Einstein summation convention — Hinglish notes

4.10.9 · Maths › Advanced Topics (Elite Level)

YEH HAI KYA?

Toh $a_{i} b_{i}$ secretly matlab hai $i = 1 \sum n a_{i} b_{i}$ , jahan $n$ dimension hai (physics mein aksar 3 hota hai).

RULES ACTUALLY KAISE KAAM KARTE HAIN (derive karo, memorise mat karo)

Teen guiding questions se poora system build karte hain.

Q1: Main object ka type kaise count karun? Free indices count karo.

0 free indices → scalar (e.g. $a_{i} b_{i}$ ).
1 free index → vector / column of numbers (e.g. $A_{ij} b_{j}$ mein free index $i$ hai).
2 free indices → matrix / rank-2 tensor (e.g. $A_{ij}$ ).

KYU? Har free index $1 \dots n$ tak jaati hai, isliye ek free index se aapko $n$ alag alag components milte hain.

Q2: Summed index exactly do baar hi kyun appear karna chahiye? Kyunki summation cheezein pair karti hai: $\sum_{i} (this_{i}) (that_{i})$ . Agar ek index teen baar appear kare toh aap nahi bata sakte ki kaun sa pair sum karna hai — notation ambiguous ho jaayega. Ek index jo 3 ya zyada baar ek term mein appear kare woh illegal hai.

Q3: Main dummy index ko freely rename kyun kar sakta hoon? Kyunki $\sum_{i} a_{i} b_{i} = \sum_{k} a_{k} b_{k}$ — summation variable ka naam matter nahi karta. Toh $a_{i} b_{i} = a_{k} b_{k}$ . Yeh trick clashes avoid karne ke liye zaroori hai.

Do indispensable helper objects

Forecast-then-Verify drill

Common mistakes (steel-manned)

Recall Feynman: ek 12-saal ke bachche ko samjhao

Ek recipe imagine karo jo kehti hai "har matching jurabon ki jodi ke liye, unke sizes add karo." Tumhe har baar "add up" likhna nahi padta — matching hi add karne ka signal hai. Einstein ki trick bhi yehi hai: jab bhi koi label ek hi chunk mein do baar aaye, woh ek secret "sab add karo" command hai. Jo label sirf ek baar aaye woh bas bata raha hai "yeh component number itna hai," jaise ghar ka number — use rakh lo. Toh dekh ke ki kaun se letters double hain, tum turant jaante ho kya add karna hai aur kaisa answer (number, list, ya grid) milega.

Flashcards

In Einstein convention, what does a repeated index in one term mean?

Uss index ko uski poori range par sum karo (

\sum

implied hai).

What is a free index and what rule must it obey?

Exactly ek baar appear karne wala index; equation ke dono sides par same hona chahiye aur sum nahi hota.

How many free indices does a scalar / vector / matrix have?

Krama se 0 / 1 / 2.

Write the dot product

a \cdot b

in index notation.

a_{i} b_{i}

Write matrix multiplication

C = A B

in index notation.

C_{ik} = A_{ij} B_{j k}

State the substitution property of the Kronecker delta.

δ_{ij} a_{j} = a_{i}

What is

δ_{ii}

n

dimensions?

n

(dimension).

Write the

i

-th component of

a \times b

(a \times b)_{i} = ε_{ij k} a_{j} b_{k}

State the epsilon–delta identity.

ε_{ij k} ε_{i l m} = δ_{j l} δ_{k m} - δ_{j m} δ_{k l}

Why is an index appearing three times illegal?

Repeated index ek pairwise sum mean karta hai; teen copies pairing ko ambiguous bana dete hain.

Simplify

δ_{ij} δ_{j k}

δ_{ik}

What does

ε_{ij k}

equal if two indices are equal?

0

Connections

Dot product aur Cross product — dono convention ke zariye compact form mein likhe jaate hain.
Kronecker delta aur Levi-Civita symbol — index algebra ke do engines.
Matrix multiplication — ek shared repeated index se encode hoti hai.
Tensors — convention upper/lower (contravariant/covariant) indices tak generalise hoti hai.
Determinants — 3D mein $det A = ε_{ij k} A_{1 i} A_{2 j} A_{3 k}$ .
Vector calculus identities — epsilon–delta identity se seconds mein derive hoti hain.