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Notation help (summations)

  1. Jun 28, 2010 #1
    1. The problem statement, all variables and given/known data

    Hi, I see something like this in my book:

    [tex]\sum_{i,j} A_{i,j}[/tex]... et c

    does this mean that the index values of i and j are both accumulated at the same time? or that i or j gets accumulated first? I'm not sure

  2. jcsd
  3. Jun 28, 2010 #2


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    Generally that really means you have a double summation, that i and j are independent of each other and both range over the natural numbers (or whatever domain makes sense).

    You can also think of it as summing over all pairs (i,j) in [tex] \mathbb{N}[/tex]x[tex] \mathbb{N}[/tex].

    Either way, implicitly stated is that the summation is independent of order (since none is given)
  4. Jun 28, 2010 #3


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    It is usually shorthand for
    [tex]\sum_i \sum_j \cdots \sum_n A_{ij\cdots n},[/tex]
    i.e. a nested summation.
  5. Jun 28, 2010 #4


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    If, for example, i and j can both range from 1 to 3, that is
    [tex]A_{11}+ A_{12}+ A_{13}+ A_{21}+ A_{22}+ A_{23}+ A_{31}+ A_{32}+ A_{33}[/tex].

    That is, [itex]A_{ij}[/itex] has 3(3)= 9 values and this is the sum of all of them. Since addition of numbers is commutative, the order does not matter so the order in which you take i or j does not matter.

    More generally, if i ranges from 1 to m and j ranges from 1 to n, [itex]A_{ij}[/itex] can have mn values and [itex]\sum_{i, j} A_{ij}[/itex] is the sum of all of them.
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