4-Vector Cross Product

[Philosophaie]

Could someone tell me if this 4-Vector cross product is correct:

i j k t
dx dy dz 1/c*dt
Ex Ey Ez Et
=[(dy(Ez)-dz(Ey))-(dy(Et)-1/c*dt(Ey))+(dz(Et)-1/c*dt(Ez))]*i
-[(d(E)-d(E))-(d(E)-d(E))+(d(E)-d(E))]*j
+[(d(E)-d(E))-(d(E)-d(E))+(d(E)-d(E))]*k
-[(d(E)-d(E))-(d(E)-d(E))+(d(E)-d(E))]*t

 

[Orodruin]

There is no such thing as a 4-vector cross product. The closest thing you will get is the wedge product between two 1-forms (or rather, its dual), which is a 2-form.

 

[Philosophaie]

How do you do the wedge product between 2 1-forms.
I searched for Wedge Product. I got Exterior Algebra. Here are some Equations could someone explain the "^" operator to me.

?

where k=4

?

 

[Orodruin]

I believe a better path is to identify your purpose, what do you need a generalisation of the cross product for? Once we have established that, we may be able to point you in an appropriate direction.

 

[Philosophaie]

I am trying to solve Maxwell Equations in 4 dimensions.

 

[Orodruin]

Maxwell’s equations are already in 4 space-time dimensions. Their appearance in the tensor formalism of special relativity takes the form $\partial_\mu F^{\mu\nu} = J^\nu$ and $\epsilon^{\mu\nu\sigma\rho}\partial_\nu F_{\sigma\rho}=0$, where F is the antisymmetric rank 2 field tensor.

 

[Philosophaie]

To solve for for some Maxwell Equation it takes the cross product:
grad=dx,dy,dz,1/c*dt
dx = partial d/dx
dy = partial d/dy
dz = partial d/dz
dt = partial d/dt

grad X E(r,t) = -dt(B(r,t))

grad X H(r,t) = dt(D(r,t)) + J