Tensor analysis — scalars, vectors, rank-2 tensors

4.10.7 · Maths › Advanced Topics (Elite Level)

WHAT — rank ke hisaab se hierarchy

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

HOW — transformation rules ko scratch se derive karna

Hum coordinates change karte hain $x^{i} \to \tilde{x}^{i}$ . Jacobian matrices define karte hain:

J^{i}_{k} = \frac{\partial x ~ ^{i}}{\partial x ^{k}}, (J^{- 1})^{k}_{j} = \frac{\partial x ^{k}}{\partial x ~ ^{j}} .

Step 1 — Scalars (rank 0)

Ek scalar ek point par coordinates se independent same number hota hai.

\tilde{ϕ} (\tilde{x}) = ϕ (x) .

Kyun? Room ke kone par temperature nahi badalti kyunki aapne apna ruler ghuma liya.

Step 2 — Vectors, do flavours (rank 1)

Ek aisi cheez se vector banao jiske baare mein hum pehle se jaante hain ki woh sahi transform hoti hai.

Contravariant (upper index) — jaise ek displacement. $d x^{k}$ lo. Chain rule se,

d \tilde{x}^{i} = \frac{\partial x ~ ^{i}}{\partial x ^{k}} d x^{k} = J^{i}_{k} d x^{k} .

Yeh step kyun? $d \tilde{x}^{i}$ literally hai hi $\partial \tilde{x}^{i} / \partial x^{k}$ times $d x^{k}$ — yeh sirf total differentiation hai. Toh jo bhi is tarah transform hota hai woh contravariant hai:

\tilde{V}^{i} = \frac{\partial x ~ ^{i}}{\partial x ^{k}} V^{k} .

Covariant (lower index) — jaise ek gradient. $\partial ϕ / \partial x^{k}$ lo. Chain rule se,

\frac{\partial ϕ}{\partial x ~ ^{i}} = \frac{\partial x ^{k}}{\partial x ~ ^{i}} \frac{\partial ϕ}{\partial x ^{k}} .

Toh jo bhi is tarah transform hota hai woh covariant hai:

\tilde{W}_{i} = \frac{\partial x ^{k}}{\partial x ~ ^{i}} W_{k} .

Step 3 — Rank-2 tensors (sirf do rank-1 rules jodo)

Ek rank-2 tensor har index ke liye ek Jacobian factor ke saath transform hota hai, har factor apne index ke type se match karta hai. Teen flavours:

\tilde{T}^{ij} = \frac{\partial x ~ ^{i}}{\partial x ^{k}} \frac{\partial x ~ ^{j}}{\partial x ^{l}} T^{k l} (contravariant)

\tilde{T}_{ij} = \frac{\partial x ^{k}}{\partial x ~ ^{i}} \frac{\partial x ^{l}}{\partial x ~ ^{j}} T_{k l} (covariant), \tilde{T}^{i}_{j} = \frac{\partial x ~ ^{i}}{\partial x ^{k}} \frac{\partial x ^{l}}{\partial x ~ ^{j}} T^{k}_{l} (mixed) .

Kyun? Ek rank-2 tensor ek aisi machine hai jo do vector slots khaati/deti hai; har slot ko apna transformation factor milta hai. Rotation $R$ ke liye matrix form mein (orthonormal, $R^{- 1} = R^{⊤}$ ):

\tilde{T} = R T R^{⊤} .

Key operations (aur kyun yeh tensor character preserve karte hain)

Worked examples

Common mistakes (Steel-manned)

Flashcards

Numbers ka ek grid tensor kya banata hai (sirf matrix nahi)?

Isko coordinates change hone par tensor transformation law obey karna hoga; components Jacobian factors ke saath transform hote hain.

Contravariant vector

V^{i}

ka transformation rule

\tilde{V}^{i} = \frac{\partial x ~ ^{i}}{\partial x ^{k}} V^{k}

(ek forward Jacobian).

Covariant vector

W_{i}

ka transformation rule

\tilde{W}_{i} = \frac{\partial x ^{k}}{\partial x ~ ^{i}} W_{k}

(inverse Jacobian).

Rank-2 contravariant tensor kaise transform hota hai?

\tilde{T}^{ij} = \frac{\partial x ~ ^{i}}{\partial x ^{k}} \frac{\partial x ~ ^{j}}{\partial x ^{l}} T^{k l}

(rotations ke liye

R T R^{⊤}

Contraction

T^{i}_{i}

scalar kyun hai?

Do Jacobian factors chain rule se

δ_{k}^{l}

mein combine ho jaate hain, koi transformation factor nahi bachta.

Quotient theorem kya hai?

Agar

A^{ij} V_{j}

har vector

V_{j}

ke liye ek tensor hai, toh

A^{ij}

khud bhi ek tensor hai.

n

dimensions mein rank-

r

tensor ke components ki sankhya

n^{r}

\partial_{k} V^{i}

tensor kyun nahi hai?

Derivative position-dependent Jacobian par bhi act karti hai, jo ek extra non-tensorial term produce karti hai; covariant derivative isko fix karta hai.

Metric tensor

g_{ij}

ki role

d s^{2} = g_{ij} d x^{i} d x^{j}

define karta hai aur indices raise/lower karta hai:

V_{i} = g_{ij} V^{j}

Rank-2 tensor ki kaun si quantities rotation-invariant hain?

Trace, determinant, aur eigenvalues.

Recall Feynman: ek 12-saal ke bacche ko explain karo

Ek map par khaazane ki kalpana karo. Khaazana kahan hai kabhi nahi badalte — lekin agar tum map ghuma do, toh "3 kadam east, 4 kadam north" wali instructions kuch aur ho jaati hain, bhale hi woh same jagah point kar rahi hon. Ek tensor woh rulebook hai ki jab tum map ghumaate ho toh woh instructions kaise badalti hain. Ek scalar khaazane ka weight hai (chahe tum map kaise bhi ghumaao, same rehta hai). Ek vector khaazane ki taraf jaane wala arrow hai (instructions ek simple tarike se badalti hain). Ek rank-2 tensor us jagah push-aur-twist describe karne wali stretchy rubber sheet jaisi hai (instructions do linked tareekon se badalti hain). Jaadu yeh hai: asli khaazana (physics) hamesha same hota hai — sirf hamare descriptions ghoomte hain.