Proving Relationship: Epsilon-Delta Decomposition for Tensors

In summary, tensors index notation is a mathematical notation used to represent tensors, which are multi-dimensional arrays of numbers used to describe the properties of physical systems. It differs from regular notation by using a more compact and systematic approach, with each index representing a specific component of the tensor. The benefits of using tensors index notation include easier manipulation and analysis of tensors, as well as a better understanding of their symmetries and transformations. However, there are rules that must be followed when using this notation, such as the use of index pairs and the Einstein summation convention. Finally, tensors index notation can be applied to tensors of any rank, making it a flexible and versatile tool for representing complex systems.
  • #1
vortmax
19
1

Homework Statement



Prove the following relationship:

[tex]\epsilon[/tex]pqi[tex]\epsilon[/tex]pqj = 2[tex]\delta[/tex]ij

Homework Equations





The Attempt at a Solution



All I have so far is the decomposition using the epsilon-delta

[tex]\epsilon[/tex]pqi[tex]\epsilon[/tex]pqj = [tex]\epsilon[/tex]qip[tex]\epsilon[/tex]pqj
[tex]\epsilon[/tex]qip[tex]\epsilon[/tex]pqj = [tex]\delta[/tex]qp[tex]\delta[/tex]iq - [tex]\delta[/tex]qj[tex]\delta[/tex]iq

have no idea where to turn next
 
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  • #2
Hi vortmax, do you have a definition for the eijk (levi cevita) and understand the summation?

You could do it reasonably simply just by evaluating ij cases for both the levi cevitas & kronecka deltas (probably all you really need to do is i=j & i<>j cases)
 

FAQ: Proving Relationship: Epsilon-Delta Decomposition for Tensors

What is tensors index notation?

Tensors index notation is a mathematical notation used to represent tensors, which are multi-dimensional arrays of numbers used to describe the properties of physical systems. It involves using indices to represent the different components of a tensor and allows for easier manipulation and analysis of these tensors.

How is tensors index notation different from regular notation?

In regular notation, tensors are represented using subscripts, superscripts, and commas to indicate the different components. Tensors index notation, on the other hand, uses a more compact and systematic approach, where each index represents a specific component of the tensor. This makes it easier to perform operations on tensors and understand their properties.

What are the benefits of using tensors index notation?

Tensors index notation allows for a more concise and systematic representation of tensors, making it easier to perform operations and analyze their properties. It also helps in visualizing and understanding the symmetries and transformations of tensors, which are important in many areas of physics and engineering.

Are there any rules for using tensors index notation?

Yes, there are certain rules that must be followed when using tensors index notation. For example, indices must always appear in pairs, and the order of the indices is important for maintaining the correct tensor symmetry. Additionally, the Einstein summation convention is often used, which states that if an index appears twice in a term, it is being summed over all possible values.

Can tensors index notation be applied to tensors of any rank?

Yes, tensors index notation can be applied to tensors of any rank. The number of indices used corresponds to the rank of the tensor, so a rank-2 tensor would have two indices, while a rank-3 tensor would have three indices, and so on. This notation is flexible and can be used to represent tensors of any dimension or complexity.

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