Question on Elementary Index Notation

[Void123]

I have a question regarding the attached file. How do you get those indicies when you multiply the kronecker deltas with A, B, and C? For instance, C - subscript m remains the same on the left side of the expression, but then becomes C subscript i on the right side.

How does this logically work out? What are the rules for these operations?

Thanks.

 

[Astronuc]

$$\varepsilon_{kij}\varepsilon_{klm}A_jB_lC_m\,=\,(\delta_{il}\delta_{jm}\,-\,\delta_{im}\delta_{jl})(A_jB_lC_m)\\ =\,A_mB_iC_m\,-\,A_lB_lC_i\,=\,B_iA_mC_m\,-\,C_iA_lB_l\,=\,B_i(\bold{A}\cdot{\bold{C}})\,-\,C_i(\bold{A}\cdot{\bold{B}})$$

The key is $$\varepsilon_{kij}\varepsilon_{klm}\,=\,(\delta_{il}\delta_{jm}\,-\,\delta_{im}\delta_{jl})$$

Is that clear?

 

[HallsofIvy]

Do you understand that they are using the "summation convention"? That, since j, l, and m are repeated, there is an implied sum as j, l, and m take on values 1, 2, and 3. The final result cannot depend on j, l, or m.