Rank 3 tensor created by taking the derivative of electromagnetic field tensor

In summary, the given tensor S_{\alpha \beta \gamma} can be shown to be completely antisymmetric by comparing it to the definition of a completely antisymmetric tensor. This means that it must satisfy certain conditions on all pairs of indices, which can be demonstrated by comparing S_{\alpha \beta \gamma} to S_{\alpha \gamma \beta} and other similar expressions.
  • #1
mjordan2nd
177
1

Homework Statement



Show that the rank 3 tensor [tex]S_{\alpha \beta \gamma}=F_{\alpha \beta , \gamma} + F_{\beta \gamma , \alpha} + F_{\gamma \alpha , \beta}[/tex] is completely antisymmetric.

I just don't feel comfortable doing this stuff at all. Each of the three terms seems like they should be exactly the same to me. Could someone show me how I would start doing something like this please? Furthermore, if this is a rank 3 tensor, what would it mean if this tensor equals 0?

Thanks. :-\
 
Physics news on Phys.org
  • #2
If [itex]S_{\alpha\beta\gamma}[/itex] is completely antisymmetric, then

[tex]S_{\alpha\beta\gamma}=-S_{\alpha\gamma\beta}[/tex]
[tex]S_{\alpha\beta\gamma}=-S_{\beta\alpha\gamma}[/tex]

and

[tex]S_{\alpha\beta\gamma}=-S_{\gamma\beta\alpha}[/tex]

That is, [itex]S_{\alpha\beta\gamma}[/itex] is antisymmetric on all pairs of indices...

So, start by comparing [itex]S_{\alpha\beta\gamma}[/itex] to [itex]S_{\alpha\gamma\beta}[/itex] using the definition you posted...
 

1. What is a rank 3 tensor?

A rank 3 tensor is a mathematical object that represents a multilinear map between three vector spaces. It is commonly used in physics and engineering to describe physical quantities that have both magnitude and direction in three-dimensional space.

2. How is a rank 3 tensor created by taking the derivative of an electromagnetic field tensor?

The electromagnetic field tensor is a rank 2 tensor that describes the electric and magnetic fields in space. Taking the derivative of this tensor results in a rank 3 tensor, where the first index represents the direction of the derivative and the other two indices represent the components of the original tensor.

3. What is the significance of a rank 3 tensor in electromagnetic theory?

A rank 3 tensor is used to describe the relationship between the electric and magnetic fields in three-dimensional space. It is an important tool in electromagnetic theory as it allows for a more comprehensive understanding of the behavior and interactions of these fields.

4. Can a rank 3 tensor be visualized in physical space?

No, a rank 3 tensor cannot be directly visualized in physical space as it represents a mathematical object. However, it can be represented graphically using diagrams and matrices to aid in understanding its components and relationships.

5. What other applications does a rank 3 tensor have besides electromagnetic theory?

Rank 3 tensors have many other applications in physics, engineering, and mathematics. They are used to describe physical quantities in fluid mechanics, elasticity, and quantum mechanics, among others. They are also used in machine learning and computer vision for image and pattern recognition.

Similar threads

  • Advanced Physics Homework Help
Replies
8
Views
1K
  • Advanced Physics Homework Help
Replies
3
Views
814
  • Advanced Physics Homework Help
Replies
2
Views
357
  • Advanced Physics Homework Help
Replies
0
Views
979
  • Linear and Abstract Algebra
Replies
1
Views
1K
  • Special and General Relativity
Replies
25
Views
884
  • Advanced Physics Homework Help
Replies
6
Views
1K
  • Advanced Physics Homework Help
Replies
1
Views
3K
  • Advanced Physics Homework Help
Replies
1
Views
2K
  • Special and General Relativity
Replies
1
Views
633
Back
Top