02. Special Tensors

δ Tensor

The δ tensor, like the Kronecker delta, for $δ_{i,j}$ equals 0 if i≠j. If i=j, Einstein summation is used. FIXME.

Is a completely antisymmetric tensor (differential form).

Defined to be 1 for all even permutations, especially:

$ε_{1,2,3,4,...,n}=1$

Swaps sign if you swap two indices:

$ε_{1,3,2,4,...,n}=-1$

$ε_{3,1,2,4,...,n}=1$

If two indices are the same, it will be forced to become 0.

If two ε Tensors have one index in common, they can be reduced to a combination of δ Tensors:

$ε_{i,j,k}⋅ε_{i,l,m}=δ_{j,l}⋅δ_{k,m}-δ_{j,m}⋅δ_{k,l}$

Author: Danny (remove the ".nospam" to send)

Last modification on: Sat, 04 May 2024 .