Curl — definition, physical meaning (rotation): Question bank

4.4.25 · D5 · Maths › Multivariable Calculus › Curl — definition, physical meaning (rotation)

Yeh bank parent note Curl ka conceptual sibling hai. Yahan tum se koi determinant grind nahi karaaya jaata (woh D3/D4 computation banks mein hai) — yahan poochha jaata hai ki machinery jo kehti hai woh kyun kehti hai.

Shuru karne se pehle, teen words earn karne honge taaki neeche ke reveals kabhi surprise na karein:

True ya false — justify karo

True ya false: Agar

F

ke saare arrows ek hi direction mein point karein, toh curl zero hona chahiye.

False — agar wheel ke across speed change ho (shear), toh woh phir bhi ghoomega.

F = (- y, 0, 0)

ke arrows parallel hain phir bhi curl

= \hat{k}

hai, kyunki neeche wala flow upar wale se tez hai.

True ya false: Curl ek single number hai, bilkul divergence ki tarah.

False — divergence ek scalar hai (net outflow), lekin 3D mein rotation ke liye ek axis chahiye, isliye curl ek vector hai. 2D mein hum sirf

\hat{k}

component rakhte hain, jo logon ko lagta hai yeh scalar hai.

True ya false: Ek field ka ek hi point pe zero curl aur nonzero divergence ho sakta hai.

True — yeh dono independent sawaalon ke jawab dete hain. Radial field

F = (x, y, 0)

ka divergence

2

hai (bahar failna) lekin curl

0

hai (wheel kuch nahi twists hota).

True ya false: Ek field ka nonzero curl lekin zero divergence har jagah ho sakta hai.

True — pure rotation

F = (- y, x, 0)

ka curl

2 \hat{k}

hai aur divergence

\partial_{x} (- y) + \partial_{y} (x) = 0

hai. Spread kiye bina swirl.

True ya false: Har gradient field irrotational hoti hai.

True —

\nabla \times (\nabla ϕ) = 0

kyunki har component

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

hai, equality of mixed partials se. Downhill flow swirl nahi kar sakta.

True ya false: Har irrotational field kisi scalar ka gradient hoti hai.

Sirf simply-connected domain pe — ek hole ke saath (jaise

z

-axis ke around wala whirlpool field), curl wahan zero ho sakta hai jahan field defined hai, phir bhi koi global potential exist nahi karta. Dekho Gradient and Conservative Fields.

True ya false: Agar curl exactly ek point pe zero hai, toh wahan rakha paddlewheel nahi ghoomta.

Us point pe True hai, lekin value local hai — ek point ka irrotational hona uske neighbors ke baare mein kuch nahi kehta, aur limiting definition mein wheel ka size zero hota hai.

True ya false: Curl sirf is baat par depend karta hai ki arrows diagram pe kitne curved dikhte hain.

False — curl ek sihrte loop ka circulation per unit area hai, jo velocity differences se set hota hai, visual curvature se nahi. Straight arrows curl kar sakte hain; radial (spreading) straight arrows nahi karte.

True ya false: Poori field ko reverse karne par (

F \to - F

) curl bhi reverse ho jaata hai.

True — curl

F

mein linear hai, isliye

\nabla \times (- F) = - \nabla \times F

. Wheel ulti direction mein ghoomta hai, toh axis flip ho jaata hai.

True ya false: Ek stationary (constant) field

F = (a, b, c)

ka curl zero hota hai.

True — constant ka har partial derivative

0

hota hai, isliye curl ke teeno components vanish ho jaate hain. Uniform flow kuch nahi ghoomata.

Error dhundho

F = (- y, 0, 0)

ke straight arrows hain, isliye iska curl zero hai." — galti kahan hai?

Reasoner ne "straight arrows" ko "no rotation" se equate kar diya. Curl shear

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

padhta hai; wheel isliye ghoomta hai kyunki neeche wali stream upar wali se aage nikal jaati hai.

"Curl ka

z

-component

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

hai." — theek karo.

Yeh galat pairing hai. Yeh

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

hai: agle axis ke component ka derivative pehle, cyclic order

x \to y \to z

mein.

"Curl of a gradient zero hai kyunki gradients hamesha zero vectors hote hain." — theek karo.

Gradient zero nahi hona chahiye; reason yeh hai ki iska curl vanish hota hai, kyunki

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

Clairaut se. Field khud generally nonzero hoti hai.

"Divergence of a curl zero hai, isliye har divergence-free field ek curl hai." — overreach dhundho.

Identity

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

sirf yeh kehti hai ki curls divergence-free hote hain; converse (har divergence-free field ek curl hai) ke liye extra topological conditions chahiye aur woh ek alag theorem hai.

"Curl ki magnitude paane ke liye main paddlewheel ki angular speed leta hoon." — theek karo.

Magnitude local angular speed ki do guna hai.

(- y, x, 0)

ke liye wheel angular speed

1

se ghoomta hai lekin curl magnitude

2

hai.

"Choonki field origin ke baare mein symmetric dikhti hai, isliye iska curl wahan zero hona chahiye." — galti dhundho.

Visual symmetry test nahi hai.

F = (- y, x, 0)

rotationally symmetric hai phir bhi curl

2 \hat{k}

hai — tumhe wheel ke across speeds compare karni hain, na ki symmetry dekhni hai.

"Maine ek bade loop ke liye

\oint_{C} F \cdot d r = 0

compute kiya, isliye curl andar zero hai." — theek karo.

Ek vanishing loop integral se pointwise curl ka zero hona force nahi hota; positive aur negative swirl ek bade region pe cancel ho sakte hain. Curl sihrte loops ki limit hai. (Yahi Stokes' Theorem / Green's Theorem ka content hai.)

Why questions

Curl ko per unit area circulation kyun define karte hain, sirf circulation kyun nahi?

Raw circulation loop size ke saath badhta hai, isliye woh kisi point ki property nahi hai; area se divide karke aur loop ko shrink karke local swirl density isolate hoti hai, jo ek point-sized wheel feel karta hai.

Hum definition

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

mein limit

A \to 0

kyun lete hain?

Ek finite loop poore region mein swirl ko average karta hai; sirf use point tak shrink karne se woh averaging hati hai aur curl exactly us point pe milta hai.

Curl ko ek direction

\overset{n}{^}

kyun chahiye jabki divergence ko nahi?

Rotation ek plane mein hota hai aur ek spin axis hota hai, isliye tumhe batana hoga kaunsa plane —

\overset{n}{^}

woh pick karta hai. Outflow (divergence) ki koi preferred direction nahi hai, isliye scalar kaafi hai. Dekho Divergence — definition and physical meaning.

Counterclockwise loop

+ \hat{k}

curl kyun deta hai,

- \hat{k}

nahi?

Counterclockwise chosen positive orientation hai; right-hand rule se iska normal page se bahar

+ \hat{k}

ki taraf point karta hai, isliye positive circulation

+ \hat{k}

ke roop mein register hoti hai.

F = (- y, x, 0)

ke liye

\partial_{x} (x) - \partial_{y} (- y) = 2

mein dono terms cancel hone ki jagah add kyun hote hain?

Right jaana upward flow ko speed up karta hai aur upar jaana leftward flow ko speed up karta hai — yeh dono usi counterclockwise spin ko do tarafon se dekh rahe hain, isliye unke contributions reinforce karte hain.

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

ko "ek pure swirl ka koi source nahi hota" kyun interpret karte hain?

Ek curl field poori tarah circulation se bani hai, jo apne aap mein loop back karta hai; kisi bhi point se net mein kuch bahar nahi nikalta, isliye iska divergence zero hai.

Ek jaisi arrow pictures wali do fields ke different curls kyun ho sakte hain?

Ek diagram usually ek coarse grid pe arrows sample karta hai aur arrows ke beech speed kaise change hoti hai yeh chhupa deta hai; curl exactly woh hidden gradient padhta hai, isliye equal-looking arrow plots alag shears encode kar sakte hain.

Edge cases

Zero field

F = 0

: iska curl aur divergence kya hain, aur kya yeh irrotational hai?

Curl aur divergence dono

0

0

hain; yeh (trivially) irrotational aur source-free hai — kuch flow nahi karta, kuch nahi ghoomta.

Sirf ek plane pe defined field

F = (F_{1} (x, y), F_{2} (x, y), 0)

: kya iska curl automatically

\hat{k}

ke along hai?

Haan —

\hat{i}

aur

\hat{j}

components ko

z

-dependence ya

F_{3}

chahiye, dono absent hain, isliye sirf

(\partial_{x} F_{2} - \partial_{y} F_{1}) \hat{k}

bachta hai. Yahi Green's Theorem ki 2D setting hai.

Us point pe jahan streamline ke along flow speed maximum ho lekin wheel ke dono sides pe equal ho: wahan curl?

Us plane mein zero — curl ko wheel ke across difference chahiye, flow ke along change nahi. Dono flanks pe symmetric speed balanced torque deti hai.

Whirlpool field

F = \frac{( - y , x , 0 )}{x ^{2} + y ^{2}}

(origin pe undefined):

(x, y) \neq = (0, 0)

ke liye iska curl kya hai?

Har jagah exactly

0

jahan yeh defined hai — yeh axis ke bahar irrotational hai, phir bhi origin ko enclosing kisi bhi loop ke around circulation

2 π

hai. Origin ek singular point hai jise domain se exclude karna hoga.

Jab paddlewheel ek point ki taraf shrink hota hai, toh iska measured circulation zero kyun nahi ho jaata?

Circulation area ki tarah shrink hoti hai (order

Δ x Δ y

), isliye ratio circulation/area ek finite nonzero limit pe pohonchta hai — wahi finite limit curl component hai.

Woh field jo har jagah irrotational aur divergence-free ho: uska kya special naam aur structure hai?

Woh harmonic hai — locally

F = \nabla ϕ

with

\nabla^{2} ϕ = 0

, zero swirl (gradient) aur zero source (Laplace) combine karke. Aisi fields steady, idealized flow model karti hain.

Recall One-line survival kit

Curl ek tiny wheel ke across speed differences padhta hai, ek vector return karta hai (axis = spin direction), angular speed ki do guna hai, gradients ko khatam karta hai, aur divergence se khatam hota hai — inme se kuch bhi arrows dikhne mein kitne curved hain isse judge nahi kar sakte.

Connections

Parent: Curl — definition & physical meaning
Divergence — definition and physical meaning
Stokes' Theorem · Green's Theorem · Line Integrals and Circulation
Gradient and Conservative Fields · Equality of Mixed Partial Derivatives (Clairaut)