Index Notation Help: Understanding Swapping Symbols in AxB=-BxA

[genericusrnme]

Homework Statement

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

it went along the lines of;
Let;
C=AxB=C_ie_i
D=BxA=D_ie_i
for i=1,2,3
and we know
C_i=e_ijkA_jB_k
D_i=e_ijkB_jA_k
we can manipulate B and A to give
B_j=B_sDelta_sj
B_k=A_mDelta_mk

so we find;
D_i=e_ijkDelta_sjDelta_mkB_sA_m = e_ismB_sA_m

then it says replace s by k and m by j to find;
D_i=e_ikjB_kA_j=-e_ijkA_jB_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

 

[lanedance]

first try this for tex
C = A \times B = C_i = \epsilon_{ijk} A_j B_k