- #1

- 613

- 1

## Homework Statement

I was following along with a proof of AxB=-Bxa

it went along the lines of;

Let;

C=AxB=C

_{i}e

_{i}

D=BxA=D

_{i}e

_{i}

for i=1,2,3

and we know

C

_{i}=e

_{ijk}A

_{j}B

_{k}

D

_{i}=e

_{ijk}B

_{j}A

_{k}

we can manipulate B and A to give

B

_{j}=B

_{s}Delta

_{sj}

B

_{k}=A

_{m}Delta

_{mk}

so we find;

D

_{i}=e

_{ijk}Delta

_{sj}Delta

_{mk}B

_{s}A

_{m}= e

_{ism}B

_{s}A

_{m}

then it says replace s by k and m by j to find;

D

_{i}=e

_{ikj}B

_{k}A

_{j}=-e

_{ijk}A

_{j}B

_{k}=-C

_{i}

I don't understand why you can just swap s to k and m to j, what's stopping you just replacing s to j and m to k and finding that D

_{i}=C

_{i}?

(every time I tried to use the delta from the Latex reference I just got a superscript 1 so that's why I write Delta for the Kronecker Deltas)

Thanks in advance