Foundations — Einstein summation convention

This is the foundations child of Einstein summation convention. The parent note uses words like "index", "component", "dummy", "rank", "${\delta }_{ij}$", "${\epsilon }_{ijk}$" as if you already own them. Here we build every single one from the ground floor, in an order where each brick rests on the previous one. A smart 12-year-old who has never seen a subscript should finish this page ready to read the parent.

1. A number vs. a list of numbers

Before any fancy notation, the whole subject is about two kinds of "thing".

• A scalar is a single number. Your age, the temperature, the length of a stick. One value, nothing else.
• A vector (in the plain "arrow" sense) is a list of numbers stacked together. In 3D, an arrow pointing somewhere needs three numbers: how far along the $x$-direction, how far along $y$, how far along $z$.

Why do we need a name for "one entry"? Because the entire Einstein machinery is a language for talking about one component at a time and then quietly summing over all of them. You cannot describe the trick without first being able to point at a single entry.

2. The subscript: an index

How do we point at the second entry of a list using symbols? We attach a small number below the letter:

$a_1=4,a_2=2,a_3=5.$

That little "$1$", "$2$", "$3$" is the index. Read $a_2$ as "the number in slot 2 of the list $a$".

3. The summation sign $\sum$

Now the operation everything is built on: adding up a whole list.

Suppose we want $a_1+a_2+a_3$. Writing three dots gets clumsy for long lists, so mathematicians invented shorthand — the capital Greek letter Sigma, $\sum$ ("S" for "Sum"):

${\sum }_{i=1}^3{a}_i=a_1+a_2+a_3.$

Read the pieces out loud once and it sticks: "sum, for $i$ from $1$ to $3$, of $a_i$."

Here $i$ is a temporary counter — once the sum is done, $i$ is gone; it doesn't survive into the answer. This "used up and thrown away" behaviour is the seed of what the parent calls a dummy index.

4. Einstein's observation: twice means sum

Look back at every sum you actually meet in vector algebra:

• Dot product ${\sum }_i{a}_i{b}_i$ — the index $i$ appears twice.
• Matrix times vector ${\sum }_j{A}_{ij}{x}_j$ — the summed index $j$ appears twice.
• Matrix times matrix ${\sum }_j{A}_{ij}{B}_{jk}$ — the summed index $j$ appears twice.

Every single time, the index you sum over is the one appearing twice. So the $\sum$ sign is redundant — the doubling itself already announces "add me up". Delete it:

$\boxed{a_i{b}_i\equiv \sum _{i=1}^3{a}_i{b}_i.}$

Why does the count of survivors matter so much? Because it tells you what kind of object you built:

That is the parent's "count the free indices to know the rank" rule — and now you can see why it works: each free index is a live "for every value" loop, and $k$ nested loops over $n$ values produce an $n\times n\times \cdots$ grid of numbers.

5. Why "exactly twice" and never three times

A sum pairs things: ${\sum }_i({\text{factor}}_i)({\text{factor}}_i)$ glues two factors together, slot by slot. If a label showed up three times, $a_i{b}_i{c}_i$, which two of the three do you pair for the sum? There's no unique answer — the notation would be ambiguous. So three copies of one index in one term is illegal. If you genuinely want that triple sum, you must write ${\sum }_i{a}_i{b}_i{c}_i$ explicitly. This is why the parent forbids $a_i{b}_i{c}_i$.

6. Helper object one — the Kronecker delta ${\delta }_{ij}$

We need a symbol that says "are these two indices equal?". Enter $\delta$ (lowercase Greek "delta"), carrying two indices:

${\delta }_{ij}=\{\begin{array}{ll}1& i=j\\ 0& i\ne j\end{array}$

Its superpower — the substitution rule — falls straight out of section 3's summation: ${\delta }_{ij}{a}_j={\sum }_j{\delta }_{ij}{a}_j=a_i,$ because every term with $j\ne i$ is killed by the $0$, and only $j=i$ (where $\delta =1$) survives. In words: "a delta eats a repeated index and renames it." See Kronecker delta for more.

7. Helper object two — the Levi-Civita symbol ${\epsilon }_{ijk}$

We also need a symbol encoding orientation / order — needed for cross products and Determinants. That is $\epsilon$ (Greek "epsilon"), carrying three indices in 3D:

${\epsilon }_{ijk}=\{\begin{array}{ll}+1& (i,j,k)\text{ is }(1,2,3),(2,3,1),(3,1,2)\\ -1& (i,j,k)\text{ is }(3,2,1),(1,3,2),(2,1,3)\\ 0& \text{any two indices equal}\end{array}$

8. Two objects side by side

$\delta$ answers "are these the same?" (a symmetric yes/no). $\epsilon$ answers "in what order / orientation?" (an antisymmetric signed answer). The parent's master identity ${\epsilon }_{ijk}{\epsilon }_{ilm}={\delta }_{jl}{\delta }_{km}-{\delta }_{jm}{\delta }_{kl}$ is just these two helpers talking to each other — but you can only read it once every symbol above is second nature.

Prerequisite map

Equipment checklist

Test yourself — cover the right side and answer out loud.