Covariant and Contravariant Kronecker Delta operating on Tensor

In summary, the operation mathbf{M}_{ij} \delta_{ij} produces mathbf{M}_{ii} or mathbf{M}_jj, but mathbf{M}_{ij} \delta^i{}_j does not result in a transformed tensor M. The correct contraction is mathbf{M}_{ij} \delta^j{}_i = mathbf{M}_{ii}. Additionally, \delta_i^i is usually assumed to be summed over, with the dimension of the vector space specified as the upper limit of summation.
  • #1
lewis198
96
0
I am aware that the following operation:

[itex] mathbf{M}_{ij} \delta_{ij} [/itex]

produces

[itex] mathbf{M}_{ii} or mathbf{M}_jj[/itex]


However, if we have the following operation:

[itex] mathbf{M}_{ij} \delta^i{}_j [/itex]

will the tensor M be transformed at all?


Thank you for your time.
 
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  • #2
I can't read your latex well but [itex]M_{ij}\delta _{ij} \neq M_{ii}, M_{ij}\delta _{ij} \neq M_{jj}[/itex] as there is no implied summation. In order to contract indeces you have to employ the summation convention i.e. [itex]M_{ij}\delta ^{ij} = M^{i}_{i} = M^{j}_{j}[/itex] which is just the trace of the tensor M and [itex]M_{ij}\delta ^{i}_{j} = M_{jj}[/itex].
 
  • #3
is M_{ij}\delta ^{i}_{j} = M_{ii} also valid?
 
  • #4
No, [itex]M_{ij}\delta ^{i}_{j}\neq M_{ii} [/itex] because you are contracting with the i index which is what the summation is implied over; contract the i's and move in the j where the i is. However, [itex]M_{ij}\delta ^{j}_{i}= M_{ii} [/itex].
 
  • #5
Ok thanks, got it. By the way, how come your Latex is showing and mine isn't?
 
  • #6
lewis198 said:
Ok thanks, got it. By the way, how come your Latex is showing and mine isn't?
You typed the tags wrong in the first post and used no tags at all in the second. You need to type itex or tex, not latex. Hit the quote button next to a post with math, and you'll see how it's done.

WannabeNewton said:
No, [itex]M_{ij}\delta ^{i}_{j}\neq M_{ii} [/itex] because you are contracting with the i index which is what the summation is implied over; contract the i's and move in the j where the i is. However, [itex]M_{ij}\delta ^{j}_{i}= M_{ii} [/itex].
That first claim is wrong. [itex]M_{ij}\delta ^{i}_{j}[/itex] is definitely [itex]=M_{ii} [/itex]. Edit: Uh, wait, if the convention is that there's no implied summation when both indices are downstairs, you're right and I was wrong. We have [itex]M_{ij}\delta^i_j=M_{jj}[/itex].
 
  • #7
Thanks - I have another question if you don't mind- Is [tex]\delta_{i}^{i}[/tex] summed over? i.e Is the above equal to [tex]\displaystyle\sum\limits_{i=0}^n \delta_{i}^{i}[/tex] If so what determines n?
 
  • #8
Usually yes, the summation is assumed. The dimension of the vector space (or the upper limit of summation) should also be specified.
 

What is the Kronecker Delta symbol and how is it used in tensors?

The Kronecker Delta symbol, often denoted as δ, is a mathematical symbol used to represent a function that takes two values, typically indices, and returns 1 if they are equal and 0 if they are not. In tensors, the Kronecker Delta symbol is used to represent the identity matrix, which is a special type of tensor that has 1s along the main diagonal and 0s everywhere else.

What is the difference between covariant and contravariant Kronecker Delta?

In tensors, covariant and contravariant are two types of transformations that describe how a tensor changes when the coordinates used to represent it are changed. The covariant Kronecker Delta, denoted as δij, is used to transform a covariant tensor, while the contravariant Kronecker Delta, denoted as δij, is used to transform a contravariant tensor.

How do you operate on a tensor using the Kronecker Delta symbol?

To operate on a tensor using the Kronecker Delta symbol, you must first identify the type of tensor (covariant or contravariant) and then use the appropriate Kronecker Delta symbol to perform the transformation. This involves multiplying the tensor by the Kronecker Delta symbol and summing over the repeated indices. The resulting tensor will have the same type as the original tensor.

What is the significance of the Kronecker Delta operating on a tensor?

The Kronecker Delta symbol is an important mathematical tool used in tensor analysis to simplify calculations and express relationships between tensors. By operating on a tensor using the Kronecker Delta symbol, you can transform the tensor into a new form that may be easier to work with or that has a specific physical interpretation.

What are some applications of the Kronecker Delta operating on tensors in scientific research?

The Kronecker Delta symbol is used extensively in various fields of science and engineering, including physics, mathematics, and computer science. In physics, it is used to represent the metric tensor in general relativity and the Kronecker Delta symbol also plays a crucial role in the manipulation of tensors in quantum mechanics. In engineering, it is used to solve problems related to fluid dynamics and elasticity. In computer science, it is used in algorithms and data structures for efficient indexing and searching.

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