Einstein Summation Convention

Definitions

Expression, Terms, Indices

\(\begin{equation} A_{im}x_{m}+b_i+C_{ikq}x_{k}y_{q} \end{equation}\)

The expression above contains three terms. The symbols $i,m,k,q$ denote indices. They typically run from one to number of space dimensions, i.e., two or three.

The complexity of the terms can be indefinitely increased. \(\begin{equation} a_{ij} x_ix_jx_k \end{equation}\)

Free and Dummy Indices

• Definition. An index that appears only once is a called a free index
• Definition. An index that appears exactly twice is a called a dummy index

\[\begin{equation} \underbrace { A_{im}x_{m} } _{ \text{Free}=\{i\}\\ \text{Dummy}=\{m\} } +\underbrace{ b_i} _{ \text{Free}=\{i\}\\ \text{Dummy}=\emptyset }+\underbrace{C_{ikq} x_{k} y_{q}}_{ \text{Free}=\{i\}\\ \text{Dummy}=\{k,q\} } \end{equation}\]

Rules

Rule 1.a: A dummy index implies a sum w.r.t that index.

Rule 1.b: In ESC, an index cannot appear more than twice

Examples

1. \[\begin{align} x_{i}y_{i} & = x_1y_1+ x_2 y_2+ x_3 y_3 \end{align}\]
2. \[\begin{align} m^{j}v_{j} & = m^{1}v_{1}+m^{2}v_{2}+m^{3}v_{3} \end{align}\]
3. \[\begin{align} a_{ij}x_i y_j&=x_ia_{i1} y_1+x_ia_{i2} y_2+x_ia_{i3} y_3\\ &=(x_1a_{11}+x_2a_{21}+x_3a_{31}) y_1 +(x_1a_{12}+x_2a_{22}+x_3a_{32}) y_2+ (x_1a_{13}+x_2a_{23}+x_3a_{33}) y_3\\ &=(a_{11}x_1y_1+a_{21}x_2y_1+a_{31}x_3y_1 +a_{12}x_1y_2+a_{22}x_2y_2+a_{32}x_3y_2 +a_{13}x_1y_3+a_{23}x_2y_3+a_{33}x_3y_3 \end{align}\]
4. \[\begin{align}a_{ijk}x_ix_j x_k&=\sum_{i=1}^{3} \sum_{j=1}^{3} \sum_{k=1}^{3} a_{ijk}x_ix_j x_k \end{align}\]
5. \[\begin{align} x_i a_j x_j a_i&=\sum_{i=1}^{3} \sum_{j=1}^{3} x_i a_j x_j a_i\\ &= \sum_{i=1}^{3}x_i a_i \sum_{j=1}^{3} x_j a_j \end{align}\]

Non-examples

\(\begin{equation} x_{jj}y_j \quad \text{(Not okay; j repeated thrice)} \end{equation}\)

Rule 2 (Renaming of dummy indices): A dummy index can be renamed, as long as the new name does not conflict with the names of the other dummy or free indices in the same term

Examples

\(\begin{align} x_iy_i&\Rightarrow x_j y_j\\ \end{align}\)

\[\begin{align} a_ix_jy_j&\Rightarrow a_ix_k y_k, \quad (\text{OK, no conflict with free index}) \\ a_ix_jy_j&\not\Rightarrow a_ix_i y_i \quad (\text{conflict with free index})\\ \end{align}\] \[\begin{align} A_{im}x_{m}+b_i+C_{ikq}x_{k}y_{q} & \Rightarrow A_{im}x_{m}+b_i+C_{ipq}x_{p}y_{q}\quad (\text{OK, no conflict}) \\ A_{im}x_{m}+b_i+C_{ikq}x_{k}y_{q} & \Rightarrow A_{im}x_{m}+b_i+C_{imq}x_{m}y_{q}\quad (\text{OK, no conflict, even though matches dummy index in a different term}) \\ A_{im}x_{m}+b_i+C_{ikq}x_{k}y_{q} & \not\Rightarrow A_{im}x_{m}+b_i+C_{iiq}x_{i}y_{q}\quad (\text{conflict with free index}) \\ A_{im}x_{m}+b_i+C_{ikq}x_{k}y_{q} & \not\Rightarrow A_{im}x_{m}+b_i+C_{iqq}x_{q}y_{q}\quad (\text{conflict with dummy index}) \\ A_{im}x_{m}+b_i+C_{ikq}x_{k}y_{q} &\not \Rightarrow A_{ii}x_{i}+b_i+C_{ikq}x_{k}y_{q}\quad (\text{conflict with free index}) \\ \end{align}\]

Rule 2.b: Different terms can have same dummy indices.

Examples:

\[\begin{equation} \underbrace { A_{im}x_m } _{ \{i\}\\ \{m\} } +\underbrace{ b_i} _{ \{i\}\\ \emptyset }+\underbrace{C_{ikq} x_{k} y_{q}}_{ \{i\}\\ \{k,q\} } \end{equation}\]

$\Rightarrow$

\[\begin{equation} \underbrace { A_{im}x_m } _{ \{i\}\\ \{m\} } +\underbrace{ b_i} _{ \{i\}\\ \emptyset }+\underbrace{C_{ikq} x_{k} y_{q}}_{ \{i\}\\ \{k \rightarrow m,q\} } \end{equation}\] \[\begin{equation} \underbrace { A_{im}x_m } _{ \{i\}\\ \{m\} } +\underbrace{ b_i} _{ \{i\}\\ \emptyset }+\underbrace{C_{imq} x_{m} y_{q}}_{ \{i\}\\ \{m,q\} } \end{equation}\]

$\Rightarrow$

\[\begin{equation} \underbrace { A_{im}x_m } _{ \{i\}\\ \{m\} } +\underbrace{ b_i} _{ \{i\}\\ \emptyset }+\underbrace{C_{imq} x_{m} y_{q}}_{ \{i\}\\ \{m \rightarrow i,q\} } \end{equation}\]

$\Rightarrow$

\(\begin{equation} \underbrace { A_{im}x_m }_{ \{i\}\\ \{m\} } +\underbrace{ b_i} _{ \{i\}\\ \emptyset }+\underbrace{C_{iiq} x_{i} y_{q}}_{ \{i\}\\ \{i,q\} } \end{equation}\) (Not okay - conflict with free index)

Rule 3 (Free Indices) A free index denotes a set of $n_{\rm sd}$ terms

Examples

\[\begin{align} a_{i}&\Rightarrow\{a_1,a_2,a_3\}\\ a_{i}=b_{i}&\Rightarrow\{a_1=b_1,a_2=b_2,a_3=b_3\}\\ a_{i}+b_{i}&\Rightarrow\{a_1+b_1,a_2+b_2,a_3+b_3\}\\ \end{align}\]

Denoting ordered sets

\[(\left(A_{ij})_{j\in \mathcal{I}}\right)_{i\in \mathcal{I}}=(A_{ij})_{i,~j\in \mathcal{I}}\] \[\left(\left(A_{1j}\right)_{j \in \mathcal{I}}, \left(A_{2j}\right)_{j\in \mathcal{I}}, \left(A_{3j}\right)_{j\in \mathcal{I}}\right)\]

Notice that in each expression, all terms have the same set of free indices. That is, take a look at the expression,

\[\begin{equation} \underbrace { A_{im}x_{m} } _{ \text{Free}=\{i\}\\ \text{Dummy}=\{m\} } +\underbrace{ b_i} _{ \text{Free}=\{i\}\\ \text{Dummy}=\emptyset }+\underbrace{C_{ikq} x_{k} y_{q}}_{ \text{Free}=\{i\}\\ \text{Dummy}=\{k,q\} } \end{equation}\]

This is a consequence of the following rule:

Rule 4: All terms in an expression should have the same set of free indices.

Examples

\(\begin{align} a_i +b_i\quad (\text{OK})\\ a_j +b_j\quad (\text{OK})\\ a_i +b_j\quad (\text{Not OK})\\ M_i =C_{mi}b_m\quad (\text{OK})\\ M_i =C_{mj}b_m\quad (\text{Not OK})\\ \end{align}\)

(the power of computers) - Viola Mathematica

Vector dot-product

Definition

$\boldsymbol{a}, \boldsymbol{b} \mapsto \text{Dot}(\boldsymbol{a},\boldsymbol{b}) \in \mathbb{R}$, where $\mathbb{R}$ is the set of real numbers, and $\text{Dot}(\boldsymbol{a},\boldsymbol{b}) = \text{Dot}(\boldsymbol{b},\boldsymbol{a})$.

\[\begin{align} \hat{\boldsymbol{\mathsf{E}}}_1\cdot\hat{\boldsymbol{\mathsf{E}}}_1&=1,\\ \hat{\boldsymbol{\mathsf{E}}}_2\cdot\hat{\boldsymbol{\mathsf{E}}}_2&=1,\\ \hat{\boldsymbol{\mathsf{E}}}_3\cdot\hat{\boldsymbol{\mathsf{E}}}_3&=1,\\ \hat{\boldsymbol{\mathsf{E}}}_1\cdot\hat{\boldsymbol{\mathsf{E}}}_2 &= 0 & \hat{\boldsymbol{\mathsf{E}}}_2\cdot\hat{\boldsymbol{\mathsf{E}}}_1 &= 0, \\ \hat{\boldsymbol{\mathsf{E}}}_2\cdot\hat{\boldsymbol{\mathsf{E}}}_3 &= 0 & \hat{\boldsymbol{\mathsf{E}}}_3\cdot\hat{\boldsymbol{\mathsf{E}}}_2 &= 0, \\ \hat{\boldsymbol{\mathsf{E}}}_3\cdot\hat{\boldsymbol{\mathsf{E}}}_1 &= 0 & \hat{\boldsymbol{\mathsf{E}}}_1\cdot\hat{\boldsymbol{\mathsf{E}}}_3 &= 0. \\ \end{align}\]

That is, in summary, for an orthonromal basis we have \(\hat{\boldsymbol{\mathsf{E}}}_i \cdot \hat{\boldsymbol{\mathsf{E}}}_j=\delta_{ij}.\)

For two arbitrary vectors $\boldsymbol{\mathsf{a}}$ and $\boldsymbol{\mathsf{b}}$, such that $[\boldsymbol{\mathsf{a}}]_i=a_i$ and $[\boldsymbol{\mathsf{b}}]_i=b_i$, we have that

\[\begin{align} (a_i \hat{\boldsymbol{\mathsf{E}}}_i) \cdot (b_j \hat{\boldsymbol{\mathsf{E}}}_j) = & a_1 b_1 \hat{\boldsymbol{\mathsf{E}}}_1 \cdot \hat{\boldsymbol{\mathsf{E}}}_1+ a_1 b_2 \hat{\boldsymbol{\mathsf{E}}}_1 \cdot \hat{\boldsymbol{\mathsf{E}}}_2+ a_1 b_3 \hat{\boldsymbol{\mathsf{E}}}_1 \cdot \hat{\boldsymbol{\mathsf{E}}}_3+\\ & a_2 b_1 \hat{\boldsymbol{\mathsf{E}}}_2 \cdot \hat{\boldsymbol{\mathsf{E}}}_1+ a_2 b_2 \hat{\boldsymbol{\mathsf{E}}}_2 \cdot \hat{\boldsymbol{\mathsf{E}}}_2+ a_2 b_3 \hat{\boldsymbol{\mathsf{E}}}_2 \cdot \hat{\boldsymbol{\mathsf{E}}}_3+ \\ & a_3 b_1 \hat{\boldsymbol{\mathsf{E}}}_3 \cdot \hat{\boldsymbol{\mathsf{E}}}_1+ a_3 b_2 \hat{\boldsymbol{\mathsf{E}}}_3 \cdot \hat{\boldsymbol{\mathsf{E}}}_2+ a_3 b_3 \hat{\boldsymbol{\mathsf{E}}}_3 \cdot \hat{\boldsymbol{\mathsf{E}}}_3 \\ =& a_1 b_1 + a_2 b_2 + a_3 b_3 \\ =&a_i b_i \end{align}\]

Kronecker-Delta symbol

Kronecker delta, denoted by $\delta_{ij}$, is defined as:

\[\begin{equation} \delta_{ij} = \begin{cases} 1 & i=j \\ 0 & i \neq j \end{cases} \end{equation}\]

That is,

$\delta_{11}=1$ $\delta_{12}=0$ $\delta_{13}=0$ $\delta_{21}=0$ $\delta_{22}=1$ $\delta_{23}=0$ $\delta_{31}=0$ $\delta_{32}=0$ $\delta_{33}=1$

For example, $\delta_{12}=0$ and $\delta_{33}=1$.

Identity Matrix

\(\mathbb{\mathsf{I}} = \left(\delta_{ij}\right)= \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix}\\ \text{Read paranthesis as ``matrix representation of the Kronecker-delta symbol``}\) \(\left[\delta_{ij}\right]= \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0\\ 0 & 0 & 1 \end{bmatrix}\)

Properties of the Kronecker Delta

1. Replacement Property:

\[\begin{align} \delta_{1m}a_m =\delta_{11}a_1+\delta_{12}a_2+\delta_{13}a_3\ \\ =1a_1+0a_2+0a_3\ \\ =a_1 \end{align}\] \[\begin{align} \delta_{2m}a_m = \delta_{21}a_1+\delta_{22}a_2+0a_3\ \\ =a_2 \end{align}\] \[\begin{align} \delta_{3m}a_m & = \delta_{31}a_1+\delta_{32}a_2+\delta_{33}a_3\ \\ =0a_1+0a_2+a_3\ \\ =a_3 \end{align}\]

So, \(\delta_{im}a_m = a_i\),\((\delta_{1m}a_m, \delta_{2m}a_m , \delta_{3m}a_m) = (a_1, a_2, a_3)\),\(\delta_{1m}a_m = a_i\).

\[\begin{align} \delta_{1m}T_{mj} & = \delta_{11}T_{1j}+ \delta_{12}T_{2j}+\delta_{13}T_{3j} =T_{1j} \\ \delta_{2m}T_{mj} & = \delta_{21}T_{1j}+ \delta_{22}T_{2j}+\delta_{23}T_{3j} =T_{2j} \\ \delta_{3m}T_{mj} & = \delta_{31}T_{1j}+ \delta_{32}T_{2j}+\delta_{33}T_{3j} =T_{3j} \end{align}\]

\(\begin{bmatrix} \delta_{1m}T_{mj} = T_{1j} \\ \delta_{2m}T_{mj} = T_{2j} \\ \delta_{3m}T_{mj} = T_{3j} \end{bmatrix}\) $\Rightarrow$ $\delta_{1m}T_{mj} = T_{1j}, \delta_{2m}T_{mj} = T_{2j}, \delta_{3m}T_{mj}=T_{3j}$. \(\delta_{im}T_{mj} = T_{ij}\)

We can generalize the last two results. If, in a term, the Kronecker delta and another symbol share an index, the index in that symbol can be replaced by the Kronecker delta’s other index in its symbol. For example:

1. $\delta_{im}a_{mb}$ $\Rightarrow$ $\delta_{ib}$
2. $\delta_{mi}a_{mb}$ $\Rightarrow$ $a_{ib}$
3. $\delta_{im}a_{pqms}$ $\Rightarrow$ $a_{pqis}$
4. $\delta_{im}\delta_{mj}$ $\Rightarrow$ $\delta_{ij}$
5. $\delta_{im}\delta_{mn}\delta_{nj}$ $\Rightarrow$ $\delta_{in}\delta_nj$ $\Rightarrow$ $\delta_{ij}$

\[\begin{align} \delta_{m1}T_{mj} &= \delta_{11}T_{1j} + \delta_{21}T_{2j}+ \delta_{31}T_{3j} = T_{1j} \\ \delta_{m2}T_{mj} &= \delta_{12}T_{1j} + \delta_{22}T_{2j}+ \delta_{32}T_{3j} = T_{2j} \\ \delta_{m3}T_{mj} &= \delta_{13}T_{1j} + \delta_{23}T_{2j}+ \delta_{33}T_{3j} = T_{3j} \end{align}\] \[\begin{align} \delta_{m1}T_{mj} &= T_{1j} \\ \delta_{m2}T_{mj} &= T_{2j} \\ \delta_{m3}T_{mj} &= T_{3j} \\ \delta_{mi}T_{mj} &= T_{ij} \end{align}\]

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