Understnding this index notation

[kajalschopra]

Homework Statement

I shall be grateful if someone can help me understand this notation:

http://files.engineering.com/getfile.aspx?folder=340bee11-1ba4-49b2-9a31-1a747012d69b&file=1.gif

I know that this notation will finally/should finally give me the below six equations

http://files.engineering.com/getfile.aspx?folder=bced2393-e971-40fe-ad29-fba6ac509567&file=2.gifBut I do not know how to expand the above notation.

I have uploaded the imapge file to engineering.com as I couldn't upload it here

Homework Equations

As stated / described abov

The Attempt at a Solution

I know the answer but struggling o expand. Tried with notation/calculus but couldn't arrive at it.

 

[Mr-R]

Fix $i$, $j$ and then run over $k$.
For example for $i=x$ and $j=y$ $$\epsilon_{xy,kk}=\epsilon_{xy,xx}+\epsilon_{xy,yy}+\epsilon_{xy,zz}$$
Where a comma denotes a partial derivative. ($A_{,x}=\frac{\partial A}{\partial x}$)
Hope this helps.

 

[kajalschopra]

Sorry..but I'm not able to exactly get what you meant? I mean what do the "###" indicate. Didn't really what you meant by "fix and run over"

I know comma denotes a partial derivative but just not able to get the expansion

 

[Mr-R]

what do the "###" indicate

You are not supposed to see those. Are you sure that your browser can render Latex?
I gave an example for you, but if your browser can't render it then there is no point of typing again. Please check that first.
Meanwhile, read about Einstein summation convention. (it seems that this is your problem).

 

[phinds]

Sorry..but I'm not able to exactly get what you meant? I mean what do the "###" indicate. Didn't really what you meant by "fix and run over"

I know comma denotes a partial derivative but just not able to get the expansion

Here's what it should look like:

 

[kajalschopra]

Thanks a lot. I have been reading about Einstein summation convention from here: http://mathworld.wolfram.com/EinsteinSummation.html

Thus, I have:

ε_ij,_kk + ε_kk,_ij = ε_ik,_jk+ε_jk,_ik

So, will I get 6 equations from the above?

I think, I will get 9 equations as per Einsteins summation convention?

To get the first equation,

1) To get the first equation, put i = x and j = x and sum over k (the repeated index)
2) To get the second equation , put i = x and j=y and sum over k
3) To get third equation put i = x and j=z and sum over k
4) To get the fourth equation put i =y and j=x and sum over k
5) To get the fifth equation put i =y and j=y and sum over k
6) To get the sixth equation put i =y and j=z and sum over k
7) To get the seventh equation put i =z and j=x and sum over k
8) To get the eigthh equation put i =z and j=y and sum over k
9) To get the ninth equation put i =z and j=z and sum over k

Will it give nine equations? for the above term?

 

[kajalschopra]

I shall be grateful if someone can help with the above?