The equation you've provided appears to be an identity involving the Levi-Civita symbol (also known as the antisymmetric tensor $\epsilon_{ijk}$) and Kronecker delta $\delta_{ij}$.

The equation is:

$\sum_{k=1}^{3} \epsilon_{ijk} \epsilon_{mnk} = \delta_{im} \delta_{jn} - \delta_{in} \delta_{jm}$

Let's break this down step by step:

Step 1: Understanding the Levi-Civita Symbol $\epsilon_{ijk}$

The Levi-Civita symbol $\epsilon_{ijk}$ is defined as:

• $\epsilon_{ijk} = 1$ if $(i, j, k)$ is an even permutation of (1, 2, 3),
• $\epsilon_{ijk} = -1$ if $(i, j, k)$ is an odd permutation of (1, 2, 3),
• $\epsilon_{ijk} = 0$ if any two indices are equal.

Step 2: Understanding the Kronecker Delta $\delta_{ij}$

The Kronecker delta $\delta_{ij}$ is defined as:

• $\delta_{ij} = 1$ if $i = j$,
• $\delta_{ij} = 0$ if $i \neq j$.

Step 3: Expanding the Summation

The summation $\sum_{k=1}^{3} \epsilon_{ijk} \epsilon_{mnk}$ involves summing over the index $k$. To expand this, we consider all possible values for $k = 1, 2, 3$.

For each fixed pair of indices $i, j$ and $m, n$, you calculate the product $\epsilon_{ijk} \epsilon_{mnk}$ for each value of $k$ and sum the results.

Step 4: Applying the Identity

The identity is well-known in tensor calculus, and the result of this summation is:

$\sum_{k=1}^{3} \epsilon_{ijk} \epsilon_{mnk} = \delta_{im} \delta_{jn} - \delta_{in} \delta_{jm}$

This identity shows how the Levi-Civita symbol (antisymmetric) relates to the Kronecker delta (symmetric), and it is useful in vector calculus, particularly in cross products and determinant calculations.

Step 5: Geometric Interpretation

This equation often arises in vector and tensor algebra, where it is used to express the relationship between determinants (from the Levi-Civita symbol) and metric properties of Euclidean space (represented by the Kronecker delta).

Summary

• The equation expresses a relationship between the Levi-Civita symbol and Kronecker delta.
• It is a standard result used in vector calculus and linear algebra.

Would you like further details on the derivation or a specific example where this identity is applied?

Here are some related questions you might find interesting:

1. How is the Levi-Civita symbol used in cross products?
2. What are the properties of the Kronecker delta in higher dimensions?
3. Can this identity be generalized to four dimensions?
4. How does the Levi-Civita symbol relate to the determinant of a matrix?
5. How is this identity used in tensor calculus?

Tip: The Levi-Civita symbol plays a crucial role in operations involving determinants and cross products in three-dimensional space.