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I'm having trouble working with suffix notation (under the summation convention). I am trying to follow working that (a x b) . (c x d) = (a.c)(b.d)-(a.d)(b.c) (where . = scalar product, x = vector product)

I have got to the stage where:

(axb)_{i}(cxd)_{i} =(∂_{jl}∂_{km} - ∂_{jk}∂_{lm})a_{j}b_{k}c_{l}d_{m}

I dont get how you go from there to the next line:

= a_{j}b_{k}c_{j}d_{k} - a_{j}b_{k}c_{k}d_{j}

Any ideas on why the m's and the l's were allowed to be replaced, and with those particular combinations (how do the Kronecker delta's affect it?)

I have got to the stage where:

(axb)

I dont get how you go from there to the next line:

= a

Any ideas on why the m's and the l's were allowed to be replaced, and with those particular combinations (how do the Kronecker delta's affect it?)

Willa

(∂_{jl}∂_{km} - ∂_{jk}∂_{lm})a_{j}b_{k}c_{l}d_{m}

(∂

∂

∂

Or more explicitly, when summing over j, you get ∂_{ij}b_{j} = b_{i}

Since you have ∂_{i1}b_{1} + ∂_{i2}b_{2} + ∂_{i3}b_{3}, and two of the three deltas are zero, whilst the other is 1.

I find the easiest way to remember is is that the action of a Kronecker delta is to change indexes - it always changes the repeated index on another term into the free index on the Kronecker.

Since you have ∂

I find the easiest way to remember is is that the action of a Kronecker delta is to change indexes - it always changes the repeated index on another term into the free index on the Kronecker.

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