Foundations — Curl — definition, physical meaning (rotation)

4.4.25 · D1 · Maths › Multivariable Calculus › Curl — definition, physical meaning (rotation)

Is page par kuch bhi assume nahi kiya gaya. Agar parent note mein koi symbol likha tha, toh hum use yahaan pehle banayenge. Upar se neeche padho: har idea agli ke liye ek brick hai.

1. Space mein ek point: $(x,y,z)$

Neeche ka figure ek aisi address ko flat $xy$-sheet par dikhata hai: yellow dot $x=3.5$, $y=2.5$ par hai, aur blue dashed lines dikhati hain ki har coordinate apni axis se kaise measure hoti hai. Jab bhi is page mein "ek point" aaye, woh yellow dot imagine karo.

Topic ko yeh kyun chahiye: curl ek local measurement hai — yeh ek point par rehti hai. Pehle "yahaan ki spin" kehne se pehle hume "yahaan" ko naam dena hoga. Woh naam hai $(x,y,z)$.

2. Ek vector: size aur direction wala arrow

Topic ko yeh kyun chahiye: curl jis cheez par kaam karta hai woh ek flow hai, aur ek flow ek point par exactly "ek arrow hai jo bata raha hai paani kidhar aur kitni tez chal raha hai". Woh arrow ek vector hai.

3. Ek vector field: har point par ek arrow

Figure mein field $\vec{F}=(-y,x,0)$ dikhaya gaya hai: blue arrows ka poora grid, har point par ek. Yellow dot ko $(1,0)$ par follow karo — uska arrow seedha upar point karta hai — aur pink dot ko $(0,1)$ par — uska arrow left point karta hai. Origin ke around najar ghumao aur arrows ek doosre ko counterclockwise chase karte hain, woh whirlpool jo neeche describe kiya gaya hai.

Topic ko yeh kyun chahiye: curl arrow yahaan ko arrows bilkul paas mein se compare karta hai. Yeh comparison tabhi kaam karta hai jab har neighbouring point par bhi ek arrow ho — yaani ek field, na ki koi ek vector.

4. Symbol $\partial$ — ek derivative jo doosron ko ignore karta hai

Yeh tool kyun, koi aur kyun nahi? Curl poochh ta hai "kya wheel ke bottom par flow wheel ke top par flow se alag hai?" — ek direction mein difference. Partial derivative exactly wahi tool hai jo measure karta hai "ek direction mein thoda move karne par field kitni alag ho jaati hai". Koi doosra tool is sawaal ka itna cleanly jawab nahi deta.

5. Chote steps $\Delta x,\Delta y,\Delta z$, area $A$, aur limit ${\lim }_{A\to 0}$

Topic ko yeh kyun chahiye: parent ka core formula circulation ko area $A=\Delta x\Delta y$ se divide karta hai phir loop ko shrink karta hai. Bina chote steps $\Delta x,\Delta y,\Delta z$ ke (loop aur uska area banane ke liye) aur $\lim$ ke (use ghataane ke liye) woh sentence meaningless hai.

6. Dot product $\vec{A}\cdot \vec{B}$ — "yeh dono kitne agree karte hain?"

Figure mein teen chote arrow-pairs yeh concrete banate hain: yellow aur blue almost aligned (top-left) → bada positive; right angles par (top-right) → zero; ek doosre ke against point karte hue (bottom) → negative.

Topic ko yeh kyun chahiye: circulation hai $\oint \vec{F}\cdot d\vec{r}$. Paddlewheel loop ke har edge ke along hum poochh te hain "flow $\vec{F}$ ka kitna us taraf point karta hai jis taraf main chal raha hoon $d\vec{r}$?" Woh ek dot product hai. Agar flow aur walk agree karein toh spin mein add hota hai; agar perpendicular ho toh kuch contribute nahi karta.

7. $d\vec{r}$, loop $C$, aur ${\oint }_C$ — ek tiny loop sahi tarike se chalna

Figure mein yeh pairing dikhaya gaya hai: blue arrows loop ke around counterclockwise chalte hain, aur yellow axle seedha board se bahar point karta hai — right hand ke thumb-aur-fingers.

Topic ko yeh kyun chahiye: yeh integral — apni orientation nail down ke saath — limit mein curl ki definition hai: circulation per area, sign right-hand rule se fixed. Yeh parent note ka beating heart hai.

8. Cross product aur $\nabla \times$ — curl ek vector kyun hai

9. Mixed partials ki equality (ek fact jis par hum lean karenge)

Topic ko yeh kyun chahiye: parent ki identity $\nabla \times (\nabla \varphi )=\vec{0}$ poori tarah is fact ki wajah se hai — do mixed partials cancel ho jaate hain. Dekho Equality of Mixed Partial Derivatives (Clairaut) aur Gradient and Conservative Fields.

Pieces topic ko kaise feed karte hain

Ise aise padho: locations arrows banate hain, arrows fields banate hain, fields plus partial derivatives dono circulation (dot product aur loop ke zariye) aur ek axis (cross product ke zariye) banate hain — aur woh dono milke curl ban jaate hain.

Equipment checklist

Daayein side cover karo aur khud test karo. Agar koi bhi jawab fuzzy lage, toh woh section upar se dobara padho.

Connections

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