Summation Notation: How Do I Properly Sum Up v_iw_i with i in {x,y,z}?

In summary, Niles is asking if it is correct to use the notation \sum_i{v_iw_i} to represent the sum of v_x w_x, v_y w_y, and v_z w_z, with i being an element of the set {x,y,z}. The response is that it would make more sense to introduce a bijection between the label set {x,y,z} and a corresponding index set {1,2,3} to avoid confusion and make it easier for people with a simple mathematics background to understand.
  • #1
Niles
1,866
0
Hi

Is it correct of me to say that I want to carry out the sum
[tex]
\sum_i{v_iw_i}
[/tex]
where [itex]i\in\{x,y,z\}[/itex]? Or is it most correct to say that [itex]i=\{x,y,z\}[/itex]?Niles.
 
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  • #2
If you have the sum
[tex] v_x w_x + v_y w_y + v_z w_z[/tex]
then you want [itex] i \in \{ x,y,z \} [/itex], which says sum over every element of the set [itex] \{x,y,z \}[/itex]. If you wrote
[tex] \sum_{i=\{x,y,z \}} v_i w_i[/tex] what you really just wrote is
[tex] v_{ \{x,y,z \}} w_{ \{x,y,z \}}[/tex]
which is strange because it's not a sum, and because indices are unlikely (but might be) sets of variables
 
  • #3
Thanks, that is also what I thought was the case. I see the "i={x,y,z}"-version in all sorts of books.

Best wishes,
Niles.
 
  • #4
Niles said:
Hi

Is it correct of me to say that I want to carry out the sum
[tex]
\sum_i{v_iw_i}
[/tex]
where [itex]i\in\{x,y,z\}[/itex]? Or is it most correct to say that [itex]i=\{x,y,z\}[/itex]?


Niles.

While one can interpret that, it would make more sense if associated an index set with your label set if you need to do this. So if instead of {x,y,z} just introduce the bijection {x,y,z} = {1,2,3} where the ith component of one set maps to the ith of the other.

This is just my opinion, but the reason is mostly conventional because its easier for everyone with a simple mathematics background to understand and causes less confusion.
 
  • #5
Thanks for the help, that is kind of everybody.Niles.
 

What is summation notation?

Summation notation is a mathematical notation used to represent the sum of a series of numbers or terms. It involves using the Greek letter sigma (∑) to indicate the start of the sum, followed by the expression to be summed and the upper and lower limits (subscripts) of the summation.

What is the purpose of using summation notation?

Summation notation allows us to express a sum in a concise and efficient way, especially when the number of terms in the series is large. It is also useful for representing and manipulating mathematical formulas and equations involving sums.

How do you read and interpret summation notation?

When reading summation notation, you start from the lower limit and plug in the first value into the expression. Then, you move on to the next value until you reach the upper limit. The results of each evaluation are then added together. For example, ∑n=15 n2 means to start with n=1 and square it, then move on to n=2 and square it, and so on until n=5, then add all the results together.

What are the common mistakes to avoid when using summation notation?

One common mistake is forgetting to include the limits of the summation (upper and lower subscripts). Another mistake is using the wrong variable or expression inside the summation notation. It is also important to correctly evaluate the expression for each value within the limits.

How is summation notation used in scientific research?

Summation notation is used in various scientific fields, such as physics, chemistry, and economics, to represent and solve complex equations and formulas involving sums. It allows scientists to efficiently perform calculations and analyze data, making it an essential tool in scientific research.

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