Questions about Index Notation

In summary, the notation for grad and div in index notation is not clear to me, but I understand that \sum_{j,k}\epsilon_{ijk}\partial_j\partial_kC=0.
  • #1
Peregrine
22
0
I am playing around with learning index notation for tensors, and I came across the following where C is a 0th order tensor:

[tex]E_{ijk} \partial_j \partial_k C = 0[/tex]

I believe this equates to [tex]\nabla \times \nabla C[/tex]. I don't understand why this comes out to 0. Any ideas?

Also, I am trying to understand in index notation how to represent the grad of a vector. The reason I am confused is that it seems that, taking C as a 0th order tensor, V as a 1st order tensor and T as a 2nd order tensor:

[tex]div T = \nabla \cdot T = \partial_iT_{ij}[/tex]
[tex]div V = \nabla \cdot V = \partial_iV_i[/tex]
And of course, div C does not make sense as it would be a -1st order tensor.

But, since:
[tex]grad C = \nabla C = \partial_iC[/tex]
I don't follow how to represent [tex]grad V = \nabla V[/tex] or [tex]grad T = \nabla T[/tex] in index notation; from what I have it seems there would be no difference in notation between grad and div! Any help would be greatly appreciated. Thanks!
 
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  • #2
Peregrine said:
I am playing around with learning index notation for tensors, and I came across the following where C is a 0th order tensor:

[tex]E_{ijk} \partial_j \partial_k C = 0[/tex]

I believe this equates to [tex]\nabla \times \nabla C[/tex]. I don't understand why this comes out to 0. Any ideas?

The first thing to note is that your notation here is a bit non-standard. If you assume that [itex]E_{ijk}[/itex] is just some arbitrary (0,3) tensor then [itex]\sum_{j,k}E_{ijk}\partial_j\partial_kC\ne 0[/itex] in general. However, if you take [itex]E_{ijk}=\epsilon_{ijk}[/itex], the totally antisymmetric or permutation tensor, then [itex]\sum_{j,k}\epsilon_{ijk}\partial_j\partial_kC=0[/itex] is trivially satisfied. To see why this is so, note that since partial derivatives commute we can write

[tex]\begin{equation*}
\begin{split}
\sum_{j,k}\epsilon_{ijk}\partial_j\partial_kC
&= \sum_{j,k}\frac{1}{2}\epsilon_{ijk}(\partial_j\partial_kC + \partial_k\partial_jC) \\
&= \sum_{j,k}\frac{1}{2}(\epsilon_{ijk}\partial_j\partial_kC + \epsilon_{ijk}\partial_k\partial_jC) \\
&= \sum_{j,k}\frac{1}{2}(\epsilon_{ijk}\partial_j\partial_kC + \epsilon_{ikj}\partial_j\partial_kC) \\
&= \sum_{j,k}\frac{1}{2}(\epsilon_{ijk} + \epsilon_{ikj})\partial_j\partial_kC.
\end{split}
\end{equation*}[/tex]

However, since one has [itex]\epsilon_{ijk}=-\epsilon_{ikj}[/itex] by definition, one can then write

[tex]\sum_{j,k}\epsilon_{ijk}\partial_j\partial_kC = \sum_{j,k}\frac{1}{2}(\epsilon_{ijk} - \epsilon_{ijk})\partial_j\partial_kC = 0.
[/tex]

You are then correct to say that [itex]\nabla\times\nabla C=\sum_{j,k}\epsilon_{ijk}\partial_j\partial_kC=0[/itex].

Peregrine said:
Also, I am trying to understand in index notation how to represent the grad of a vector. The reason I am confused is that it seems that, taking C as a 0th order tensor, V as a 1st order tensor and T as a 2nd order tensor:

[tex]div T = \nabla \cdot T = \partial_iT_{ij}[/tex]
[tex]div V = \nabla \cdot V = \partial_iV_i[/tex]
And of course, div C does not make sense as it would be a -1st order tensor.

Correct.

Peregrine said:
But, since:
[tex]grad C = \nabla C = \partial_iC[/tex]
I don't follow how to represent [tex]grad V = \nabla V[/tex] or [tex]grad T = \nabla T[/tex] in index notation; from what I have it seems there would be no difference in notation between grad and div! Any help would be greatly appreciated. Thanks!

You may or may not find this post helpful.
 
  • #3
Peregrine said:
I am playing around with learning index notation for tensors, and I came across the following where C is a 0th order tensor:

[tex]E_{ijk} \partial_j \partial_k C = 0[/tex]

I believe this equates to [tex]\nabla \times \nabla C[/tex]. I don't understand why this comes out to 0. Any ideas?
...

Not sure if this is the source of your confusion, but notice that the 0 is a component of a tensor with rank(0,1) since only the j and k indices are repeated. i.e.

[tex]A_{i} = E_{ijk} \partial_j \partial_k C = 0[/tex]
 
  • #4
coalquay,

You are correct that I intended [tex]\epsilon_{ijk}[/tex]. Sorry for the bad notation. But I greatly appreciate the help, I did not think of that approach. Thanks!
 
  • #5
sillyme, i hoped this was about the index of an elliptic operator, but its the same old same old.
 
  • #6
A question about index theorems would be nice, wouldn't it? I suspect, however, we'll be waiting a while before we see one...
 

Related to Questions about Index Notation

1. What is index notation?

Index notation is a mathematical notation used to represent and manipulate multi-dimensional arrays or vectors. It is also known as subscript notation, and it is commonly used in fields such as physics, engineering, and computer science.

2. How is index notation different from standard notation?

In standard notation, a mathematical expression is written using symbols and operators. In index notation, a variable is represented using a letter or symbol with a subscript, which indicates the position of the variable in a multi-dimensional array. This allows for easier manipulation and visualization of complex equations.

3. What are the advantages of using index notation?

Index notation allows for a more concise and efficient representation of multi-dimensional arrays or vectors. It also makes it easier to perform operations such as addition, subtraction, and multiplication on these arrays. Additionally, index notation allows for easier generalization and can help identify patterns in equations.

4. How many indices can a variable have in index notation?

A variable in index notation can have any number of indices, depending on the number of dimensions in the array or vector it represents. For example, a 2-dimensional array may have two indices, while a 3-dimensional array may have three indices.

5. Can index notation be used for non-numeric variables?

Yes, index notation can be used for both numeric and non-numeric variables. It is commonly used in physics to represent vectors with both numeric and non-numeric components, such as velocity or force. It can also be used in computer science to represent arrays of characters or strings.

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