsymbol{\mathbf{P}})=\nabla\psi\cdot\boldsymbol{\mathbf{P}}+\psi\nabla\cdot\boldsymbol{\mathbf{P}}$,
and noting that
$\nabla\times\boldsymbol{\mathbf{F}}(\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})=0$
and
$\nabla\cdot\boldsymbol{\mathbf{F}}(\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})=0$
we can rewrite equation (A.2.96) as:

$\displaystyle\boldsymbol{\mathbf{F}}(\boldsymbol{\mathbf{r}},t)=$
$\displaystyle\nabla\times\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\nabla
G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})\times\boldsymbol{\mathbf{F}}(\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}$
(A.2.97)
$\displaystyle-\nabla\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\boldsymbol{\mathbf{F}}(\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})\cdot\nabla
G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}$
$\displaystyle+\frac{1}{c^{2}}\frac{\partial}{\partial
t}\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\boldsymbol{\mathbf{F}}(\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})\frac{\partial}{\partial
t}G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}$

We now use identities $\nabla
G(\boldsymbol{\mathbf{r}},\boldsymbol{\mathbf{t}};\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})=-\nabla^{\prime}G(\boldsymbol{\mathbf{r}},\boldsymbol{\mathbf{t}};\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})$
and $\frac{\partial}{\partial
t}G(\boldsymbol{\mathbf{r}},\boldsymbol{\mathbf{t}};\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})=-\frac{\partial}{\partial
t^{\prime}}G(\boldsymbol{\mathbf{r}},\boldsymbol{\mathbf{t}};\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})$
to rewrite the right hand side integrals in equation (A.2.97) as:

$\displaystyle\boldsymbol{\mathbf{F}}(\boldsymbol{\mathbf{r}},t)=$
$\displaystyle-\nabla\times\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\nabla^{\prime}G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})\times\boldsymbol{\mathbf{F}}(\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}$
(A.2.98)
$\displaystyle+\nabla\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\boldsymbol{\mathbf{F}}(\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})\cdot\nabla^{\prime}G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}$
$\displaystyle-\frac{1}{c^{2}}\frac{\partial}{\partial
t}\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\boldsymbol{\mathbf{F}}(\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})\frac{\partial}{\partial
t^{\prime}}G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}$

Using vector calculus identity
$\nabla\times(\psi\boldsymbol{\mathbf{P}})=\nabla\psi\times\boldsymbol{\mathbf{P}}+\psi\nabla\times\boldsymbol{\mathbf{P}}$
and the the form of divergence theorem
$\int_{V}{\nabla\times\boldsymbol{\mathbf{P}}dV}=\oint_{\partial
V}{\boldsymbol{\mathbf{P}}\times d\boldsymbol{\mathbf{S}}}$ we rewrite the
first right hand side integral over $\mathbb{R}^{3}$ as:

$\displaystyle\int_{\mathbb{R}^{3}}{\nabla^{\prime}G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})\times\boldsymbol{\mathbf{F}}(\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}=$
$\displaystyle\oint_{\partial\mathbb{R}^{3}}{G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})\boldsymbol{\mathbf{F}}(\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})\times
d\boldsymbol{\mathbf{S}}^{\prime}}$ (A.2.99)
$\displaystyle-\int_{\mathbb{R}^{3}}{G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})\nabla^{\prime}\times\boldsymbol{\mathbf{F}}(\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}$

Note that the surface $\partial\mathbb{R}^{3}$ is an infinite surface that
bounds an infinite volume $\mathbb{R}^{3}$. Furthermore, for the surface
integral in the equation above, position vector
$\boldsymbol{\mathbf{r^{\prime}}}$ is located on infinite surface
$\partial\mathbb{R}^{3}$, i.e.
$\boldsymbol{\mathbf{r}}^{\prime}\in\partial\mathbb{R}^{3}$. Hence, if vector
function
$\boldsymbol{\mathbf{F}}(\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})$
decreases faster than
$1/\left|\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}^{\prime}\right|$ as
$\left|\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}^{\prime}\right|\rightarrow\infty$
the surface integral in equation (A.2.99) vanishes. In that case, we can
write:

$\int_{\mathbb{R}^{3}}{\nabla^{\prime}G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})\times\boldsymbol{\mathbf{F}}(\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}=-\int_{\mathbb{R}^{3}}{G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})\nabla^{\prime}\times\boldsymbol{\mathbf{F}}(\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}$
(A.2.100)

Using similar considerations, vector calculus identity
$\nabla\cdot(\psi\boldsymbol{\mathbf{P}})=\nabla\psi\cdot\boldsymbol{\mathbf{P}}+\psi\nabla\cdot\boldsymbol{\mathbf{P}}$
and standard divergence theorem
$\int_{V}{\nabla\cdot\boldsymbol{\mathbf{P}}dV}=\oint_{\partial
V}{\boldsymbol{\mathbf{P}}\cdot d\boldsymbol{\mathbf{S}}}$ it is obtained
that:

$\int_{\mathbb{R}^{3}}{\boldsymbol{\mathbf{F}}(\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})\cdot\nabla^{\prime}G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}=-\int_{\mathbb{R}^{3}}{G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}},t^{\prime})\nabla^{\prime}\cdot\boldsymbol{\mathbf{F}}(\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}$
(A.2.101)

To treat the last integral on the right hand side of equation (A.2.98) we use
the following identity:

$\boldsymbol{\mathbf{F}}(\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})\frac{\partial}{\partial
t^{\prime}}G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}},t^{\prime})=\frac{\partial}{\partial
t^{\prime}}\left(\boldsymbol{\mathbf{F}}(\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}},t^{\prime})\right)-G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}},t^{\prime})\frac{\partial}{\partial
t^{\prime}}\boldsymbol{\mathbf{F}}(\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})$
(A.2.102)

Using the identity above and noting that $t^{\prime}$ is independent of
$x^{\prime}$, $y^{\prime}$ and $z^{\prime}$ we can rewrite the last right hand
side integral of equation (A.2.98) as:

$\displaystyle\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\boldsymbol{\mathbf{F}}(\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})\frac{\partial}{\partial
t^{\prime}}G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}=$
$\displaystyle\int_{\mathbb{R}}{dt^{\prime}\frac{\partial}{\partial
t^{\prime}}\int_{\mathbb{R}^{3}}{\boldsymbol{\mathbf{F}}(\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}$
(A.2.103)
$\displaystyle-\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})\frac{\partial}{\partial
t^{\prime}}\boldsymbol{\mathbf{F}}(\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}$

By integrating over $t^{\prime}$, it can be shown that the first right hand
side integral in equation (A.2.103) vanishes :

$\displaystyle\int_{\mathbb{R}}{dt^{\prime}\frac{\partial}{\partial
t^{\prime}}\int_{\mathbb{R}^{3}}{\boldsymbol{\mathbf{F}}(\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}=\left[\int_{\mathbb{R}^{3}}{\boldsymbol{\mathbf{F}}(\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}\right]_{t^{\prime}\rightarrow-\infty}^{t^{\prime}\rightarrow\infty}$
(A.2.104)

If $t^{\prime}\rightarrow\pm\infty$, and if $t$ is finite, then from equation
(A.2.92) follows that
$G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})=0$,
thus, the right hand side of equation (A.2.104) is equal to zero. Inserting
this result into equation (A.2.103) yields:

$\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\boldsymbol{\mathbf{F}}(\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})\frac{\partial}{\partial
t^{\prime}}G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}=-\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})\frac{\partial}{\partial
t^{\prime}}\boldsymbol{\mathbf{F}}(\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}$
(A.2.105)

By inserting equations (A.2.100), (A.2.101) and (A.2.105) into equation
(A.2.98) we obtain the generalized Helmholtz theorem for vector functions of
space and time:

$\displaystyle\boldsymbol{\mathbf{F}}(\boldsymbol{\mathbf{r}},t)=$
$\displaystyle-\nabla\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\bigg{(}\nabla^{\prime}\cdot\boldsymbol{\mathbf{F}}(\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})\bigg{)}G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}$
(A.2.106) $\displaystyle+\frac{1}{c^{2}}\frac{\partial}{\partial
t}\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\left(\frac{\partial}{\partial
t^{\prime}}\boldsymbol{\mathbf{F}}(\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})\right)G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}$
$\displaystyle+\nabla\times\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\bigg{(}\nabla^{\prime}\times\boldsymbol{\mathbf{F}}(\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})\bigg{)}G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}$

The theorem is valid for functions
$\boldsymbol{\mathbf{F}}(\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})$ that
decrease faster than
$1/\left|\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}^{\prime}\right|$ as
$\left|\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}^{\prime}\right|\rightarrow\infty$.

##  Appendix B Novel vector calculus identities

In this appendix we prove two novel vector calculus identities, without which,
it would be very difficult, perhaps even not possible, to derive Maxwell’s
equations from Coulomb’s law. These two novel vector calculus identities are
given by the following two equations:

$\displaystyle\int_{V}{\psi\nabla^{2}\boldsymbol{\mathbf{P}}dV}$
$\displaystyle=\oint_{\partial
V}{\psi\left(d\boldsymbol{\mathbf{S}}\cdot\nabla\right)\boldsymbol{\mathbf{P}}}-\int_{V}{\left(\nabla\psi\cdot\nabla\right)\boldsymbol{\mathbf{P}}dV}$
(B.107) $\displaystyle\int_{V}{\boldsymbol{\mathbf{P}}\nabla^{2}\psi dV}$
$\displaystyle=\oint_{\partial V}{\boldsymbol{\mathbf{P}}\left(\nabla\psi\cdot
d\boldsymbol{\mathbf{S}}\right)}-\int_{V}{\left(\nabla\psi\cdot\nabla\right)\boldsymbol{\mathbf{P}}dV}$
(B.108)

where $\boldsymbol{\mathbf{P}}$ is differentiable vector field, $\psi$ is
differentiable scalar function, volume $V\subset\mathbb{R}^{3}$ is simply
connected volume, $\partial V$ is the bounding surface of volume $V$ and
$d\boldsymbol{\mathbf{S}}$ is differential surface element of $\partial V$
such that $d\boldsymbol{\mathbf{S}}=\boldsymbol{\mathbf{n}}dS$. Vector
$\boldsymbol{\mathbf{n}}$ is an outward unit normal to the surface $\partial
V$. In Cartesian coordinate system the product
$\psi\nabla^{2}\boldsymbol{\mathbf{P}}$ can be written in terms of Cartesian
components as:

$\psi\nabla^{2}\boldsymbol{\mathbf{P}}=\hat{\boldsymbol{\mathbf{x}}}\psi\nabla^{2}P_{x}+\hat{\boldsymbol{\mathbf{y}}}\psi\nabla^{2}P_{y}+\hat{\boldsymbol{\mathbf{z}}}\psi\nabla^{2}P_{z}$
(B.109)

where $P_{x}$, $P_{y}$ and $P_{z}$ are Cartesian components of vector
$\boldsymbol{\mathbf{P}}$ and vectors $\boldsymbol{\mathbf{\hat{x}}}$,
$\boldsymbol{\mathbf{\hat{y}}}$ and $\boldsymbol{\mathbf{\hat{z}}}$ are
Cartesian unit basis vectors. Using standard vector calculus identity
$\nabla\cdot f\boldsymbol{\mathbf{T}}=\nabla
f\cdot\boldsymbol{\mathbf{T}}+f\nabla\cdot\boldsymbol{\mathbf{T}}$, valid for
some scalar function $f$ and for some vector function
$\boldsymbol{\mathbf{T}}$, we can rewrite equation (B.109) as:

$\displaystyle\psi\nabla^{2}\boldsymbol{\mathbf{P}}$
$\displaystyle=\hat{\boldsymbol{\mathbf{x}}}\nabla\cdot\left(\psi\nabla
P_{x}\right)-\hat{\boldsymbol{\mathbf{x}}}\nabla\psi\cdot\nabla P_{x}$ (B.110)
$\displaystyle+\hat{\boldsymbol{\mathbf{y}}}\nabla\cdot\left(\psi\nabla
P_{y}\right)-\hat{\boldsymbol{\mathbf{y}}}\nabla\psi\cdot\nabla P_{y}$
$\displaystyle+\hat{\boldsymbol{\mathbf{z}}}\nabla\cdot\left(\psi\nabla
P_{z}\right)-\hat{\boldsymbol{\mathbf{z}}}\nabla\psi\cdot\nabla P_{z}$

To proceed, we now expand the identity
$\left(\nabla\psi\cdot\nabla\right)\boldsymbol{\mathbf{P}}$ in terms of its
Cartesian components as:

$\displaystyle\left(\nabla\psi\cdot\nabla\right)\boldsymbol{\mathbf{P}}$
$\displaystyle=\left(\frac{\partial\psi}{\partial x}\frac{\partial}{\partial
x}+\frac{\partial\psi}{\partial y}\frac{\partial}{\partial
y}+\frac{\partial\psi}{\partial z}\frac{\partial}{\partial
z}\right)\left(\boldsymbol{\mathbf{\hat{x}}}P_{x}+\boldsymbol{\mathbf{\hat{y}}}P_{y}+\boldsymbol{\mathbf{\hat{z}}}P_{z}\right)$
(B.111) $\displaystyle=\boldsymbol{\mathbf{\hat{x}}}\nabla\psi\cdot\nabla
P_{x}+\boldsymbol{\mathbf{\hat{y}}}\nabla\psi\cdot\nabla
P_{y}+\boldsymbol{\mathbf{\hat{z}}}\nabla\psi\cdot\nabla P_{z}$

Inserting equation (B.111) into (B.110) it is obtained that:

$\psi\nabla^{2}\boldsymbol{\mathbf{P}}=\hat{\boldsymbol{\mathbf{x}}}\nabla\cdot\left(\psi\nabla
P_{x}\right)+\hat{\boldsymbol{\mathbf{y}}}\nabla\cdot\left(\psi\nabla
P_{y}\right)+\hat{\boldsymbol{\mathbf{z}}}\nabla\cdot\left(\psi\nabla
P_{z}\right)-\left(\nabla\psi\cdot\nabla\right)\boldsymbol{\mathbf{P}}$
(B.112)

We now integrate equation (B.112) over volume $V$ and apply the divergence
theorem $\int_{V}{\nabla\cdot\boldsymbol{\mathbf{T}}dV}=\oint_{\partial
V}{\boldsymbol{\mathbf{T}}\cdot d\boldsymbol{\mathbf{S}}}$ to obtain:

$\displaystyle\int_{V}{\psi\nabla^{2}\boldsymbol{\mathbf{P}}dV}=$
$\displaystyle\hat{\boldsymbol{\mathbf{x}}}\oint_{\partial V}{\psi\nabla
P_{x}\cdot
d\boldsymbol{\mathbf{S}}}+\hat{\boldsymbol{\mathbf{y}}}\oint_{\partial
V}{\psi\nabla P_{y}\cdot
d\boldsymbol{\mathbf{S}}}+\hat{\boldsymbol{\mathbf{z}}}\oint_{\partial
V}{\psi\nabla P_{z}\cdot d\boldsymbol{\mathbf{S}}}$ (B.113)
$\displaystyle-\int_{V}{\left(\nabla\psi\cdot\nabla\right)\boldsymbol{\mathbf{P}}dV}$

The first three right hand side terms of equation (B.113) can be rewritten as:

$\hat{\boldsymbol{\mathbf{x}}}\oint_{\partial V}{\psi\nabla P_{x}\cdot
d\boldsymbol{\mathbf{S}}}+\hat{\boldsymbol{\mathbf{y}}}\oint_{\partial
V}{\psi\nabla P_{y}\cdot
d\boldsymbol{\mathbf{S}}}+\hat{\boldsymbol{\mathbf{z}}}\oint_{\partial
V}{\psi\nabla P_{z}\cdot d\boldsymbol{\mathbf{S}}}=\oint_{\partial
V}{\psi\left(d\boldsymbol{\mathbf{S}}\cdot\nabla\right)\boldsymbol{\mathbf{P}}}$
(B.114)

Inserting equation (B.114) into (B.113) yields:

$\int_{V}{\psi\nabla^{2}\boldsymbol{\mathbf{P}}dV}=\oint_{\partial
V}{\psi\left(d\boldsymbol{\mathbf{S}}\cdot\nabla\right)\boldsymbol{\mathbf{P}}}-\int_{V}{\left(\nabla\psi\cdot\nabla\right)\boldsymbol{\mathbf{P}}dV}$
(B.115)

which we intended to prove. To prove equation (B.108) we rewrite
$\boldsymbol{\mathbf{P}}\nabla^{2}\psi$ in terms of Cartesian components as:

$\boldsymbol{\mathbf{P}}\nabla^{2}\psi=\boldsymbol{\mathbf{\hat{x}}}P_{x}\nabla^{2}\psi+\boldsymbol{\mathbf{\hat{y}}}P_{y}\nabla^{2}\psi+\boldsymbol{\mathbf{\hat{z}}}P_{z}\nabla^{2}\psi$
(B.116)

By using standard differential calculus identity
$f\nabla^{2}h=\nabla\cdot(f\nabla h)-\nabla f\cdot\nabla h$, where $f$ and $h$
are differentiable scalar functions, equation (B.116) can be written as:

$\displaystyle\boldsymbol{\mathbf{P}}\nabla^{2}\psi=$
$\displaystyle\boldsymbol{\mathbf{\hat{x}}}\nabla\cdot\left(P_{x}\nabla\psi\right)-\boldsymbol{\mathbf{\hat{x}}}\nabla
P_{x}\cdot\nabla\psi+$ (B.117)
$\displaystyle\boldsymbol{\mathbf{\hat{y}}}\nabla\cdot\left(P_{y}\nabla\psi\right)-\boldsymbol{\mathbf{\hat{y}}}\nabla
P_{y}\cdot\nabla\psi+$
$\displaystyle\boldsymbol{\mathbf{\hat{z}}}\nabla\cdot\left(P_{z}\nabla\psi\right)-\boldsymbol{\mathbf{\hat{z}}}\nabla
P_{z}\cdot\nabla\psi$

Inserting equation (B.111) into equation (B.117) yields:

$\boldsymbol{\mathbf{P}}\nabla^{2}\psi=\boldsymbol{\mathbf{\hat{x}}}\nabla\cdot\left(P_{x}\nabla\psi\right)+\boldsymbol{\mathbf{\hat{y}}}\nabla\cdot\left(P_{y}\nabla\psi\right)+\boldsymbol{\mathbf{\hat{z}}}\nabla\cdot\left(P_{z}\nabla\psi\right)-\left(\nabla\psi\cdot\nabla\right)\boldsymbol{\mathbf{P}}$
(B.118)

Integrating equation (B.118) over volume $V$ and applying divergence theorem
$\int_{V}{\nabla\cdot\boldsymbol{\mathbf{T}}dV}=\oint_{\partial
V}{\boldsymbol{\mathbf{T}}\cdot d\boldsymbol{\mathbf{S}}}$ it is obtained
that:

$\int_{V}{\boldsymbol{\mathbf{P}}\nabla^{2}\psi dV}=\oint_{\partial
V}{\boldsymbol{\mathbf{P}}\left(\nabla\psi\cdot
d\boldsymbol{\mathbf{S}}\right)}-\int_{V}{\left(\nabla\psi\cdot\nabla\right)\boldsymbol{\mathbf{P}}dV}$
(B.119)

which we intended to prove.

##  Appendix C Derivation of auxiliary mathematical identities

In this appendix we derive auxiliary mathematical identities that we find
useful for the derivation of Maxwell equations from Coulomb’s law.

###  C.1 Derivation of equation (28)

Equation (24) allows us to rewrite the time derivative in the second right
hand side integral in equation (27) as:

$\frac{\partial}{\partial
t^{\prime}}\frac{q_{s}}{4\pi\epsilon}\frac{\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|^{3}}=-\frac{q_{s}}{4\pi\epsilon}\frac{\partial}{\partial
t^{\prime}}\nabla^{\prime}\frac{1}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|}$
(C.1.120)

Because coordinates $x^{\prime}$, $y^{\prime}$ and $z^{\prime}$ are
independent of time $t^{\prime}$ we can swap operator $\nabla^{\prime}$ and
time derivative with respect to time $t^{\prime}$ as:

$\frac{\partial}{\partial
t^{\prime}}\frac{q_{s}}{4\pi\epsilon}\frac{\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|^{3}}=-\frac{q_{s}}{4\pi\epsilon}\nabla^{\prime}\frac{\partial}{\partial
t^{\prime}}\frac{1}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|}$
(C.1.121)

Furthermore, because coordinates $x^{\prime}$, $y^{\prime}$ and $z^{\prime}$
are independent of time $t^{\prime}$, the time derivative of
$\boldsymbol{\mathbf{r}}^{\prime}$ is equal to zero
$\frac{\partial\boldsymbol{\mathbf{r}}^{\prime}}{\partial t^{\prime}}=0$. The
inner time derivative in the equation (C.1.121) can now be written as:

$\frac{\partial}{\partial
t^{\prime}}\frac{1}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|}=\frac{\boldsymbol{\mathbf{v}}_{s}(t^{\prime})\cdot\left(\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right)}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|^{3}}=-\nabla^{\prime}\cdot\frac{\boldsymbol{\mathbf{v}}_{s}(t^{\prime})}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|}$
(C.1.122)

where $\boldsymbol{\mathbf{v}}_{s}(t^{\prime})$ is the velocity of the source
charge $q_{s}$ at time $t^{\prime}$ given by equation (A.1.84). Inserting
equation (C.1.122) into equation (C.1.121) yields:

$\frac{\partial}{\partial
t^{\prime}}\frac{q_{s}}{4\pi\epsilon}\frac{\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|^{3}}=\frac{q_{s}}{4\pi\epsilon}\nabla^{\prime}\left(\nabla^{\prime}\cdot\frac{\boldsymbol{\mathbf{v}}_{s}(t^{\prime})}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|}\right)$
(C.1.123)

To proceed with derivation, we now make use of standard vector calculus
identity
$\nabla\times\nabla\times\boldsymbol{\mathbf{P}}=\nabla\left(\nabla\cdot\boldsymbol{\mathbf{P}}\right)-\nabla^{2}\boldsymbol{\mathbf{P}}$,
valid for any differentiable vector function $\boldsymbol{\mathbf{P}}$. This
identity allows us to rewrite the equation (C.1.123) as:

$\frac{\partial}{\partial
t^{\prime}}\frac{q_{s}}{4\pi\epsilon}\frac{\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|^{3}}=\frac{q_{s}}{4\pi\epsilon}\nabla^{\prime}\times\nabla^{\prime}\times\frac{\boldsymbol{\mathbf{v}}_{s}(t^{\prime})}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|}+\frac{q_{s}}{4\pi\epsilon}\nabla^{\prime
2}\frac{\boldsymbol{\mathbf{v}}_{s}(t^{\prime})}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|}$
(C.1.124)

Since Laplacian operator $\nabla^{\prime 2}$ does not have effect on velocity
vector $\boldsymbol{\mathbf{v}}_{s}(t^{\prime})$ the last right hand side term
in equation (C.1.124) can be written using 3D Dirac’s delta function as:

$\frac{q_{s}}{4\pi\epsilon}\nabla^{\prime
2}\frac{\boldsymbol{\mathbf{v}}_{s}(t^{\prime})}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|}=\frac{q_{s}}{\epsilon}\boldsymbol{\mathbf{v}}_{s}(t^{\prime})\nabla^{\prime
2}\frac{1}{4\pi}\frac{1}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|}=-\frac{q_{s}}{\epsilon}\boldsymbol{\mathbf{v}}_{s}(t^{\prime})\delta\left(\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right)$
(C.1.125)

Hence, replacing the last right hand side term of equation (C.1.124) with
equation (C.1.125) yields:

$\frac{\partial}{\partial
t^{\prime}}\frac{q_{s}}{4\pi\epsilon}\frac{\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|^{3}}=\frac{q_{s}}{4\pi\epsilon}\nabla^{\prime}\times\nabla^{\prime}\times\frac{\boldsymbol{\mathbf{v}}_{s}(t^{\prime})}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|}-\frac{q_{s}}{\epsilon}\boldsymbol{\mathbf{v}}_{s}(t^{\prime})\delta\left(\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right)$
(C.1.126)

which proves equation (28).

###  C.2 Derivation of equation (3)

To derive equation (3) we make use of standard vector calculus identity
$\nabla\times\left(\psi\boldsymbol{\mathbf{P}}\right)=\nabla\psi\times\boldsymbol{\mathbf{P}}+\psi\nabla\times\boldsymbol{\mathbf{P}}$,
where $\psi$ is a scalar function and $\boldsymbol{\mathbf{P}}$ is a vector
function, to rewrite the integrand in the last right hand side term of
equation (30) as:

$\displaystyle\left[\nabla^{\prime}\times\nabla^{\prime}\times\frac{\boldsymbol{\mathbf{v}}_{s}(t^{\prime})}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|}\right]G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})=$
$\displaystyle=\nabla^{\prime}\times\left[\left(\nabla^{\prime}\times\frac{\boldsymbol{\mathbf{v}}_{s}(t^{\prime})}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|}\right)G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})\right]-\nabla^{\prime}G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})\times\left[\nabla^{\prime}\times\frac{\boldsymbol{\mathbf{v}}_{s}(t^{\prime})}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|}\right]$

Integrating equation (C.2) with respect to variables $x^{\prime}$,
$y^{\prime}$, $z^{\prime}$ and $t^{\prime}$, and making use of a standard form
of divergence theorem
$\int_{V}{\nabla\times\boldsymbol{\mathbf{P}}dV}=\oint_{\partial
V}{d\boldsymbol{\mathbf{S}}\times\boldsymbol{\mathbf{P}}}$ it is obtained
that:

$\displaystyle\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\left[\nabla^{\prime}\times\nabla^{\prime}\times\frac{\boldsymbol{\mathbf{v}}_{s}(t^{\prime})}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|}\right]G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}=$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}}{dt^{\prime}\oint_{\partial\mathbb{R}^{3}}{d\boldsymbol{\mathbf{S}}^{\prime}\times\left(\nabla^{\prime}\times\frac{\boldsymbol{\mathbf{v}}_{s}(t^{\prime})}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|}\right)G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})}}$
$\displaystyle-\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\nabla^{\prime}G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})\times\left[\nabla^{\prime}\times\frac{\boldsymbol{\mathbf{v}}_{s}(t^{\prime})}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|}\right]dV^{\prime}}}$

where $dV=dx^{\prime}dy^{\prime}dz^{\prime}$, $\partial\mathbb{R}^{3}$ is an
infinite surface that bounds $\mathbb{R}^{3}$, and $d\boldsymbol{\mathbf{S}}$
is differential surface element of the surface $\partial\mathbb{R}^{3}$.
Because $\partial\mathbb{R}^{3}$ is an infinite surface, the first right hand
side integral vanishes. To see this, we can use standard vector identity
$\nabla\times(\psi\boldsymbol{\mathbf{P}})=\nabla\psi\times\boldsymbol{\mathbf{P}}+\psi\nabla\times\boldsymbol{\mathbf{P}}$
and using $\nabla^{\prime}\times\boldsymbol{\mathbf{v}}_{s}(t^{\prime})=0$ to
rewrite the first term in the first right hand side integrand as:

$\nabla^{\prime}\times\frac{\boldsymbol{\mathbf{v}}_{s}(t^{\prime})}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|}=-\frac{\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|^{3}}\times\boldsymbol{\mathbf{v}}_{s}(t^{\prime})$
(C.2.129)

Because $\boldsymbol{\mathbf{r}}^{\prime}\in\partial\mathbb{R}^{3}$ this means
that $\left|\boldsymbol{\mathbf{r^{\prime}}}\right|\rightarrow\infty$.
Provided that charge $q_{s}$ is moving with finite velocity
$\boldsymbol{\mathbf{v}}_{s}(t^{\prime})$ it is clear that right hand side
term of equation (C.2.129) vanishes as
$\left|\boldsymbol{\mathbf{r^{\prime}}}\right|\rightarrow\infty$. Thus, we can
now write equation (C.2) as:

$\displaystyle\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\left[\nabla^{\prime}\times\nabla^{\prime}\times\frac{\boldsymbol{\mathbf{v}}_{s}(t^{\prime})}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|}\right]G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}=$
$\displaystyle=$
$\displaystyle-\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\nabla^{\prime}G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})\times\left[\nabla^{\prime}\times\frac{\boldsymbol{\mathbf{v}}_{s}(t^{\prime})}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|}\right]dV^{\prime}}}$

There is another useful property of Green’s function
$G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})$
which enables us to proceed with the derivation of equation (3). This property
can be written as follows:

$\nabla^{\prime}G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})=-\nabla
G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})$
(C.2.131)

Using this property and standard vector calculus identity
$\nabla\times(\psi\boldsymbol{\mathbf{P}})=\nabla\psi\times\boldsymbol{\mathbf{P}}+\psi\nabla\times\boldsymbol{\mathbf{P}}$
allows us to rewrite the integrand in equation (C.2) as:

$\displaystyle\nabla^{\prime}G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})\times\left[\nabla^{\prime}\times\frac{\boldsymbol{\mathbf{v}}_{s}(t^{\prime})}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|}\right]=$
$\displaystyle=-\nabla
G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})\times\left[\nabla^{\prime}\times\frac{\boldsymbol{\mathbf{v}}_{s}(t^{\prime})}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|}\right]=$
$\displaystyle=-\nabla\times\left[G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})\nabla^{\prime}\times\frac{\boldsymbol{\mathbf{v}}_{s}(t^{\prime})}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|}\right]$

Equation above is valid because operator $\nabla$ does not act on velocity
vector $\boldsymbol{\mathbf{v}}_{s}(t^{\prime})$, nor does it act on position
vectors $\boldsymbol{\mathbf{r}}^{\prime}$ and
$\boldsymbol{\mathbf{r}}_{s}(t^{\prime})$. It only acts on Green’s function
$G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})$
because it is a function of position vector $\boldsymbol{\mathbf{r}}$.
Inserting equation (C.2) into equation (C.2) yields:

$\displaystyle\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\left[\nabla^{\prime}\times\nabla^{\prime}\times\frac{\boldsymbol{\mathbf{v}}_{s}(t^{\prime})}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|}\right]G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}=$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\nabla\times\left[G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})\nabla^{\prime}\times\frac{\boldsymbol{\mathbf{v}}_{s}(t^{\prime})}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|}\right]dV^{\prime}}}$

Because differential volume element is
$dV^{\prime}=dx^{\prime}dy^{\prime}dz^{\prime}$ and because operator $\nabla$
does not act on variables $x^{\prime}$, $y^{\prime}$, $z^{\prime}$ and
$t^{\prime}$ we can write operator $\nabla$ in front of the integral:

$\displaystyle\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\left[\nabla^{\prime}\times\nabla^{\prime}\times\frac{\boldsymbol{\mathbf{v}}_{s}(t^{\prime})}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|}\right]G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}=$
$\displaystyle=$
$\displaystyle\nabla\times\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})\nabla^{\prime}\times\frac{\boldsymbol{\mathbf{v}}_{s}(t^{\prime})}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|}dV^{\prime}}}$

Using the same trick again, that is, by using standard vector calculus
identity
$\nabla\times(\psi\boldsymbol{\mathbf{P}})=\nabla\psi\times\boldsymbol{\mathbf{P}}+\psi\nabla\times\boldsymbol{\mathbf{P}}$,
using
$\nabla^{\prime}G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})=-\nabla
G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})$ and
noting that operator $\nabla$ does not act on variables $x^{\prime}$,
$y^{\prime}$ and $z^{\prime}$ we can rewrite the integrand in equation (C.2)
as:

$\displaystyle
G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})\nabla^{\prime}\times\frac{\boldsymbol{\mathbf{v}}_{s}(t^{\prime})}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|}=$
$\displaystyle=$
$\displaystyle\nabla^{\prime}\times\left[G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})\frac{\boldsymbol{\mathbf{v}}_{s}(t^{\prime})}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|}\right]+\nabla\times\left[G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})\frac{\boldsymbol{\mathbf{v}}_{s}(t^{\prime})}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|}\right]$

Then, by inserting equation (C.2) into equation (C.2) and using a form of
standard divergence theorem
$\int_{V}{\nabla\times\boldsymbol{\mathbf{P}}dV}=\oint_{\partial
V}{d\boldsymbol{\mathbf{S}}\times\boldsymbol{\mathbf{P}}}$ we obtain that:

$\displaystyle\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\left[\nabla^{\prime}\times\nabla^{\prime}\times\frac{\boldsymbol{\mathbf{v}}_{s}(t^{\prime})}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|}\right]G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}=$
$\displaystyle=$
$\displaystyle\nabla\times\int_{\mathbb{R}}{dt^{\prime}\oint_{\partial\mathbb{R}^{3}}{d\boldsymbol{\mathbf{S}}^{\prime}\times\frac{\boldsymbol{\mathbf{v}}_{s}(t^{\prime})}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|}}}+$
$\displaystyle\nabla\times\
\int_{\mathbb{R}}{dt^{\prime}\int_{\partial\mathbb{R}^{3}}{\nabla\times\left[G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})\frac{\boldsymbol{\mathbf{v}}_{s}(t^{\prime})}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|}\right]dV^{\prime}}}$

Because surface $\partial\mathbb{R}^{3}$ is an infinite surface the magnitude
of position vector $\boldsymbol{\mathbf{r}}^{\prime}$ is infinite, hence the
surface integral in the first right hand side term of equation (C.2) vanishes.
Furthermore, because operator $\nabla$ does not act on variables $x^{\prime}$,
$y^{\prime}$, $z^{\prime}$ and $t^{\prime}$ we can write operator $\nabla$ in
front of the second right hand side space-time integral. Hence, equation (C.2)
can be written as:

$\displaystyle\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\left[\nabla^{\prime}\times\nabla^{\prime}\times\frac{\boldsymbol{\mathbf{v}}_{s}(t^{\prime})}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|}\right]G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}=$
$\displaystyle=$