Physics foundations

Einstein summation notation: the rules, symbols, and identities that make tensor expressions readable.

Read it in order: rules, simplification, symbols, and applications.

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Start with the rules See simplification Back to Physics page

Reading order

1. Read the rules: Learn what counts as a legal index expression.
2. Simplify carefully: Keep free indices visible and contract the rest.
3. Keep identities nearby: Delta and Levi-Civita are the main tools you will use repeatedly.

Step 1

The rule set comes first.

How Einstein notation behaves

Rule	Meaning	Example
Repeated index	One repeated index in a term means sum over it.	\(A_iB_i = \sum_i A_iB_i\)
Free index	Free indices must match on both sides of an equation.	\(T_{ij}v_j = w_i\)
Dummy index	A dummy index is local and may be renamed.	\(A_iB_i = A_jB_j\)
Index count	No index may appear more than twice in one term.	\(A_iB_iC_i\) is invalid
Index position	Upper and lower indices matter in tensor calculus.	\(a_i = g_{ij}a^j\)

Step 2

A reliable simplification sequence

Move	What it does	Example
Mark free indices	These must survive to the final expression.	\(T_{ij}v_j\) keeps the free index \(i\)
Identify dummy pairs	Each repeated pair stands for one sum.	\(\delta_{ij}A_j = A_i\)
Apply identities early	Kronecker delta usually collapses terms first.	\(\delta_{ij}\delta_{jk}A_k = A_i\)
Use symmetry	Symmetric and antisymmetric pieces can cancel.	\(\epsilon_{ijk}a_ja_k = 0\)

Worked checks

Expression	Result	Reason
\(A_i + B_i\)	Valid	Same free index structure.
\(A_i + B_j\)	Invalid	The free indices do not match.
\(A_iB_iC_i\)	Invalid	Index \(i\) appears three times in one term.
\(\delta_{ii}\)	\(n\)	Trace of the identity tensor.

Step 3

Symbol sheet

Symbol	Definition	Common use
\(\delta_{ij}\)	1 when \(i=j\), otherwise 0.	Acts like the identity matrix.
\(\epsilon_{ijk}\)	Completely antisymmetric in 3D.	Encodes cross products and orientation.

\delta_{ij}A_j = A_i, \qquad \delta_{ij}\delta_{jk} = \delta_{ik}, \qquad \frac{\partial x_i}{\partial x_j}=\delta_{ij}

\epsilon_{ijk}=\begin{cases} +1 & \text{even permutation of }(123)\\ -1 & \text{odd permutation of }(123)\\ 0 & \text{if any indices repeat} \end{cases}

(a\times b)_i = \epsilon_{ijk}a_jb_k, \qquad (\nabla\times A)_i = \epsilon_{ijk}\partial_j A_k

\epsilon_{ijk}\epsilon_{imn}=\delta_{jm}\delta_{kn}-\delta_{jn}\delta_{km}

Step 4

Identity sheet

Identity	Use
\(\epsilon_{i_1\cdots i_n}\epsilon_{j_1\cdots j_n} = \det[\delta_{i_a j_b}]\)	General antisymmetric contraction in any dimension.
\(\epsilon_{i_1\cdots i_n}\epsilon_{i_1\cdots i_n}=n!\)	Norm contraction of the Levi-Civita symbol.
\(\epsilon_{ij}\epsilon_{kj}=\delta_{ik}\)	Compact 2D contraction identity.
\(T_{ii}=\mathrm{tr}(T), \qquad \partial_i T_i\)	Trace and divergence are both contractions.
\(S_{ij}\epsilon_{ijk}=0\) when \(S_{ij}=S_{ji}\)	A symmetric tensor vanishes against an antisymmetric one.

Step 5

Where the notation shows up

Area	Example	Why it helps
Classical mechanics	\(L_i=\epsilon_{ijk}x_jp_k\)	Keeps angular momentum structure compact.
Electromagnetism	\((\nabla\times E)_i=\epsilon_{ijk}\partial_jE_k\)	Turns vector identities into index checks.
Continuum mechanics	\(\partial_j\sigma_{ij}+f_i=\rho a_i\)	Free-index consistency becomes immediate.
Relativity	\(a_i=g_{ij}a^j\)	Shows how the metric moves indices.

Final check

Final checklist

Check	Pass condition
Free indices	They match on both sides.
Dummy indices	Each appears exactly twice in one term.
Mixed roles	No symbol is both free and dummy in the same term.
Index position	Upper and lower placement is consistent when relevant.
Symmetry	Use symmetry and antisymmetry to remove zero terms early.

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