Foundations — Einstein summation convention

4.10.9 · D1 · Maths › Advanced Topics (Elite Level) › Einstein summation convention

Yeh Einstein summation convention ka foundations child hai. Parent note use karta hai words jaise "index", "component", "dummy", "rank", "${\delta }_{ij}$", "${\epsilon }_{ijk}$" — jaise ye tumhare paas pehle se hain. Yahan hum har ek cheez ground floor se build karenge, ek aisi order mein jahan har brick pichli brick par rakhti hai. Ek smart 12-saal-ka bacha jo kabhi subscript nahi dekha, is page ke baad parent note padh sakta hai.

1. Ek number vs. numbers ki list

Kisi bhi fancy notation se pehle, poora subject do tarah ki "cheez" ke baare mein hai.

• Ek scalar ek single number hota hai. Tumhari umar, temperature, ek stick ki length. Ek value, kuch nahi.
• Ek vector (plain "arrow" sense mein) numbers ki ek list hoti hai jo ek saath stacked hoti hai. 3D mein, kisi taraf point karne wale arrow ko teen numbers chahiye: $x$-direction mein kitna, $y$ mein kitna, $z$ mein kitna.

Hume "ek entry" ke liye naam kyun chahiye? Kyunki poori Einstein machinery ek language hai ek component at a time ke baare mein baat karne ke liye, aur phir quietly unhe saari values par sum karne ke liye. Tum is trick ko describe nahi kar sakte jab tak tum pehle ek single entry par point nahi kar sakte.

2. Subscript: ek index

Hum symbols use karke list ki doosri entry par point kaise karte hain? Hum letter ke neeche ek chhota number attach karte hain:

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

Woh chhota "$1$", "$2$", "$3$" hi index hai. $a_2$ ko padho "list $a$ ke slot 2 mein number".

3. Summation sign $\sum$

Ab woh operation jis par sab kuch build hota hai: poori list ko add karna.

Maan lo hum chahte hain $a_1+a_2+a_3$. Teen dots likhna lambi lists ke liye clumsy ho jaata hai, isliye mathematicians ne shorthand banaya — capital Greek letter Sigma, $\sum$ ("S" for "Sum"):

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

Ise ek baar zor se padho aur yeh yaad ho jaayega: "sum, for $i$ from $1$ to $3$, of $a_i$."

Yahan $i$ ek temporary counter hai — ek baar sum ho jaane ke baad, $i$ chala jaata hai; woh answer mein nahi bachta. Yeh "use hoke throw away" wala behaviour hi woh seed hai jise parent dummy index kehta hai.

4. Einstein ka observation: do baar matlab sum

Vector algebra mein jo bhi sums actually milte hain unhe wapas dekho:

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

Har baar, jis index par sum karte ho woh do baar aane wala hota hai. Toh $\sum$ sign redundant hai — doubling khud hi announce kar deta hai "mujhe add karo". Ise delete karo:

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

Bachnewaalon ki count itni zyaada kyun matter karti hai? Kyunki yeh tumhe batati hai tumne kaunsa object banaya:

Yahi parent ka "rank jaanne ke liye free indices count karo" rule hai — aur ab tum dekh sakte ho kyun yeh kaam karta hai: har free index ek live "for every value" loop hai, aur $n$ values par $k$ nested loops $n\times n\times \cdots$ numbers ka grid produce karte hain.

5. "Exactly do baar" kyun, teen baar kyun nahi

Ek sum cheezein pair karta hai: ${\sum }_i({\text{factor}}_i)({\text{factor}}_i)$ do factors ko slot by slot glue karta hai. Agar ek label teen baar aata, $a_i{b}_i{c}_i$, toh teeno mein se kaun se do ko sum ke liye pair karte? Koi unique answer nahi hoga — notation ambiguous ho jaata. Isliye ek term mein ek index ke teen copies illegal hain. Agar tumhe sach mein woh triple sum chahiye, toh tumhe ${\sum }_i{a}_i{b}_i{c}_i$ explicitly likhna hoga. Isliye parent $a_i{b}_i{c}_i$ ko forbid karta hai.

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

Hume ek aisa symbol chahiye jo kahe "kya yeh do indices equal hain?". Enter $\delta$ (lowercase Greek "delta"), jo do indices carry karta hai:

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

Uski superpower — substitution rule — seedha section 3 ki summation se nikalti hai: ${\delta }_{ij}{a}_j={\sum }_j{\delta }_{ij}{a}_j=a_i,$ kyunki $j\ne i$ wale har term ko $0$ kill kar deta hai, aur sirf $j=i$ (jahan $\delta =1$) bachta hai. Words mein: "ek delta ek repeated index kha jaata hai aur usse rename kar deta hai." Zyaada ke liye Kronecker delta dekho.

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

Hume ek aisa symbol bhi chahiye jo orientation / order encode kare — cross products aur Determinants ke liye zaroori. Woh hai $\epsilon$ (Greek "epsilon"), jo 3D mein teen indices carry karta hai:

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

8. Do objects side by side

$\delta$ jawab deta hai "kya yeh same hain?" (ek symmetric yes/no). $\epsilon$ jawab deta hai "kis order / orientation mein?" (ek antisymmetric signed answer). Parent ki master identity ${\epsilon }_{ijk}{\epsilon }_{ilm}={\delta }_{jl}{\delta }_{km}-{\delta }_{jm}{\delta }_{kl}$ bas yeh do helpers ek doosre se baat kar rahe hain — lekin tum ise tab tak padh nahi sakte jab tak upar ke har symbol second nature nahi ban jaate.

Prerequisite map

Equipment checklist

Khud ko test karo — right side cover karo aur zor se jawab do.