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4.4.25Multivariable Calculus

Curl — definition, physical meaning (rotation)

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🇮🇳 Yeh note Hinglish mein padho →

WHY does curl exist?

WHAT is curl? (definition)

A field with ∇×F⃗=0⃗\nabla\times\vec F = \vec 0 everywhere is called irrotational.

HOW do we DERIVE it from first principles?

We derive curl from circulation, not memorise the determinant.

Step 1 — Build a tiny rectangle in the xyxy-plane with corner (x,y)(x,y), sides Δx\Delta x, Δy\Delta y. Walk counterclockwise.

Why this step? Counterclockwise = positive orientation, so a positive answer means "spins like a left-hand turn", i.e. +k^+\hat k.

Step 2 — Sum F⃗⋅dr⃗\vec F\cdot d\vec r on the four edges. Only the xx-motion uses F1F_1; only the yy-motion uses F2F_2.

  • Bottom edge (left→right), F⃗⋅dr⃗≈F1(x,y) Δx\vec F\cdot d\vec r \approx F_1(x,y)\,\Delta x
  • Top edge (right→left), ≈−F1(x,y+Δy) Δx\approx -F_1(x,y+\Delta y)\,\Delta x
  • Right edge (up), ≈F2(x+Δx,y) Δy\approx F_2(x+\Delta x,y)\,\Delta y
  • Left edge (down), ≈−F2(x,y) Δy\approx -F_2(x,y)\,\Delta y

Why this step? Going backward along an edge flips the sign of dr⃗d\vec r, hence the minus signs on the return edges.

Step 3 — Add and factor. ∮F⃗⋅dr⃗≈[F2(x+Δx,y)−F2(x,y)]Δy−[F1(x,y+Δy)−F1(x,y)]Δx\oint \vec F\cdot d\vec r \approx \big[F_2(x+\Delta x,y)-F_2(x,y)\big]\Delta y - \big[F_1(x,y+\Delta y)-F_1(x,y)\big]\Delta x

Step 4 — Use the definition of partial derivative (F(x+Δx)−F(x)≈∂xF ΔxF(x+\Delta x)-F(x)\approx \partial_x F\,\Delta x): ∮F⃗⋅dr⃗≈(∂xF2−∂yF1)Δx Δy\oint \vec F\cdot d\vec r \approx \left(\partial_x F_2 - \partial_y F_1\right)\Delta x\,\Delta y

Step 5 — Divide by the area Δx Δy\Delta x\,\Delta y and take the limit: (∇×F⃗)z=∂xF2−∂yF1(\nabla\times\vec F)_z = \partial_x F_2 - \partial_y F_1

Why this step? "Circulation per unit area" is exactly the swirl density. Repeating the same argument in the yzyz- and zxzx-planes gives the other two components. The right-hand rule fixes the cyclic pattern x→y→z→xx\to y\to z\to x.

Figure — Curl — definition, physical meaning (rotation)

Worked Examples

Common Mistakes (Steel-manned)

Key identities (and WHY)

Recall Feynman: explain to a 12-year-old

Imagine a river and you toss in a tiny toy waterwheel. If the water on one side pushes harder than the other, the wheel spins. Curl is just a number-and-arrow that says how fast the wheel spins and which way the axle points. Even if the water flows in a straight line, the wheel can still spin if one side is faster than the other — that's the surprising part!

Active Recall

What does curl measure physically?
The local rotation (spin) of a vector field; its direction is the spin axis (right-hand rule), magnitude is twice the local angular speed.
Coordinate-free definition of curl in direction n^\hat n?
(∇×F⃗)⋅n^=lim⁡A→01A∮CF⃗⋅dr⃗(\nabla\times\vec F)\cdot\hat n=\lim_{A\to0}\frac1A\oint_C\vec F\cdot d\vec r — circulation per unit area of a shrinking loop with normal n^\hat n.
zz-component of ∇×F⃗\nabla\times\vec F?
∂xF2−∂yF1\partial_x F_2-\partial_y F_1.
Curl of (−y,x,0)(-y,x,0)?
(0,0,2)(0,0,2).
Why does (−y,0,0)(-y,0,0) have nonzero curl despite straight arrows?
Because speed varies across the paddlewheel (shear): ∂y(−y)=−1\partial_y(-y)=-1 gives curl 11. Curl detects velocity differences, not curved arrows.
What is an irrotational field?
One with ∇×F⃗=0⃗\nabla\times\vec F=\vec 0 everywhere.
Why is ∇×(∇ϕ)=0⃗\nabla\times(\nabla\phi)=\vec0?
Mixed partials are equal, so each component like ∂x∂yϕ−∂y∂xϕ=0\partial_x\partial_y\phi-\partial_y\partial_x\phi=0.
Why is ∇⋅(∇×F⃗)=0\nabla\cdot(\nabla\times\vec F)=0?
Cancelling mixed partial derivatives; a pure swirl has no source.
Is curl a scalar or vector, and why?
A vector — rotation in 3D needs an axis direction.
Curl vs divergence in one line?
Divergence = how much field flows OUT (scalar); curl = how much field swirls AROUND (vector).

Connections

Concept Map

spreads/converges

circulates/rotates

visualizes

per unit area gives

computed via

expands to

shrink loop derives

axis set by

equals zero means

Vector field F

Divergence scalar

Curl vector

Circulation loop integral

Paddlewheel spin

Determinant nabla cross F

Component formulas

Irrotational field

Right-hand rule

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Curl ka simple matlab hai: vector field kitna "ghoom" raha hai ek point ke around. Socho tum ek chhota paddlewheel (paani ka pankha) flow me daal do — agar wo ghoomne lage, toh curl non-zero hai. Curl ek vector hai: uski direction wo axis batati hai jiske around spin ho raha hai (right-hand rule se), aur magnitude batati hai kitni tezi se spin ho raha hai. Yaad rakho: divergence batata hai field bahar ki taraf phail raha hai ya nahi (scalar), aur curl batata hai field swirl kar raha hai ya nahi (vector) — dono alag cheezein hain.

Formula ratne ki zaroorat nahi — hum ise circulation se derive karte hain. Ek tiny rectangle ke charo taraf ∮F⃗⋅dr⃗\oint \vec F\cdot d\vec r nikaalo, area se divide karo, limit lo — bas wahi hai (∇×F)z=∂xF2−∂yF1(\nabla\times F)_z = \partial_x F_2 - \partial_y F_1. Yeh hi Stokes theorem ka chhota roop hai: curl = circulation per unit area.

Sabse important trick wala point: agar arrows bilkul straight ho tab bhi curl ho sakta hai! Jaise F=(−y,0)F=(-y,0) — saare arrows sirf x-direction me hain, lekin neeche flow tez aur upar slow hai, isliye paddlewheel torque khaata hai aur ghoomta hai, curl =1=1. Toh curl arrows ke curve dekhne se nahi, balki flow ki speed ka difference (shear) se aata hai. Ye baat exam me trap karti hai, isliye yaad rakhna.

Do mast identities: ∇×(∇ϕ)=0\nabla\times(\nabla\phi)=0 (gradient field kabhi swirl nahi karta) aur ∇⋅(∇×F)=0\nabla\cdot(\nabla\times F)=0 (pure swirl ka koi source nahi hota). Dono mixed partial derivatives barabar hone ki wajah se aate hain. In do facts ka 80/20 value bahut high hai — interview aur exam dono me kaam aate hain.

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