Solving Confusion with Summation Convention - Ian

In summary, summation convention is a mathematical notation technique that simplifies expressions involving repeated sums or products. It works by omitting the summation or product symbol and understanding that the repeated index is summed over all possible values. The purpose of using summation convention is to make equations easier to read and write, as well as to reduce the number of terms in an expression. The rules of summation convention include summing over all possible values for repeated indices, only allowing each index to appear twice, understanding a single index as being summed over all possible values, and allowing for the renaming of indices without changing the expression's value. Examples of using summation convention include rewriting <code>\sum_{i=1}^n a_i b_i
  • #1
iansullivan88
5
0
Hello, I think I am fundamentally confused with summation convention. For example, if I have

[itex]
\epsilon_{ijk}x_j\delta_{jk}
[/itex]

Can I sift the levi civita and get

[itex]
\epsilon_{ijj}x_j=0
[/itex]

or sift x and get

[itex]
\epsilon_{ijk}x_k\not=0
[/itex]

Each gives a different answer. What mistake am I making here?
Thank you,

Ian
 
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  • #2
The fundamental confusion is that you've got an index repeated three times. This is incorrect: an index can only appear once or twice, the latter meaning it is summed over.
 
  • #3
Ah I see - thanks very much

Ian
 

What is summation convention?

Summation convention is a mathematical notation technique used to simplify expressions involving repeated sums or products. It allows for the omission of the summation or product symbol, making equations easier to read and write.

How does summation convention work?

In summation convention, the repeated index in an expression is understood to be summed over all possible values. This means that instead of writing out each term in the sum, the index is simply repeated and the summation symbol is omitted.

What is the purpose of using summation convention?

The purpose of using summation convention is to simplify mathematical expressions and make them easier to read and write. It also helps to reduce the number of terms in an expression, which can make it more manageable to work with.

What are the rules of summation convention?

The rules of summation convention include:

  • Repeated indices are summed over all possible values.
  • Each index can only appear twice in an expression.
  • If an index appears only once, it is understood to be summed over all possible values.
  • Indices can be renamed without changing the value of the expression.

What are some examples of using summation convention?

An example of using summation convention is the expression \sum_{i=1}^n a_i b_i, which can be rewritten as a_i b_i using summation convention. Another example is the expression \sum_{j=0}^\infty c_j, which can be rewritten as c_j using summation convention.

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