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Above is the question and answer to question (b) (ignore (a))... I don't get where the final implication comes from. Why can we use c as a (covariant) index on the LHS... surely we must use d or something different to c. Is the implication obvious, or do we have to justify all the possible combinations in order to understand the implication?

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Since each side is vector valued, It doesn't matter what you call the index. For instance, say you called one $c$ and the second $d$, you still would have that: $$LHS_{c1}=LHS_{d1},~ LHS_{c2}=LHS_{d2}, ~LHS_{c3}=LHS_{d3}$$ So it these are essentially the same.

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Ah right, c on the LHS is a dummy suffix? Can we always use this logic? Are you sure there aren't similar examples which fail? (the "failing" part based on the idea that c is used on the RHS) – Adam Rubinson Apr 18 '12 at 14:54
Ah okay... I think you can. You can think of it like this: Write the LHS with d to be safe. Then sum over the repeated suffixes other than d on LHS and d on RHS. Then equate coefficients. So it doesn't matter... – Adam Rubinson Apr 18 '12 at 15:01

You can use the same dummy index on both sides of an equation; the Einstein summation convention applies on each side separately.

The only thing that happens in the last impliciation is that the left-hand side is rewritten using $\mathbf e_a\times \mathbf e_b = \epsilon_{abc}\mathbf e_c$ (which is close to being a definition of the cross product).

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Yeah, I know... the only thing that had me worried was that there is a c on the RHS, and usually we use different dummy variables to be safe. But I'm fine about it now. Thanks for the clarification anyway – Adam Rubinson Apr 18 '12 at 15:10

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