1.3 Example: Maxwell’s Equation

Equipped with the machinery provided by the DeRhamComplex we try to verify some calculations done in [4] , namely Maxwell’s field equations in differential form notation. The ordinary well known form of the equations reads as follows, where the first set is Faraday’s and Ampere’s law and the second set are the continuity laws:

\[\operatorname {curl} E = - \frac{1}{c} \frac{\partial B}{\partial t}, \hspace{2.4em} \operatorname {curl} H = \frac{4 \pi}{c} J + \frac{1}{c} \frac{\partial D}{\partial t},\]

and

\[\operatorname{div} D = 4 \pi \rho, \hspace{1.8em} \operatorname{div} B = 0.\]

The vector functions E,H,B,D and J are called, in this order, electric- and magnetic field, magnetic induction, dielectric displacement and electric current. The only scalar function ρ is the charge density. Each of the vector functions is a function \(\mathbb{R}^4\rightarrow \mathbb{R}^3\), and ρ is \(\mathbb{R}^4\rightarrow \mathbb{R}\), a scalar function in the variables \((x,y,z,t)\). And c denotes the speed of light. Flanders defines the following differential forms:

\[\alpha = (E_1 d x + E_2 d y + E_3 d z) \wedge c d t + (B_1 d y \wedge d z + B_2 d z \wedge d x + B_3 d x \wedge d y)\]

and

\[\beta = - (H_1 d x + H_2 d y + H_3 d z) \wedge c d t + (D_1 d y \wedge d z + D_2 d z \wedge d x + D_3 d x \wedge d y)\]

and

\[\gamma = (J_1 d y \wedge d z + J_2 d z \wedge d x + J_3 d x \wedge d y) \wedge d t - \rho d x \wedge d y \wedge d z.\]

Then it is claimed that the four vector equations above are equivalent to the following two differential form equations:

\[d \alpha = 0 \hspace{1.8em} \mathrm{and} \hspace{1.8em} d \beta + 4 \pi \gamma = 0.\]

Now let us prove this. We choose \(c=1\) for simplicity.

R:=Expression(Integer)
M := DERHAM(Integer,[x,y,z,t])
X := [x,y,z,t]::List R
dX := [dx,dy,dz,dt] := [generator(i)$M for i in 1..4]
d ==> exteriorDifferential

_{Type: List(DeRhamComplex(Integer,[x,y,z,t]))}

After these preparations we define the various fields and functions:

E:=vector [(operator E[j]) X for j in 1..3]
D:=vector [(operator D[j]) X for j in 1..3]
B:=vector [(operator B[j]) X for j in 1..3]
H:=vector [(operator H[j]) X for j in 1..3]
J:=vector [(operator J[j]) X for j in 1..3]

rho := (operator 'rho) X

Now we may define the forms \(\alpha,\beta,\gamma\):

alpha:=(E.1*dx + E.2*dy + E.3*dz)*dt+ B.1*dy*dz + B.2*dz*dx + B.3*dx*dy

\[\begin{split}{{{E _ {3}}\left({x, \: y, \: z, \: t}\right)}\ dz \ dt}+{{{E _ {2}} \left({x, \: y, \: z, \: t}\right)}\ dy \ dt}+{{{B _ {1}} \left({x, \: y, \: z, \: t}\right)}\ dy \ dz}+ \\ {{{E _ {1}}\left({x, \: y, \: z, \: t}\right)}\ dx \ dt} -{{{B _ {2}} \left({x, \: y, \: z, \: t}\right)}\ dx \ dz}+{{{B _ {3}} \left({x, \: y, \: z, \: t}\right)}\ dx \ dy}\end{split}\]

_{Type: DeRhamComplex(Integer,[x,y,z,t])}

beta:=-(H.1*dx + H.2*dy + H.3*dz)*dt + D.1*dy*dz + D.2*dz*dx + D.3*dx*dy

and finally

gamma:=(J.1*dy*dz + J.2*dz*dx + J.3*dx*dy)*dt - rho*dx*dy*dz

Now let us calculate \(d\alpha\)

d alpha

giving the output

  (E   (x,y,z,t) - E   (x,y,z,t) + B   (x,y,z,t))dy dz dt
    3,2             2,3             1,4
+
  (E   (x,y,z,t) - E   (x,y,z,t) - B   (x,y,z,t))dx dz dt
    3,1             1,3             2,4
+
  (E   (x,y,z,t) - E   (x,y,z,t) + B   (x,y,z,t))dx dy dt
    2,1             1,2             3,4
+
  (B   (x,y,z,t) + B   (x,y,z,t) + B   (x,y,z,t))dx dy dz
    3,3             2,2             1,1

                                    Type: DeRhamComplex(Integer,[x,y,z,t])

Thus, if we extract the coefficients of each term, we get for dα=0:

\[\begin{split}\begin{array}{c} \frac{\partial B_1}{\partial t} + E_{3, 2} - E_{2, 3}=0\\ \\ - \frac{\partial B_2}{\partial t} + E_{3, 1} - E_{1, 3}=0\\ \\ \frac{\partial B_3}{\partial t} + E_{2, 1} - E_{1, 2}=0\\ \\ B_{1, 1} + B_{2, 2} + B_{3, 3}=0 \end{array}\end{split}\]

which is just the first and last of the Maxwell equations. In the same manner we see that dβ+4πγ corresponds to the second and third one.

-> d beta + 4 * %pi * gamma

resulting in

(- H   (x,y,z,t) + H   (x,y,z,t) + D   (x,y,z,t) + 4%pi J (x,y,z,t))dy dz dt
    3,2             2,3             1,4                  1
 +
(- H   (x,y,z,t) + H   (x,y,z,t) - D   (x,y,z,t) - 4%pi J (x,y,z,t))dx dz dt
    3,1             1,3             2,4                  2
 +
(- H   (x,y,z,t) + H   (x,y,z,t) + D   (x,y,z,t) + 4%pi J (x,y,z,t))dx dy dt
    2,1             1,2             3,4                  3
 +
(D   (x,y,z,t) + D   (x,y,z,t) + D   (x,y,z,t) - 4%pi rho(x,y,z,t)) dx dy dz
  3,3             2,2             1,1

                                     Type: DeRhamComplex(Integer,[x,y,z,t])

[4]	Harley Flanders and Mathematics. Differential Forms with Applications to the Physical Sciences. Dover Pubn Inc, Auflage: Revised. edition.