Double Summations

**vlai9180** · March 23rd 2010, 06:09 PM

I posted this in calculus, but that may have been the wrong location...

How would you expand the double summation:

$\sum\limits_{i}^2{\sum\limits_{j}^2{y_{i}y_{j}a_{i j}}}$

I understand that:

$\sum\limits_{i}^2{\sum\limits_{j}^2{y_{i}y_{j}}}$

Gets you something like:

$<br /> y_{1}^{2}+y_{12}+ y_{21}+y_{2}^{2}<br />$

and

$\sum\limits_{i}^2{\sum\limits_{j}^2{a_{ij}}}$

is

$<br /> a_{11}+a_{12}+a_{21}+a_{22}<br />$

But how can you combine these two? Is it as simple as foiling the entire thing?

Any help would be greatly appreciated

**TKHunny** · March 23rd 2010, 06:15 PM

1) What does the notation mean? Does j always start at 1? That isn't clear. You can show it, $\sum_{j = 1}^{2}$ ...

2) I can't think of a single reason why you would get anything squared. You wouldn't square anything to populate a 2x2 matrix, would you?

3) Really, abandon this awkward notation and just fill in a 2x2 matrix. After staring at it for a moment, you should be able to write the summations.

**vlai9180** · March 23rd 2010, 06:24 PM

Sorry, yes, it would be

$\sum\limits_{i=1}^2{\sum\limits_{j=1}^2{y_{i}y_{j} a_{ij}}}$

I don't know where you would get a matrix from, though, or how to fill it.

It's sort of weird to explain - my thermodynamics book explains van der waals parameters for a 2-component mixture as $a_{mix} = \sum\limits_{i=1}^2{\sum\limits_{j=1}^2{y_{i}y_{j} a_{ij}}}$ , and then suddenly somehow comes up with

$a_{1}y_{1}^{2} + a_{12}a_{21}y_{1}y_{2} + a_{2}y_{2}^{2}$

without any real explanation.

Does it make any sense to you?

**CaptainBlack** · March 23rd 2010, 09:22 PM

Originally Posted by vlai9180

Sorry, yes, it would be

$\sum\limits_{i=1}^2{\sum\limits_{j=1}^2{y_{i}y_{j} a_{ij}}}$

I don't know where you would get a matrix from, though, or how to fill it.

It's sort of weird to explain - my thermodynamics book explains van der waals parameters for a 2-component mixture as $a_{mix} = \sum\limits_{i=1}^2{\sum\limits_{j=1}^2{y_{i}y_{j} a_{ij}}}$ , and then suddenly somehow comes up with

$a_{1}y_{1}^{2} + a_{12}a_{21}y_{1}y_{2} + a_{2}y_{2}^{2}$

without any real explanation.

Does it make any sense to you?

Are you sure it does not have:

$a_{1,1}y_{1}^{2} + (a_{1,2}+a_{2,1})y_{1}y_{2} + a_{2,2}y_{2}^{2}$

which is nothing more than writing the sum out longhand.

CB

**mr fantastic** · March 23rd 2010, 11:12 PM

Originally Posted by vlai9180

I posted this in calculus, but that may have been the wrong location...

[snip]

Then report the post to a Moderator (use the Report post tool - click on the small triangle in the top right corner of any post). And edit the post so that other people don't waste their time helping with something that has already been dealt with elsewhere.

Asked and answered here: http://www.mathhelpforum.com/math-he...ummations.html

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