Mathematical Methods of Physics/Summation convention

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 $$ \sum_{j} A_j B_j $$ we simply write $$ A_j B_j $$ and the summation over j is implied.

A couple of common tensors used with this notation are
 * $$ \delta_{a}^{b} $$ is only nonzero for the case that $$ a = b $$
 * $$ \epsilon_{i j k} $$ is zero if $$ i = j \lor j = k \lor i = k $$ . For odd permutations (i.e. $$ \epsilon_{j i k} = - \epsilon_{i j k} = - \epsilon_{k i j} $$).  In other words, swapping any two indices flips the sign of the tensor.
 * These are related by $$ \epsilon_{ijk} \epsilon^{imn} = \delta_j^m \delta_k^n - \delta_j^n \delta_k^m $$ (convince yourself that this is true)

Now we can write some common vector operations :
 * Scalar (Dot) Product $$ \vec{A} \cdot \vec{B} = A_i B_i $$
 * Vector (Cross) Product $$ \vec{A} \times \vec{B} = \epsilon_{ijk} A_j B_k $$

Examples Prove that $$ \vec{A} \cdot (\vec{B} \times \vec{C}) = \vec{B} \cdot (\vec{C} \times \vec{A}) $$
 * Bulleted list item

$$ A_i (\epsilon_{ijk} B_j C_k) $$ from here we can swap the indices (i <-> j) and get $$ B_j (- \epsilon_{jik} A_i C_k) $$. Note the sign flip. In order to get a positive sign again we can just swap the indices (i <-> k) and get $$ B_j (\epsilon_{jki} C_k A_i) = \vec{B} \cdot (\vec{C} \times \vec{A}) $$ as desired.


 * Prove that $$ \vec{A} \times (\vec{B} \times \vec{C}) = (\vec{A} \cdot \vec{C}) \vec B - (\vec{A} \cdot \vec{B}) \vec{C} $$

$$ \vec{A} \times (\vec{B} \times \vec{C}) \equiv \epsilon_{ijk} A_j (\epsilon_{klm} B_l C_m) $$ (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) $$ - \epsilon_{kji} \epsilon_{klm} A_j B_l C_m = - (\delta_{j l} \delta_{i m} - \delta_{j m} \delta_{i l}) A_j B_l C_m $$. Now we can make the observation that the first term is only non-zero if j=i and l=m, so $$ \delta_{j l} \delta_{i m} A_j B_l C_m = A_i B_l C_l $$ Note that this is just the dot product $$ (\vec{B} \cdot \vec{C}) A_i $$. The second term is only non-zero if k = m and j = l, so $$ \delta_{k m} \delta_{j l} A_j B_l C_m = A_j B_j C_k = (\vec{A} \cdot \vec{B}) C_k $$

Combining these we are left with $$ (\vec{A} \cdot \vec{C}) B_k - (\vec{A} \cdot \vec{B}) C_k =(\vec{A} \cdot \vec{C}) \vec B - (\vec{A} \cdot \vec{B}) \vec{C} $$ as desired.

Tensor Notation When tensors are used then a distinct difference between an upper and lower index becomes important as well as the ordering. $$ T^{a}_{b} v^b $$ will be contracted into a new vector, but $$ T^{a}_{ b} v_b $$ will not.

Definitions that may prove useful : If a tensor is symmetric, then it satisfies the property that $$ T_{a b} = T_{b a} $$ If a tensor is antisymmetric, then $$ T_{a b} = - T_{b a} $$ 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.