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4.10.7Advanced Topics (Elite Level)

Tensor analysis — scalars, vectors, rank-2 tensors

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🇮🇳 Yeh note Hinglish mein padho →

WHAT — the hierarchy by rank

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

HOW — deriving the transformation rules from scratch

We change coordinates xi→x~ ix^i \to \tilde x^{\,i}. Define the Jacobian matrices

J ik=∂x~ i∂xk,(J−1)kj=∂xk∂x~ j.J^{\,i}{}_{k}=\frac{\partial \tilde x^{\,i}}{\partial x^{k}},\qquad (J^{-1})^{k}{}_{j}=\frac{\partial x^{k}}{\partial \tilde x^{\,j}}.

Step 1 — Scalars (rank 0)

A scalar is the same number at a point regardless of coordinates.

ϕ~(x~)=ϕ(x).\tilde\phi(\tilde x)=\phi(x).

Why? Temperature at a room corner doesn't change because you rotated your ruler.

Step 2 — Vectors, two flavours (rank 1)

Build a vector from something we already trust to transform correctly.

Contravariant (upper index) — like a displacement. Take dxkdx^k. By the chain rule,

dx~ i=∂x~ i∂xk dxk=J ik dxk.d\tilde x^{\,i}=\frac{\partial \tilde x^{\,i}}{\partial x^{k}}\,dx^{k}=J^{\,i}{}_{k}\,dx^{k}.

Why this step? dx~ id\tilde x^{\,i} literally is ∂x~ i/∂xk\partial\tilde x^{\,i}/\partial x^k times dxkdx^k — that's just total differentiation. So anything that transforms this way is contravariant:

V~ i=∂x~ i∂xk Vk.\boxed{\tilde V^{\,i}=\frac{\partial \tilde x^{\,i}}{\partial x^{k}}\,V^{k}.}

Covariant (lower index) — like a gradient. Take ∂ϕ/∂xk\partial\phi/\partial x^k. By the chain rule,

∂ϕ∂x~ i=∂xk∂x~ i ∂ϕ∂xk.\frac{\partial\phi}{\partial \tilde x^{\,i}} =\frac{\partial x^{k}}{\partial \tilde x^{\,i}}\,\frac{\partial\phi}{\partial x^{k}}.

So anything transforming this way is covariant:

W~i=∂xk∂x~ i Wk.\boxed{\tilde W_{i}=\frac{\partial x^{k}}{\partial \tilde x^{\,i}}\,W_{k}.}

Step 3 — Rank-2 tensors (just glue two rank-1 rules)

A rank-2 tensor transforms with one Jacobian factor per index, each factor matching that index's type. Three flavours:

T~ij=∂x~i∂xk∂x~j∂xlTkl(contravariant)\boxed{\tilde T^{ij}=\frac{\partial \tilde x^{i}}{\partial x^{k}}\frac{\partial \tilde x^{j}}{\partial x^{l}}T^{kl}} \quad\text{(contravariant)} T~ij=∂xk∂x~i∂xl∂x~jTkl(covariant),T~ij=∂x~i∂xk∂xl∂x~jTkl (mixed).\tilde T_{ij}=\frac{\partial x^{k}}{\partial \tilde x^{i}}\frac{\partial x^{l}}{\partial \tilde x^{j}}T_{kl} \quad\text{(covariant)},\qquad \tilde T^{i}{}_{j}=\frac{\partial \tilde x^{i}}{\partial x^{k}}\frac{\partial x^{l}}{\partial \tilde x^{j}}T^{k}{}_{l}\ \text{(mixed)}.

Why? A rank-2 tensor is a machine eating/feeding two vector slots; each slot gets its own transformation factor. In matrix form for a rotation RR (orthonormal, R−1=R⊤R^{-1}=R^{\top}):

T~=R T R⊤.\tilde T = R\,T\,R^{\top}.

Key operations (and why they preserve tensor character)

Worked examples

Common mistakes (Steel-manned)

Flashcards

What makes a grid of numbers a tensor (not just a matrix)?
It must obey the tensor transformation law under change of coordinates; the components transform with Jacobian factors.
Transformation rule for a contravariant vector ViV^i
V~i=∂x~i∂xkVk\tilde V^i=\dfrac{\partial\tilde x^i}{\partial x^k}V^k (one forward Jacobian).
Transformation rule for a covariant vector WiW_i
W~i=∂xk∂x~iWk\tilde W_i=\dfrac{\partial x^k}{\partial\tilde x^i}W_k (inverse Jacobian).
How does a rank-2 contravariant tensor transform?
T~ij=∂x~i∂xk∂x~j∂xlTkl\tilde T^{ij}=\dfrac{\partial\tilde x^i}{\partial x^k}\dfrac{\partial\tilde x^j}{\partial x^l}T^{kl} (for rotations RTR⊤RTR^\top).
Why is the contraction TiiT^i{}_i a scalar?
The two Jacobian factors combine to δkl\delta^l_k via the chain rule, leaving no transformation factors.
What is the quotient theorem?
If AijVjA^{ij}V_j is a tensor for every vector VjV_j, then AijA^{ij} is itself a tensor.
Number of components of a rank-rr tensor in nn dimensions
nrn^r.
Why isn't ∂kVi\partial_k V^i a tensor?
The derivative also acts on the position-dependent Jacobian, producing an extra non-tensorial term; the covariant derivative fixes this.
Role of the metric tensor gijg_{ij}
Defines ds2=gijdxidxjds^2=g_{ij}dx^i dx^j and raises/lowers indices: Vi=gijVjV_i=g_{ij}V^j.
Which quantities of a rank-2 tensor are rotation-invariant?
Trace, determinant, and eigenvalues.
Recall Feynman: explain to a 12-year-old

Imagine a treasure on a map. Where the treasure is never changes — but if you turn the map, the "3 steps east, 4 steps north" instructions change into something else, even though they point to the same spot. A tensor is the rulebook for how those instructions must change when you turn the map. A scalar is the treasure's weight (same no matter how you turn the map). A vector is the arrow to the treasure (instructions change in one simple way). A rank-2 tensor is like a stretchy rubber sheet describing push-and-twist at the spot (instructions change in two linked ways). The magic: the real treasure (the physics) is always the same — only our descriptions rotate.

Concept Map

motivates

defined by

classified by

r=0

r=1

r=2

derives from

acts in

splits into

splits into

uses J

uses inverse J

one Jacobian per index

contracted with

gives

Coordinate independence of physics

Tensor object

Transformation rule

Rank = number of indices

Rank 0 scalar phi

Rank 1 vector

Rank 2 tensor Tij

Jacobian and inverse

Contravariant Vi upper

Covariant Wi lower

Contraction Wi Vi

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, tensor ka asli funda ye hai: physics ka rule axes pe depend nahi karta. Aap apna graph paper kitna bhi ghumao, treasure wahi rahega — bas "3 step east, 4 step north" wala instruction badal jaayega. Tensor isi badalne ke "rule" ko define karta hai. Scalar (rank 0) ek number hai jo har frame mein same — jaise temperature. Vector (rank 1) ek arrow hai, jiske components ek Jacobian factor ke saath transform hote hain. Rank-2 tensor matrix jaisa dikhta hai par har slot apna ek Jacobian factor leke chalta hai, isliye rotation mein rule banta hai T~=R T R⊤\tilde T = R\,T\,R^\top.

Sabse important point: har matrix tensor nahi hota. Tensor banne ke liye uske components ko transformation law follow karna padta hai. Numbers ka random table tensor nahi — kyunki uska koi transformation rule nahi hai. Upper index (contravariant) basis ke ulta scale karta hai, lower index (covariant) basis ke saath. Isi liye jab aap WiViW_i V^i ka contraction lete ho, Jacobian aur uska inverse cancel ho jaate hain, aur answer ek pure scalar nikalta hai jo har frame mein same.

Practical fayda? Stress tensor ko rotate karke aap principal axes dhoondh lete ho jahaan shear zero ho jaata hai — eigenvalues, trace aur determinant kabhi nahi badalte, wahi asli physics hai. Aur ek warning: ordinary derivative ∂kVi\partial_k V^i tensor nahi hota (kyunki derivative position-dependent Jacobian pe bhi lagta hai); uske liye covariant derivative chahiye. Yahi cheez aage General Relativity aur continuum mechanics mein backbone banti hai.

Go deeper — visual, from zero

D1FoundationsEvery symbol from zero · 2,173 wordsD2Visual walkthroughThe derivation in pictures · 2,091 wordsD3Worked examplesEvery scenario covered · 3,190 wordsD4ExercisesGraded, full solutions · 2,430 wordsD5Question bankConcept traps, quick reveals · 1,969 words

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Connections

Metric tensor — raising - lowering indicesMaths · 4.10.10Christoffel symbols — introMaths · 4.10.11Chain rule for multivariable functions — all casesMaths · 4.4.7Einstein summation conventionMaths · 4.10.9