Foundations — Tensor analysis — scalars, vectors, rank-2 tensors

4.10.7 · D1 · Maths › Advanced Topics (Elite Level) › Tensor analysis — scalars, vectors, rank-2 tensors

Is page par assume kiya gaya hai ki tumne kuch nahi dekha. Hum har letter, arrow, subscript aur superscript se milenge jo parent note use karta hai, ek aisi order mein jahan har ek cheez sirf un cheezon se bani hai jo pehle se explain ho chuki hain.

0. Coordinates — woh grid jis par tum likhte ho

Notice karo ki number upar baith ta hai: $x^1$, $x^2$. Ise "x-one, x-two" padho — yeh "x squared" nahi hai. Yeh raised label ek index hai, notation ka hamara pehla tukda.

1. Index — ek slot jo count karta hai

• $n$ = dimensions ki sankhya (kitne coordinates hain), isliye $i$ $1$ se $n$ tak chalta hai.
• Upper index jaise $V^i$ aur lower index jaise $W_i$ aage jaake alag-alag tarah ki cheezon ka matlab niklega (Section 6). Abhi ke liye: yeh ek slot number hai, aur iski height ek flag hai jo hum padhna seekhenge.

2. Rank — kisi cheez mein kitne index-slots hain

Numbers ki count $n^{\text{rank}}$ hai: ek scalar mein $n^0=1$ hai, ek vector mein $n^1=n$, ek rank-2 tensor mein $n^2$. Yeh wahi poori "hierarchy" hai jo parent note draw karta hai.

3. Scalar, vector, aur "meaningful grid"

Component ka matlab sirf "shadow-numbers mein se ek" hai. Arrow asli cheez hai; components woh hai jo axes ki ek particular choice report karti hai.

4. Basis change — same arrow, naye numbers

Figure dekho. Ek fixed orange arrow. Black axes par uski shadows $(V^1,V^2)$ hain; teal rotated axes par shadows $({\tilde{V}}^1,{\tilde{V}}^2)$ hain — alag numbers, same arrow. Kisi symbol ke upar tilde $\stackrel{~}{}$ hamesha matlab hai "naye coordinates mein measure kiya gaya".

5. Derivative aur Jacobian — change ka exact rule

Yeh precisely kehne ke liye ki naye numbers purane se kaise relate karte hain, hume woh tool chahiye jo measure karta hai "ek quantity kitni badlati hai jab doosri thodi si hile". Woh tool hai derivative.

Yeh parent note ka central engine hai: har transformation law hai "har index ke liye ek Jacobian factor attach karo". Machine khud ke liye dekho Jacobian and the multivariable chain rule.

6. Upper vs lower index — contravariant vs covariant

Ab index ki height apna kaam karna shuru karti hai.

$V^i$ aur $V_i$ ke beech distinction sirf bent ya stretched coordinates mein numerical ho jaata hai, jahan tum unhe metric use karke ek doosre mein convert karte ho — dekho Metric tensor and Riemannian geometry.

7. Summation convention — gayab hota $\sum$

8. Kronecker delta — identity, index clothes mein

Yeh isliye appear hota hai kyunki Jacobian aur uska inverse milkar iske barabar ho jaate hain: $\frac{\partial {\tilde{x}}^i}{\partial x^k}\frac{\partial x^k}{\partial {\tilde{x}}^j}={\delta }_j^i$ — yeh statement hai "naye coordinates mein jao aur wapas aao, aur kuch nahi hua".

9. Symmetry — ek property jo har coordinate change survive karti hai

Do grids picture karo: ek jo main diagonal ke across reflect karne par same dikhta hai, ek jo apna khud ka negative ban jaata hai. Kyunki transformation law dono indices ke saath identically treat karta hai, yeh mirror-property har frame mein same hai — ek coordinate-free fact, isliye physicists ise trust karte hain (stress kisi bhi axes mein symmetric hai — Stress and strain tensors (continuum mechanics)).

10. Yeh bricks kahan ja rahe hain

Ab tumhare paas har woh symbol hai jo parent note use karta hai. Do cheezein is foundation se bahar hain aur baad mein apne khud ke pages pe milenge — yahan sirf flag ki gayi hain taaki tum jaano ki woh exist karti hain:

• Covariant derivative ${\nabla }_k$ (ordinary ${\partial }_k$ ek tensor nahi hai) — Covariant derivative and Christoffel symbols.
• Bada payoff, curved spacetime as a tensor equation — General Relativity — Einstein field equations.

Equipment checklist

Khud ko test karo — har cheez obvious lagni chahiye parent note tackle karne se pehle.