# Thread: Maxwell's Equations in 4 Dimensions

1. I can write the electromagnetic 4-force simply as:

f = e/c (vxY)

with Y a new 4-vector function and v and f being 4-vectors. This Y allows me to write Maxwell's Equations as a single equation. I can also show that time flows faster or slower in a spatially varying magnetic field.  2.

3. You can ASSERT that these things are such, but you are continually shown to be false, and ignorant of physics, so the thread is pointless  4. I was hoping this thread was going to be about how to extend Maxwell's equations into four spatial dimensions. Two of the equations involve taking the curl of a vector space, which I don't think has a four-dimensional analogue.  5. If you'd read the OP's other "contributions" you wouldn't have got your hopes up!  6. I assure you: curl has a four dimensional analog: the 3x4 determinant. Together with my computation method for nxm determinants. These are perfectly consistent with other mathematics and serve analogous purposes.  7. You can make all the assurances you like, it doesn't alter the fact that in every thread you've posted it is obvious you don't know what you are talking about... You're an idiot...  8. Originally Posted by talanum46 I assure you: curl has a four dimensional analog: the 3x4 determinant. Together with my computation method for nxm determinants. These are perfectly consistent with other mathematics and serve analogous purposes.
Consider the vector field in R^4 given by f((x, y, z, w) = (x^3 + 2y, 5x - w, xy^2 + z, x + 2y + 3z + 4w). What is its curl? Furthermore, what is this curl useful for?  9. You really need to call this a vector function. Set: �� (��) = f(x,y,z,w) = x^3 + 2y, 5x - w, xy^2 + z, x + 2y + 3z + 4w.i.e. set: F_1(��)=x^3+2y,F_2(��)=5x-w,F_3(��)=xy^2+z,F_4(��)=x+2y+3z+4w.
I explain first how to compute the nxm determinant:
1.1 Method: nxm Determinant
The general way to compute a nxm matrix A's determinant is stated first. Say n > m. Then compute the determinant as usual until reaching the rx2 stage, (r < n). Then develop the remaining determinants as follows. Explicitly delete any row(s) (such that 2 remains) and replace the entries by []. Then move the rows upwards so that 2x2 determinants remain. Take the sum of all combinations of deleted rows. If an odd number of moves was required, multiply the term by -1, otherwise let the term plus together with the other terms. Alternatively one can start at the nxm determinant and write this as a sum of mxm determinants containing all combinations of retained rows (with the row we are going to develop with never deleted), with the same explicit deletion of rows and shifting up the rows (and multiply a term by -1 if an odd amount of shifts ocurred). For n < m do similarly, just with "column" replacing "row" and “leftwards” replacing “upwards” and “2xr” replacing “rx2” in this paragraph.
Now we may compute curl ��(��) as follows:
∇_4× ��(��)=|
�� �� �� ��
∂/(∂x_) ∂/(∂y) ∂/(∂z) ∂/(∂w)
F_1 (��) F_2(��) F_3(��) F_4(��)
|
Set this equal to LS.
Develop the determinant by row 1. One may impose a signature on the first row. In the �� direction LS equals:
��[|
∂/(∂y) ∂/(∂z) []
F_2(��)F_3(��) []
|+ |
∂/(∂y) [] ∂/(∂w)
F_2(��)[] F_4(��)
| + |
[] ∂/(∂z) ∂/(∂w)
[] F_3(��)F_4(��)
|]=��[|
∂/(∂y) ∂/(∂z)
F_2(��)F_3(��)
|- |
∂/(∂y) ∂/(∂w)
F_2(��)F_4(��)
| + |
∂/(∂z) ∂/(∂w)
F_3(��)F_4(��)
|]
=��[∂/(∂y) F_3(��) - ∂/(∂z) F_2(��) -(∂/(∂y) F_4(��) - ∂/(∂w) F_2(��)) + ∂/(∂z) F_4(��)- ∂/(∂w) F_3(��)]
=��[∂/(∂y) (xy^2+z) - ∂/(∂z)(5x-w)- (∂/(∂y) (x+2y+3z+4w) - ∂/(∂w)(3x-w)) + ∂/(∂z) (x+2y+3z+4w) - ∂/(∂w) (xy^2 + z)]
=��[2xy-0 - (2 +1) + 3 - 0 ]=��2xy .
In the �� direction:
LS_2=[|
∂/(∂x) ∂/(∂z) []
F_1(��) F_3(��) []
| + |
∂/(∂x) [] ∂/(∂w)
F_1(��) []F_4(��)
| + |
[] ∂/(∂z) ∂/(∂w)
[] F_3(�� ) F_4(��)
|]=[|
∂/(∂x) ∂/(∂z)
F_1(��) F_3(��)
| - |
∂/(∂x) ∂/(∂w)
F_1(��) F_4(��)
| + |
∂/(∂z) ∂/(∂w)
F_3(�� ) F_4(��)
|]

=∂/(∂x) F_3(��) - ∂/(∂z) F_1(��) - (∂/(∂x) F_4(��) - ∂/(∂w) F_1(��))+∂/(∂z) F_4(��) - ∂/(∂w) F_3(��)
=∂/(∂x) (xy^2 + z) - ∂/(∂z)(x^3 + 2y)-∂/(∂x) (x+2y+3z+4w) + ∂/(∂w) (x^3 +2y)+∂/(∂z) (x+ 2y + 3z + 4w) - ∂/(∂w)(xy^2+z)=y^2- 0- 1+ 0 + 3 - 0=y^2 +2.

In the �� direction:
LS_3=|
∂/(∂x) ∂/(∂y) []
F_1(��) F_2(��) []
| + |
∂/(∂x) [] ∂/(∂w)
F_1(��) [] F_4(��)
|+|
[] ∂/(∂y) ∂/(∂w)
[] F_2(��) F_4(��)
|=|
∂/(∂x) ∂/(∂y)
F_1(��) F_2(��)
| - |
∂/(∂x) ∂/(∂w)
F_1(��) F_4(��)
|+ |
∂/(∂y) ∂/(∂w)
F_2(��) F_4(��)
|=
∂/(∂x) F_2(��) - ∂/(∂y) F_1(��) - ∂/(∂x) F_4(��) + ∂/(∂w) F_1(��) + ∂/(∂y) F_4(��) - ∂/(∂w) F_2(��)
=∂/(∂x)(5x-w) - ∂/(∂y) (x^3+2y)-∂/(∂x) (x+2y+3z+4w) + ∂/(∂w)(x^3+2y)+ ∂/(∂y)(x + 2y +3z+4w) - ∂/(∂w)(5x-w)=5- 2-1+0+2 +1=5.
And in the �� direction:
LS_4=|
∂/(∂x) ∂/(∂y) []
F_1(��) F_2(��)[]
|+ |
[] ∂/(∂y) ∂/(∂z)
[] F_2(��)F_3(��)
| + |
∂/(∂x) [] ∂/(∂z)
F_1(��) [] F_3(��)
|=|
∂/(∂x) ∂/(∂y)
F_1(��) F_2(��)
|+ |
∂/(∂y) ∂/(∂z)
F_2(��)F_3(��)
| - |
∂/(∂x) ∂/(∂z)
F_1(��) F_3(��)
|=5- 2 + 2yx - 0- y^2 + 0=-y^2+2xy+3.

So:

∇_4 ×��(��)=2xy, y^2_+2, 5, -y^2+2xy+3.

It is useful in reformulating Maxwell's Equations. It is also useful to change between an integral along an area and an integral along the closed curve (boundary of the surface).    11. Originally Posted by talanum46   