The Basic Notation
Often when working with tensors there is a large amount of summation that needs to occur. To make this quicker we use Einstein index notation.
The notation is simple. Instead of writing we simply write and the summation over j is implied.
A couple of common tensors used with this notation are
- is only nonzero for the case that
- is zero if . For odd permutations (i.e. ). In other words, swapping any two indices flips the sign of the tensor.
- These are related by (convince yourself that this is true)
Now we can write some common vector operations :
- Scalar (Dot) Product
- Vector (Cross) Product
from here we can swap the indices (i <-> j) and get . Note the sign flip. In order to get a positive sign again we can just swap the indices (i <-> k) and get as desired.
- Prove that
(Watch the indices closely - some students inadvertently add too many)
We know we want to get a dot product out of this. In order to do that we will have to use the expansion of the Levi-Cevita tensor in terms of the Kronecker Deltas. We want to get the Tensors to have the same first index, so we can do this by swapping the indices ( i <-> k)
. Now we can make the observation that the first term is only non-zero if j=i and l=m, so Note that this is just the dot product . The second term is only non-zero if k = m and j = l, so
Combining these we are left with as desired.
When tensors are used then a distinct difference between an upper and lower index becomes important as well as the ordering.
will be contracted into a new vector, but will not.
Definitions that may prove useful :
If a tensor is symmetric, then it satisfies the property that
If a tensor is antisymmetric, then
There are many tensors that satisfy neither of these properties - so make sure it makes sense to use them before blindly applying them to some problem.