Maxwell's Equations represents the Conservation of the Electric field strength E. E is the Force per unit charge. Maxwell's Equations are usually represented by four equations, two scalar or real equations and two vector equations.

These four equations are really just a single quaternion equation. Quaternions are four dimensional and model the spacetime universe. It is time to abandon spacetime and just talk about quaternion four space. Quaternion four space contains the Real Space and the Complex Space.

Quaternions consist of a real unit (1) and 3 vector units(i, j and k). The Rules that define quaternions are:

i² = j²= k²=ijk=-1

A quaternion E-field is E= Er + iEx + jEy + kEz = Er + Ev, where Er is the real part and Ev is the vector part.

In traditional Maxwell's Equations the real part of E is not seen.

William Rowan Hamilton developed quaternion in 1843 as a means of mathematically rotating a line in three space. It took him 15 years to discover that he needed a four space to mathematically do the rotation.

Hamilton developed vectors and the vector differential operator called Del:

Del= id/dx + jd/dy + kd/dz

Hamilton's Del is not a quaternion differential operator so I added d/dr to Del to have a Quaternion Differential Operator, I call X for change:

X= d/dr + Del = d/dR + id/dx + jd/dy + kd/dz

R is the real space and is also "time" in spacetime, r=ct. dr= cdt.

Thus X= d/cdt + id/dx + jd/dy + kd/dz


With this introduction lets look at Maxwell's Equations in their complete form.

0=XE=(d/dr + Del)(Er + Ev) = (dEr/dr - Del.Ev) + (dEv/dr + DelxEv + Del Er)


Many of you may not recognize Maxwell's Equations becuase they were developed in the days before spacetime. Lets introduce time into the equations with r=ct:

0=XE=(d/cdt + Del)(Er + Ev) = (dEr/cdt - Del.Ev) + (dEv/cdt + DelxEv + Del Er)


Still don't recognize Maxwell's Equations then lets rediscover Maxwell's discovery that there is a relationship between the Electric and the Magnetic fields such that: E=cB = zH = zcD

0=XE=(d/cdt + Del)(Er + Ev) = (dBr/dt - Del.Ev) + (dBv/dt + DelxEv + Del Er)


Now things are coming into view. These are two of Maxwell's Equations. The real part is the Continuity Equation and the vector part is Faraday's Equation.

0=(dBr/dt - Del.Ev)

0=(dBv/dt + DelxEv + Del Er)

Substituting H for E gives the other two equations:

0=XE=(d/cdt + Del)(Hr + Hv) = (dDr/dt - Del.Hv) + (dDv/dt + DelxHv + Del Hr)


We really don't need four Maxwell's Equations there is really only one Equation, the Invariance of the Electric Field. The equations 0=XE is the four space invariance equation.

Let's understand this equation better. 0=XE says that E is spacetime Invariant. It also says E is is bounded, it has limits. It also says that E is continuous, it also says that E is homeostatic. The sum of the vectors being zero means the vectors are coplanar and form a closed polygon in this case a triangle. Two legs of the triangle are right angles to each other so it is a right triangle. The hypotenuse is the dEv/dr = dEv/cdt. DelxEv is called the curl of Ev and Del Er is called the Gradient of Er. The curl is perpendicular to the Gradient. under 0=XE, Del Er = Del.Ev n where n is the unit radial vector. The curl specifies rotation around the origin and the gradient specifies expansion or contraction towards the origin.

The real part of 0=XE is the Continuity Equation and dEr/dr = Del.Ev.

You can compare these Equations to the traditional and see differences.

Is there a reason to use mixed space and time units or can we just use space units with the r=ct?

Finally, the vector derivatives are not intuitive ,their geometric equivalents may help.

0=XE=(d/cdt + Del)(Er + Ev) = (dBr/dt - Del.Ev) + (dBv/dt + DelxEv + Del Er)


0=XE=(d/cdt + Del)(Er + Ev) = (dBr/dt - Ev/r cos Ev) + (dBv/dt + t Ev/r sin Ev + n Ev/r cos Ev)


Cos Ev is the cosine of the angle between Ev and radius unit vector n for normal . The transverse unit vector is t in th plane of Ev and n vectors. dBv/dt is the hypotenuse of the right triangle formed by the three vector terms.

My head a splode!!!

CJSF

__________________
Two years ago moved from my town
I was looking up past the city lights
But the city lights got in my way

See the constellation ride across the sky
No cigar, no lady on his arm
Just a guy made of dots and lines

-from "See The Constellation"
by They Might Be Giants

Yawyaw, what happened to the source term, i.e. charge and current densities?

The fields are related by E= cB = zH = zcD = D/epsilon.

The charge density Del.Dv= dDr/dR=dDr/cdt = rho charge density = d(e/4pi R^2)/dR= -2Dr/R= -e/2pi R^3.

The charge density is the real derivative of the real field or the time derivative of the real field divided by the speed of light, c.

The real change of the real surface density (Dr) is the charge density. At the boundary or continuity condition, the real change of the surface density is the divergence of the vector field. The vector boundary condition gives the current densities:
0 = dDv/dR + DelxDv + Del Dr = dDv/dt + DelxcDv + Del cDr= dDv/dt + DelxHv + Del Hr= dDv/dt + DEl xHv + Jr (current density)