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

[revolution200]

The epsilon delta rule states

$$\epsilon_{ijk}\epsilon_{pqk}=\delta_{ip}\delta_{jq}-\delta_{iq}\delta_{jp}$$

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

For example

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

This then becomes

$$A_{j}B_{i}C_{j}-A_{j}B_{j}C_{i}$$

Can anybody please explain this result?

Is it true that

$$\delta_{ij}a_{i}=a_{j}$$

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.