Curl — definition, physical meaning (rotation)

4.4.25 · Maths › Multivariable Calculus

Curl kya hai? (definition)

Woh field jiske liye $\nabla \times F = 0$ har jagah ho, use irrotational kehte hain.

Hum ise first principles se DERIVE kaise karte hain?

Hum curl ko circulation se derive karte hain, determinant ko memorise nahi karte.

Step 1 — Ek chhota rectangle banao $x y$ -plane mein corner $(x, y)$ , sides $Δ x$ , $Δ y$ ke saath. Counterclockwise chalo.

Ye step kyun? Counterclockwise = positive orientation, isliye ek positive answer matlab hai "left-hand turn ki tarah spins karta hai", yaani $+ \hat{k}$ .

Step 2 — Charon edges par $F \cdot d r$ ko sum karo. Sirf $x$ -motion $F_{1}$ use karta hai; sirf $y$ -motion $F_{2}$ use karta hai.

Bottom edge (left→right), $F \cdot d r \approx F_{1} (x, y) Δ x$
Top edge (right→left), $\approx - F_{1} (x, y + Δ y) Δ x$
Right edge (upar), $\approx F_{2} (x + Δ x, y) Δ y$
Left edge (neeche), $\approx - F_{2} (x, y) Δ y$

Ye step kyun? Ek edge ke saath ulta jaane se $d r$ ka sign flip ho jaata hai, isliye return edges par minus signs aate hain.

Step 3 — Add karo aur factor karo. $\oint F \cdot d r \approx [F_{2} (x + Δ x, y) - F_{2} (x, y)] Δ y - [F_{1} (x, y + Δ y) - F_{1} (x, y)] Δ x$

Step 4 — Partial derivative ki definition use karo ( $F (x + Δ x) - F (x) \approx \partial_{x} F Δ x$ ): $\oint F \cdot d r \approx (\partial_{x} F_{2} - \partial_{y} F_{1}) Δ x Δ y$

Step 5 — Area $Δ x Δ y$ se divide karo aur limit lo: $(\nabla \times F)_{z} = \partial_{x} F_{2} - \partial_{y} F_{1}$

Ye step kyun? "Circulation per unit area" exactly swirl density hai. Isi argument ko $y z$ - aur $z x$ -planes mein repeat karne se baaki do components milte hain. Right-hand rule cyclic pattern $x \to y \to z \to x$ ko fix karta hai.

Figure — Curl — definition, physical meaning (rotation)

Worked Examples

Common Mistakes (Steel-manned)

Key identities (aur WHY)

Recall Feynman: 12-saal ke bachche ko explain karo

Socho ek nadi hai aur tum usme ek chhota sa toy waterwheel daalo. Agar paani ek taraf doosri taraf se zyada force se push kare, toh wheel spin karta hai. Curl bas ek number-aur-arrow hai jo batata hai ki wheel kitni tez spin karta hai aur axle kis taraf point karta hai. Chahe paani seedhi line mein bahta ho, wheel phir bhi spin kar sakta hai agar ek side doosri se faster ho — ye surprising part hai!

Active Recall

Curl physically kya measure karta hai?

Ek vector field ki local rotation (spin); iska direction spin axis hai (right-hand rule), magnitude local angular speed ki do guni hai.

Direction

\overset{n}{^}

mein curl ki coordinate-free definition?

(\nabla \times F) \cdot \overset{n}{^} = lim_{A \to 0} \frac{1}{A} \oint_{C} F \cdot d r

— normal

\overset{n}{^}

wale shrinking loop ki circulation per unit area.

\nabla \times F

z

-component?

\partial_{x} F_{2} - \partial_{y} F_{1}

(- y, x, 0)

ka curl?

(0, 0, 2)

(- y, 0, 0)

mein seedhe arrows ke bawajood nonzero curl kyun hai?

Kyunki paddlewheel ke across speed vary karti hai (shear):

\partial_{y} (- y) = - 1

se curl

1

milta hai. Curl velocity differences detect karta hai, curved arrows nahi.

Irrotational field kya hota hai?

Woh jisme

\nabla \times F = 0

har jagah ho.

\nabla \times (\nabla ϕ) = 0

kyun hai?

Mixed partials equal hote hain, isliye har component jaise

\partial_{x} \partial_{y} ϕ - \partial_{y} \partial_{x} ϕ = 0

\nabla \cdot (\nabla \times F) = 0

kyun hai?

Mixed partial derivatives cancel ho jaate hain; ek pure swirl ka koi source nahi hota.

Curl scalar hai ya vector, aur kyun?

Vector — 3D mein rotation ko ek axis direction chahiye.

Curl vs divergence ek line mein?

Divergence = field kitna BAHAR flow karta hai (scalar); curl = field kitna AROUND swirl karta hai (vector).

Connections

Divergence — definition and physical meaning
Stokes' Theorem
Green's Theorem (2D curl special case)
Gradient and Conservative Fields (curl $= 0$ ⇒ conservative on simply-connected domains)
Line Integrals and Circulation
Equality of Mixed Partial Derivatives (Clairaut)