Evaluating Kronecker Deltas

**bilalsaeedkhan** · October 7th 2012, 10:45 AM

Need help with evaluating:

$\delta _{i,j}\delta _{i,j}$

$\delta _{i,j}\delta _{j,k}$

Thanks

**Plato** · October 7th 2012, 11:01 AM

Originally Posted by bilalsaeedkhan

Need help with evaluating:

$\delta _{i,j}\delta _{i,j}$

$\delta _{i,j}\delta _{j,k}$

Read This.

**bilalsaeedkhan** · October 7th 2012, 11:19 AM

My attempt at expanding $\delta _{i,j}\delta _{i,j}$ :

When i=j
$\delta _{i,i}\delta _{i,i}$
$(\delta _{1,1}+\delta _{2,2}+\delta _{3,3})$ * $(\delta _{1,1}+\delta _{2,2}+\delta _{3,3})$
(1+1+1)x(1+1+1)
9

**Plato** · October 7th 2012, 11:52 AM

Originally Posted by bilalsaeedkhan

My attempt at expanding $\delta _{i,j}\delta _{i,j}$ :
When i=j
$\delta _{i,i}\delta _{i,i}$
$(\delta _{1,1}+\delta _{2,2}+\delta _{3,3})$ * $(\delta _{1,1}+\delta _{2,2}+\delta _{3,3})$
(1+1+1)x(1+1+1)
9

Have you changed the question?
Where did $(\delta _{1,1}+\delta _{2,2}+\delta _{3,3})$ come from?
That was not part of the OP!

The OP asked for $\delta _{i,j}\delta _{i,j}=\left\{ {\begin{array}{lr} {1,} & {i = j} \\ {0,} & {i \ne j} \\\end{array}} \right.$

The second part of the OP asked $\delta _{ij} \delta _{jk} = \left\{ {\begin{array}{lr} {1,} & {i = j = k} \\ {0,} & {else} \\\end{array}}\right.$

So why did you change it?

**bilalsaeedkhan** · October 7th 2012, 12:07 PM

Originally Posted by Plato

Have you changed the question?
Where did $(\delta _{1,1}+\delta _{2,2}+\delta _{3,3})$ come from?
That was not part of the OP!

The OP asked for $\delta _{i,j}\delta _{i,j}=\left\{ {\begin{array}{lr} {1,} & {i = j} \\ {0,} & {i \ne j} \\\end{array}} \right.$

The second part of the OP asked $\delta _{ij} \delta _{jk} = \left\{ {\begin{array}{lr} {1,} & {i = j = k} \\ {0,} & {else} \\\end{array}}\right.$

So why did you change it?

I never asked for a definition of Kronecker Delta. I need help with evaluating them. I am asked to evaluate.

**Plato** · October 7th 2012, 12:40 PM

Originally Posted by bilalsaeedkhan

I never asked for a definition of Kronecker Delta. I need help with evaluating them. I am asked to evaluate.

This is your original post:

Originally Posted by bilalsaeedkhan

Need help with evaluating:
$\delta _{i,j}\delta _{i,j}$
$\delta _{i,j}\delta _{j,k}$

You did ask us to evaluate those two expressions.
You cannot say you did not.
I did exactly that in reply #4.

Why not post the actual question then?

**bilalsaeedkhan** · October 7th 2012, 01:01 PM

Originally Posted by Plato

This is your original post:

You did ask us to evaluate those two expressions.
You cannot say you did not.
I did exactly that in reply #4.

Why not post the actual question then?

I did post the question word for word as it is given to me and I did ask for help on evaluating them. I guess dropping the summation notation is what is throwing you off. We were told that the index notation implies that the quantities are being summed. Let me rewrite the problem although this is not how the question is given to us.

$\sum_{i,j,k=1}^{i,j,k=3}\delta _{i,j}\delta _{i,j}$

Similarly the second problem is being summed too.

**Plato** · October 7th 2012, 01:36 PM

Originally Posted by bilalsaeedkhan

I did post the question word for word as it is given to me and I did ask for help on evaluating them. I guess dropping the summation notation is what is throwing you off. We were told that the index notation implies that the quantities are being summed. Let me rewrite the problem although this is not how the question is given to us.

$\sum_{i,j,k=1}^{i,j,k=3}\delta _{i,j}\delta _{i,j}$

Similarly the second problem is being summed too.

$\sum_{i,j,k=1}^{i,j,k=3}\delta _{i,j}\delta _{i,j}=3$
There are twenty-seven triples $(i,j,k)$ where $i=1,2,3~j=1,2,3~k=1,2,3$ .
Only three of them have $i=j=k$ (see reply #4) so you three 1's and twenty-four 0's in that sum.

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