Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2013 / Paper 2 / 12 / a / ii

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- Solution ii

Solution

Contract the Kronecker deltas in the first term to obtain

a_{i} b_{i}

. In the second term,

δ_{ij}

identifies the two

a

components, while the two

b_{k}

components already contract. Therefore

a_{i} b_{k} δ_{km} δ_{im} + a_{i} b_{k} a_{j} b_{k} δ_{ij} = a \cdot b + (a \cdot a) (b \cdot b) .

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