Evaluating Expressions with Kronecker Delta

In summary, when simplifying expressions involving the Kronecker delta, we can use Einstein's summation convention to simplify repeated indices and remember that the delta function is an identity matrix. This allows us to quickly evaluate expressions like a)\delta_{qr}\delta_{rp}\delta_{pq} and b)\delta_{pp}\delta_{qr}\delta_{rq}.
  • #1
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Homework Statement


Simplify/Evaluate these expressions involving the Kronecker delta, using Einstein's summation convention:
a)[tex]\delta_{qr}[/tex][tex]\delta_{rp}[/tex][tex]\delta_{pq}[/tex]
b)[tex]\delta_{pp}[/tex][tex]\delta_{qr}[/tex][tex]\delta_{rq}[/tex]

Homework Equations


[tex]\delta_{ij}[/tex]=0 when i =/= j
[tex]\delta_{ij}[/tex]=1 when i = j


The Attempt at a Solution


a)[tex]\delta_{qr}[/tex][tex]\delta_{rp}[/tex][tex]\delta_{pq}[/tex]
=[tex]\delta_{qp}[/tex][tex]\delta_{pq}[/tex]
=[tex]\delta_{qq}[/tex] = 3 (summation over repeated q)

b)[tex]\delta_{pp}[/tex][tex]\delta_{qr}[/tex][tex]\delta_{rq}[/tex]
=[tex]\delta_{pp}[/tex][tex]\delta_{qq}[/tex]
=(3)(3)
=9
[Am I actually able to evaluate the [tex]\delta_{qr}[/tex][tex]\delta_{rq}[/tex] part before the [tex]\delta_{pp}[/tex] part, I mean you can't do that with matrices... :S]


Thank you
 
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  • #2
I think what you did is correct. If you think of them as matrices, remember they are identity matrices, so they commute with everything, so the order does not matter.
 

1. What is the Kronecker Delta?

The Kronecker Delta, denoted as δ, is a mathematical expression used to represent the equality or difference of two values. It takes the value of 1 if the two values are equal, and 0 if they are not equal.

2. How is the Kronecker Delta used to evaluate expressions?

The Kronecker Delta is used to evaluate expressions by replacing the variable with the Kronecker Delta, and then simplifying the expression based on the value of the Kronecker Delta. If the Kronecker Delta is equal to 1, the variable is kept in the expression. If the Kronecker Delta is equal to 0, the variable is removed from the expression.

3. What are the properties of the Kronecker Delta?

The Kronecker Delta has several properties that make it useful in evaluating expressions. These include the symmetry property (δij = δji), the substitution property (δijxj = xi), and the product property (δijδjk = δik).

4. Can the Kronecker Delta be used in any type of expression?

Yes, the Kronecker Delta can be used in any type of expression, as long as the expression involves two variables that can be compared for equality or difference. It is commonly used in linear algebra, calculus, and statistics.

5. How does the Kronecker Delta differ from the Dirac Delta?

The Kronecker Delta and the Dirac Delta are two different mathematical expressions. The Kronecker Delta is discrete and takes the value of 1 or 0, while the Dirac Delta is continuous and takes the value of infinity at a specific point and 0 everywhere else. The Kronecker Delta is used for evaluating expressions, while the Dirac Delta is used for solving differential equations.

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