How does vector calculus applies to electromagnetism?

Answer 1

Vectors are directional numbers. Calculus determines changes. Electromagnetism involves directional fields and thus vector calculus is the tool to calculate the changes in directional fields.

The training in Mathematics and Physics is deficient in that Nature involves the combination of real and vector numbers called Quaternions. Quaternions were invented by William Rowan Hamilton in 1843. Quaternions consist of a real number r and three vectors (i,j,k) such that i2 = j2 = k2 = ijk = -1.

A quaternion point is p=r + ix +jy + kz= r + v where v is the vector part.

Quaternion calculus has a derivative I call X for Khepra which consists of Hamilton's vector derivative called Del = id/dx + j d/dy + kd/dz and a real derivative d/dr = d/cdt .

X= d/dr + Del = d/dr + id/dx + jd/dy + kd/dz = d/cdt + Del = [d/dr,Del]

Using this quaternion derivative the fundamental laws of electromagnetism can be derived as th Boundary Condition, 0= XE where E is the quaternion electric field E=Er + Ev = [Er,Ev].

The First Derivative of the Electric field is

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

The Equilibrium Condition for the Electric field occurs when the the First Derivative is set to zero:

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

This is the Quaternion Equilibrium Condition Equation, notice that the Curl Term DelxEv =0 and is not in the equation. The curl is zero at Equilibrium and the remaining vector terms are "Equal and Opposite"! Equilibrium requires that the sum of the reals and vectors sum to zero. The vectors cannot sum to zero unless DelxEv=0, this happens only when the other terms are parallel or anti-parallel. Equilibrium is the anti-parallel case, thus Newton's "Equal and Opposite" Rule in his 3rd law of Motion.

This Equilibrium Condition is the Stationary and Invariant Condition and the Cauchy-Riemann Continutiy Condition.

Maxwell's Equations

dBr/dt - Del.Ev=0

dBv/dt + DelxEv=0

are incorrect in including DelxEv, it should be Del Er. DelxEv is perpendicular to dBv/dt =dEv/cdt. Vector Calculus shows DelxEv is perpendicular to dEv/dr, thus the sum of orthogonal vectors is not zero unless both vectors are zero.

This shows that Maxwell's Equations are incorrect and the proper Electromagnetism Equations are derived by Quaternion Calculus.