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###### Sum notation for vectors

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1 year ago

Hi I just saw a formula where there were two sums, one was i=1 to 3 and the other was j=1 to 3. I was just wondering why we couldn't just use i for both sums? Is this just notational convention? Or is there a deeper reason?

Thank you

Thank you

Original post by stevie67890

Hi I just saw a formula where there were two sums, one was i=1 to 3 and the other was j=1 to 3. I was just wondering why we couldn't just use i for both sums? Is this just notational convention? Or is there a deeper reason?

Thank you

Thank you

can you upload a pic?

Reply 2

1 year ago

Original post by mqb2766

can you upload a pic?

Here you go

Original post by stevie67890

Here you go

They could have (re)used "i" again for the second summation as the scope of the index variable is limited to the sum body. I guess they just thought it may cause confusion if they used "i" again, or they didnt really think or ...

Edit - must admit it looks a strange way of expressing the dot product, unless the ei/ejs are the usual unit vectors and there is a dot missing betwen the two brackets? If thats the case I guess (not posted) that in the following lines you'll want to combine the summations into a double summation where you need two different summation variables.

(edited 1 year ago)

Original post by stevie67890

Here you go

I think it's for clarity . Weaker students had be confused if they were to use i

Original post by mqb2766

They could have (re)used "i" again for the second summation as the scope of the index variable is limited to the sum body. I guess they just thought it may cause confusion if they used "i" again, or they didnt really think or ...

Edit - must admit it looks a strange way of expressing the dot product, unless the ei/ejs are the usual unit vectors and there is a dot missing betwen the two brackets? If thats the case I guess (not posted) that in the following lines you'll want to combine the summations into a double summation where you need two different summation variables.

Edit - must admit it looks a strange way of expressing the dot product, unless the ei/ejs are the usual unit vectors and there is a dot missing betwen the two brackets? If thats the case I guess (not posted) that in the following lines you'll want to combine the summations into a double summation where you need two different summation variables.

Yeah, without context, I'd guess missing dot and wanting to move into one sum (or Einstein summation convention) on the next line. Only thing is, that seems rather above the level of stuff I'd expect the OP to be dealing with, so "they didn't really think" is also a strong possibility...

Reply 6

1 year ago

Just out of curiosity, ware there any cases where the notation would have to be different? So in this case its for clarity, but are there cases where it would be incorrect to use i for both sums?

Original post by ewleeyzzzzzz

I think it's for clarity . Weaker students had be confused if they were to use i

Original post by stevie67890

Just out of curiosity, ware there any cases where the notation would have to be different? So in this case its for clarity, but are there cases where it would be incorrect to use i for both sums?

Can you post the following lines for this case, as per the above comments?

Reply 8

1 year ago

I don't have them sorry I just came across it when I was scrolling and was just curious

Original post by mqb2766

Can you post the following lines for this case, as per the above comments?

Original post by stevie67890

I don't have them sorry I just came across it when I was scrolling and was just curious

In that case, for that single line, you could use i as an index in both sums.

Reply 10

1 year ago

In what case would you have to use different letters?

Original post by mqb2766

In that case, for that single line, you could use i as an index in both sums.

Original post by stevie67890

In what case would you have to use different letters?

One example would be as per #4,6 which I suspect is the next couple of lines here, where you combine the two summations into a single one and they're setting up that operation on this line. Having two i's on this line then changing one of their names when combined wouldn't be wrong, but it would be unusual/confusing.

As per #4, the index is valid for the summation body. Anything outside that is irrelevant.

Original post by stevie67890

Just out of curiosity, ware there any cases where the notation would have to be different? So in this case its for clarity, but are there cases where it would be incorrect to use i for both sums?

This would be a fairly standard bit of maths:

$\displaystyle \vec{a} \cdot \vec{b} = \left( \sum_{i=1}^3 a_i e_i \right) \cdot \left( \sum_{j=1}^3 b_j e_j \right)$ (Eq 1)

$= \sum_{i=1}^3 \sum_{j=1}^3 a_i b_j (e_i \cdot e_j)$ (Eq 2)

$= \sum_{i=1}^3 \sum_{j=1}^3 a_i b_j \delta_{ij}$ (Eq 3)

$= \sum_{i=1}^3 a_i b_i$

In Eq 1, it doesn't matter if you replace j with i. It does matter for Eq 2 and Eq 3.

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