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 $a=b$
• $\epsilon _{ijk}$ is zero if $i=j\lor j=k\lor i=k$ . For odd permutations (i.e. $\epsilon _{jik}=-\epsilon _{ijk}=-\epsilon _{kij}$). 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

• Bulleted list item

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

$A_{i}(\epsilon _{ijk}B_{j}C_{k})$ 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) $-\epsilon _{kji}\epsilon _{klm}A_{j}B_{l}C_{m}=-(\delta _{jl}\delta _{im}-\delta _{jm}\delta _{il})A_{j}B_{l}C_{m}$ . 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 $({\vec {B}}\cdot {\vec {C}})A_{i}$. The second term is only non-zero if k = m and j = l, so $\delta _{km}\delta _{jl}A_{j}B_{l}C_{m}=A_{j}B_{j}C_{k}=({\vec {A}}\cdot {\vec {B}})C_{k}$

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. $T_{b}^{a}v^{b}$ 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.