Maxwell Equations in Tensor notation

In summary, the equation 0 = \epsilon^{\alpha \beta \gamma \delta} \frac{\partial F_{\alpha \beta}}{\partial x^\gamma} reduces to 0 = {\partial F_{\alpha\beta}\over\partial x^\gamma} + {\partial F_{\beta\gamma}\over\partial x^\alpha} + {\partial F_{\gamma\alpha}\over\partial x^\beta} when \delta is set to 0 because the fourth index on the anti-symmetric epsilon tensor is an error and should be equal to 0. Additionally, the equation is actually four separate equations, with one for each value of \delta.
  • #1
Karliski
10
0
http://en.wikipedia.org/wiki/Formul...s_in_special_relativity#Maxwell.27s_equations

Why does
[tex]0 = \epsilon^{\alpha \beta \gamma \delta} \frac{\partial F_{\alpha \beta}}{\partial x^\gamma}
[/tex]
reduce to
[tex]
0 = {\partial F_{\alpha\beta}\over\partial x^\gamma} + {\partial F_{\beta\gamma}\over\partial x^\alpha} + {\partial F_{\gamma\alpha}\over\partial x^\beta}
[/tex]
?
 
Physics news on Phys.org
  • #2
[tex]
0
= \epsilon^{\alpha \beta \gamma 0} \frac{\partial F_{\alpha \beta}}{\partial x^\gamma}
= {\partial F_{\alpha\beta}\over\partial x^\gamma} + {\partial F_{\beta\gamma}\over\partial x^\alpha} + {\partial F_{\gamma\alpha}\over\partial x^\beta} - {\partial F_{\beta\alpha}\over\partial x^\gamma} - {\partial F_{\gamma\beta}\over\partial x^\alpha} - {\partial F_{\alpha\gamma}\over\partial x^\beta}
= 2 \left( {\partial F_{\alpha\beta}\over\partial x^\gamma} + {\partial F_{\beta\gamma}\over\partial x^\alpha} + {\partial F_{\gamma\alpha}\over\partial x^\beta} \right)
[/tex]
because [tex]F_{\alpha\beta}[/itex] is antisymmetric.
 
  • #3
Why was delta set to 0?
 
  • #4
What "delta" are you talking about? There was no "delta" in your original question nor in Adriank's response.
 
  • #5
HallsofIvy said:
What "delta" are you talking about? There was no "delta" in your original question nor in Adriank's response.

Actually, delta is the fourth index on the anti-symmetric epsilon tensor in both Karliski's post and in the wikipedia equation he linked to. It is basically an error, we should have delta = 0 as in adriank's post, but if delta is not zero then we just get four copies of the same equation.
 
  • #6
The "equation" in #1 is actually four equations, one for each value of [itex]\delta[/itex]. The one with [itex]\delta=0[/itex] is the scalar equation

[tex]\nabla\cdot\vec B=0[/tex]

The one with [itex]\delta=i\neq 0[/itex] is the ith component of the vector equation

[tex]\nabla\times\vec E+\frac{\partial\vec B}{\partial t}=0[/tex]

(Edit: ...except for a factor of two).
 
Last edited:
  • #7
The equation
[tex]0 = \epsilon^{\alpha \beta \gamma \delta} \frac{\partial F_{\alpha \beta}}{\partial x^\gamma}[/tex]
is written with a [tex]\delta[/tex] only on one side, which means we can "plug in" any specific value for it. So I put [tex]\delta = 0[/tex] and explicitly wrote out the sum [tex]\epsilon_{\alpha\beta\gamma0}F_{\alpha\beta,\gamma}[/tex].
 

Related to Maxwell Equations in Tensor notation

What are Maxwell Equations in Tensor notation?

Maxwell Equations in Tensor notation are a set of four equations that describe the fundamental laws of electricity and magnetism. They were developed by James Clerk Maxwell in the 19th century and are typically written in tensor notation, which uses mathematical tensors to represent the physical quantities involved.

Why are Maxwell Equations written in Tensor notation?

Maxwell Equations are written in Tensor notation because it allows for a more concise and elegant representation of the equations. Tensors also allow for the equations to be easily manipulated and applied in various coordinate systems, making them useful in a wide range of physical problems.

What are the benefits of using Tensor notation for Maxwell Equations?

Using Tensor notation for Maxwell Equations has several benefits. It simplifies the notation and makes it easier to write and manipulate the equations. It also allows for a more general and elegant representation of the equations, making them applicable in different coordinate systems. Additionally, Tensor notation is useful in calculations involving higher dimensions, making it a powerful tool in modern physics.

How can Tensor notation be applied to Maxwell Equations?

Tensor notation can be applied to Maxwell Equations by using the appropriate tensor representations for each physical quantity in the equations. For example, the electric field and magnetic field can be represented as second-order tensors, while the electric charge and current can be represented as first-order tensors. By using the appropriate tensor operations, the equations can be easily manipulated and solved.

Are Maxwell Equations in Tensor notation used in practical applications?

Yes, Maxwell Equations in Tensor notation are used in many practical applications in physics and engineering. They are essential in understanding and predicting the behavior of electromagnetic fields, which is crucial in the design and operation of electronic devices, communication systems, and power systems. Tensor notation allows for a more general and accurate representation of the equations, making it a powerful tool in solving real-world problems.

Similar threads

  • Electromagnetism
Replies
13
Views
452
Replies
3
Views
685
Replies
3
Views
933
Replies
2
Views
1K
  • Electromagnetism
Replies
1
Views
798
  • Special and General Relativity
Replies
7
Views
1K
Replies
1
Views
622
  • Special and General Relativity
Replies
11
Views
791
  • Electromagnetism
Replies
2
Views
918
  • Electromagnetism
Replies
2
Views
1K
Back
Top