Why Does the Epsilon Delta Rule Simplify Expressions in Vector Calculus?

[revolution200]

The epsilon delta rule states

[TEX]\epsilon_{ijk}\epsilon_{pqk}=\delta_{ip}\delta_{jq}-\delta_{iq}\delta_{jp}[/TEX]

I am constantly using this but get stuck when it is applied.

For example

[TEX]\epsilon_{ijk}\epsilon_{pqk}A_{j}B_{l}C_{m}=(\delta_{ip}\delta_{jq}-\delta_{iq}\delta_{jp})A_{j}B_{l}C_{m}[/TEX]

This then becomes

[TEX]A_{j}B_{i}C_{j}-A_{j}B_{j}C_{i}[/TEX]

Can anybody please explain this result?

Is it true that

[TEX]\delta_{ij}a_{i}=a_{j}[/TEX]

If so does this not apply to the above

 

[DrGreg]

Sorry I don't know how to input equations

Just replace the word MATH by TEX. You can go back and edit your previous post within 24 hours (I think) of posting it.

 

[Entropia]

example?

The epsilon delta rule is a powerful tool used in mathematics, particularly in the study of vector calculus and tensor analysis. It is used to simplify expressions involving Levi-Civita symbols, which are commonly used to represent cross products and determinants in higher dimensions.

In the given example, the epsilon delta rule is being used to simplify the expression \epsilon_{ijk}\epsilon_{pqk}A_{j}B_{l}C_{m}. Using the rule, we can rewrite this as (\delta_{ip}\delta_{jq}-\delta_{iq}\delta_{jp})A_{j}B_{l}C_{m}. This is then expanded to A_{j}B_{i}C_{j}-A_{j}B_{j}C_{i}, which is the simplified version of the original expression. This result can be further simplified using the properties of the Kronecker delta, which states that \delta_{ij}a_{i}=a_{j}. In this case, we can rewrite A_{j}B_{i}C_{j} as A_{i}B_{i}C_{i}, and A_{j}B_{j}C_{i} as A_{j}B_{j}C_{j}. This then becomes A_{i}B_{i}C_{i}-A_{j}B_{j}C_{j}, which is the final result.

It is important to note that the delta rule is not applicable to all expressions, but only to those involving Levi-Civita symbols. Additionally, the rule can only simplify the expression, but not evaluate it. In order to fully solve an expression, other mathematical techniques and rules may need to be applied.

In conclusion, the epsilon delta rule is a useful tool in simplifying expressions involving Levi-Civita symbols. It is important to understand the properties and limitations of the rule in order to use it effectively.