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 $a=b$
• ${ϵ}_{ijk}$ is zero if $i=j\vee j=k\vee i=k$ . For odd permutations (i.e. ${ϵ}_{jik}=-{ϵ}_{ijk}=-{ϵ}_{kij}$). In other words, swapping any two indices flips the sign of the tensor.
• These are related by ${ϵ}_{ijk}{ϵ}^{imn}={\delta }_j^m{\delta }_k^n-{\delta }_j^n{\delta }_k^m$ (convince yourself that this is true)

Now we can write some common vector operations :

• Scalar (Dot) Product $\overrightarrow{A}\cdot \overrightarrow{B}=A_i{B}_i$
• Vector (Cross) Product $\overrightarrow{A}\times \overrightarrow{B}={ϵ}_{ijk}{A}_j{B}_k$

Examples

• Bulleted list item

Prove that $\overrightarrow{A}\cdot (\overrightarrow{B}\times \overrightarrow{C})=\overrightarrow{B}\cdot (\overrightarrow{C}\times \overrightarrow{A})$

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

• Prove that $\overrightarrow{A}\times (\overrightarrow{B}\times \overrightarrow{C})=(\overrightarrow{A}\cdot \overrightarrow{C})\overrightarrow{B}-(\overrightarrow{A}\cdot \overrightarrow{B})\overrightarrow{C}$

$\overrightarrow{A}\times (\overrightarrow{B}\times \overrightarrow{C})\equiv {ϵ}_{ijk}{A}_j({ϵ}_{klm}{B}_l{C}_m)$ (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) $-{ϵ}_{kji}{ϵ}_{klm}{A}_j{B}_l{C}_m=-({\delta }_{jl}{\delta }_{im}-{\delta }_{jm}{\delta }_{il})A_j{B}_l{C}_m$ . Now we can make the observation that the first term is only non-zero if j=i and l=m, so ${\delta }_{jl}{\delta }_{im}{A}_j{B}_l{C}_m=A_i{B}_l{C}_l$ Note that this is just the dot product $(\overrightarrow{B}\cdot \overrightarrow{C})A_i$. The second term is only non-zero if k = m and j = l, so ${\delta }_{km}{\delta }_{jl}{A}_j{B}_l{C}_m=A_j{B}_j{C}_k=(\overrightarrow{A}\cdot \overrightarrow{B})C_k$

Combining these we are left with $(\overrightarrow{A}\cdot \overrightarrow{C})B_k-(\overrightarrow{A}\cdot \overrightarrow{B})C_k=(\overrightarrow{A}\cdot \overrightarrow{C})\overrightarrow{B}-(\overrightarrow{A}\cdot \overrightarrow{B})\overrightarrow{C}$ 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. $T_b^a{v}^b$ will be contracted into a new vector, but $T_b^a{v}_b$ will not.

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