Visual walkthrough — Stokes' theorem — statement, curl-circulation connection

4.4.32 · D2 · Maths › Multivariable Calculus › Stokes' theorem — statement, curl-circulation connection

Hume sirf do ideas chahiye jo aap pehle se jaante hain, aur main dono ko phir se build karunga:

• Ek vector field — plane/space ke har point par ek arrow (socho: paani ki dhara ke arrows).
• Ek tiny sa movement — flow aapko kitna push karta hai jab aap ek chhota sa step lete ho.

Baaki sab kuch neeche earn kiya jayega.

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Step 1 — "Path ke saath flow" ka matlab kya hai

KYA HAI. Ek nadi ko upar se dekho. Har point par paani kisi direction mein kisi speed se move karta hai — wahi $\mathbf{F}$ ka arrow hai us point par.

KYU HAI. "Swirl" ya "circulation" ki baat karne se pehle, hume agree karna hoga ki flow aapki kitni madad karta hai jab aap chalte ho. Agar aap ek tiny amount $d\mathbf{r}$ step lete ho (ek chhota arrow jo aapka step dikhata hai), to flow ka helpful part $\mathbf{F}$ ka aapke step par projection hai: $\mathbf{F}\cdot d\mathbf{r}=\underset{\text{flow’s }x}{\underbrace{P}}\underset{\text{step’s }x}{\underbrace{dx}}+\underset{\text{flow’s }y}{\underbrace{Q}}\underset{\text{step’s }y}{\underbrace{dy}}.$ Dot $\cdot$ matching components ko multiply karta hai aur add karta hai. Agar flow aapke step ke saath point karta hai, to yeh positive hai (push); ulta, negative (drag); sideways, zero.

PICTURE. Blue arrows flow hain. Orange step arrow $d\mathbf{r}$ dikhata hai aap kis direction mein chal rahe ho; green shadow flow ka woh part hai jo actually aapki madad karta hai.

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Step 2 — Ek tiny rectangle ke around circulation

KYA HAI. Sabse chhota loop lo jiske baare mein hum exactly soch sakte hain: ek chhota rectangle jiske corners $(x_0,y_0)$ par hain, width $\Delta x$, height $\Delta y$. Ise counter-clockwise chalo: bottom se right, right side se upar, top se left, aur left side se neeche.

KYU HAI. Curved boundary mushkil hoti hai, lekin rectangle ke chaar straight sides hain jahan flow barely change karta hai. Agar hum ek tiny rectangle ke liye circulation calculate kar sakein, to baad mein hum kisi bhi surface ko unse tile kar sakte hain.

PICTURE. Chaar orange arrows ko order mein follow karo. Notice karo ki bottom aur top opposite $x$-directions mein travel hote hain, aur isi tarah dono vertical sides.

Ek chakkar chaar straight pieces mein split hota hai: $\oint \mathbf{F}\cdot d\mathbf{r}=\underset{y=y_0,\to }{\underbrace{\int Pdx{|}_{\text{bottom}}}}+\underset{x=x_0+\Delta x,\uparrow }{\underbrace{\int Qdy{|}_{\text{right}}}}+\underset{y=y_0+\Delta y,\leftarrow }{\underbrace{\int Pdx{|}_{\text{top}}}}+\underset{x=x_0,\downarrow }{\underbrace{\int Qdy{|}_{\text{left}}}}.$ Har label batata hai: kaun sa side, kaun sa coordinate fixed hai, aur arrow kis taraf point karta hai.

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Step 3 — Opposite sides ko pair karne se derivatives aate hain

KYA HAI. Horizontal sides (bottom aur top) aur vertical sides (right aur left) ko pair karo.

KYU HAI. Top leftward chala jaata hai, isliye uska contribution rightward version ka minus hota hai. Bottom aur top same $x$-range pe run karte hain lekin alag heights $y_0$ aur $y_0+\Delta y$ par. Unka difference exactly hai "jab humne $y$ ko raise kiya to $P$ kitna change hua" — aur rate of change wahi hai jo ek partial derivative measure karta hai.

Horizontal pair (bottom minus top, kyunki top backward chalta hai): $\int Pdx{|}_{y_0}+\int Pdx{|}_{y_0+\Delta y}\approx -\underset{\text{how }P\text{ changes with height}}{\underbrace{\frac{\partial P}{\partial y}}}\underset{\text{height}}{\underbrace{\Delta y}}\underset{\text{width}}{\underbrace{\Delta x}}.$

Vertical pair (right minus left): $\int Qdy{|}_{x_0+\Delta x}+\int Qdy{|}_{x_0}\approx +\underset{\text{how }Q\text{ changes across}}{\underbrace{\frac{\partial Q}{\partial x}}}\underset{\text{width}}{\underbrace{\Delta x}}\underset{\text{height}}{\underbrace{\Delta y}}.$

PICTURE. Color-coded arrows do pairs dikhate hain; chhote $\Delta$-labels seedha un sides par baithe hain jo woh measure karte hain.

Dono pairs add karo: ${\oint }_{\text{tiny rect}}\mathbf{F}\cdot d\mathbf{r}\approx \left(\underset{\text{one number: the swirl}}{\underbrace{\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}}}\right)\underset{\text{area }\Delta A}{\underbrace{\Delta x\Delta y}}.$

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Step 4 — Swirl ko naam dena: curl

KYA HAI. Humne abhi prove kiya: (tiny circulation) $\approx$ (curl) $\times$ (area).

KYU HAI. Yahi poora engine hai. Yeh ek loop quantity (circulation) ko ek area quantity (curl × area) mein convert karta hai. Loops ko hum chain kar sakte hain; areas ko hum add kar sakte hain — Step 6 isi ko exploit karta hai.

PICTURE. Do chhote paddle-wheels. Jahan $\partial Q/\partial x$ aur $-\partial P/\partial y$ sign mein agree karte hain, wheel fast spin karta hai (bada curl); jahan woh fight karte hain, woh barely muda karta hai.

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Step 5 — 3D mein jaana: curl ek vector ban jaata hai

KYA HAI. Space mein $\mathbf{F}=(P,Q,R)$ aur ek tiny loop kisi bhi plane mein ho sakta hai, ek unit normal $\hat{\mathbf{n}}$ se tila hua (ek length 1 ka arrow jo loop se seedha bahar nikalta hai).

KYU HAI. Flat leta hua loop sirf vertical axis ke baare mein swirl pakadta hai; tilted loop apni axis ke baare mein swirl pakadta hai. Isliye swirl ko ek direction chahiye — yeh ek vector hai, number nahi. Tiny-circulation law ban jaata hai: ${\oint }_{\text{tiny loop}}\mathbf{F}\cdot d\mathbf{r}\approx \underset{\text{swirl vector}}{\underbrace{(\nabla \times \mathbf{F})}}\cdot \underset{\text{loop ki axis}}{\underbrace{\hat{\mathbf{n}}}}\underset{\text{loop ka area}}{\underbrace{\Delta A}}.$ Dot loop ki axis ke saath aligned swirl ka part pick karta hai — exactly woh part jo circulation feel karta hai.

Right-hand rule kyu? "Counter-clockwise" tab hi kuch matlab rakhta hai jab aap batao kis side se dekh rahe ho. Apna right thumb $\hat{\mathbf{n}}$ ki taraf point karo; aapki curled fingers positive lap direction dikhate hain. $\hat{\mathbf{n}}$ flip karo aur lap direction bhi flip ho jaati hai — yahi wajah hai ki dono sides ka sign hamesha matched rehta hai. (Dekho Orientation & the Right-Hand Rule.)

PICTURE. Same physical spin, opposite normals se dekha gaya: ek $\hat{\mathbf{n}}$ se CCW milta hai, reversed $\hat{\mathbf{n}}$ se CW milta hai.

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Step 6 — Tiling: andar ke edges cancel hote hain, sirf rim bachta hai

KYA HAI. Surface $S$ ko tiny loops ki ek fine grid se cover karo, sab ek hi tarah chale jaate hain (CCW jaise $\hat{\mathbf{n}}$ se dekha jaaye). Har tiny circulation ko add karo.

KYU magic kaam karta hai. Koi bhi inner edge do neighbouring tiles se share hoti hai. Ek tile ise left-to-right chalti hai; padosi same edge ko right-to-left chalta hai. Same flow, opposite direction → dono contributions exact negatives hain aur cancel ho jaate hain. Outer boundary $C$ par ek edge ka koi padosi nahi hota, isliye kuch cancel nahi hota. Tiny loops ka sum = $C$ ke around ek chakkar.

PICTURE. Do tiles par zoom karo: shared middle edge mein do opposing orange arrows hain (woh annihilate ho jaate hain); outer edges mein ek surviving arrow hai.

$\underset{\text{sirf rim bacha}}{\underbrace{{\oint }_C\mathbf{F}\cdot d\mathbf{r}}}={\sum }_{\text{tiles}}{\oint }_{\text{tile}}\mathbf{F}\cdot d\mathbf{r}\approx {\sum }_{\text{tiles}}(\nabla \times \mathbf{F})\cdot \hat{\mathbf{n}}\Delta A.$

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Step 7 — Limit: sum surface integral ban jaata hai

KYA HAI. Har tile ko shrink karo ($\Delta A\to 0$). Taylor step ka "$\approx$" exact ho jaata hai, aur sum integral ban jaata hai.

Limit legal kyu hai. Har tile mein error higher-order hai (proportional to $\Delta A^{3/2}$ ya chhoti). Lagbhag $1/\Delta A$ tiles hain, isliye total error $\sim \Delta A^{1/2}\to 0$. Leading terms bachte hain; errors vanish ho jaate hain.

$\boxed{{\oint }_C\mathbf{F}\cdot d\mathbf{r}={\iint }_S(\nabla \times \mathbf{F})\cdot d\mathbf{S}}$

Left se right padhte hain: rim ke around circulation = skin ke through total curl-flux.

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Step 8 — Edge aur degenerate cases (reader ko kabhi girne mat do)

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Ek-picture summary

Is poore page ki ek hi frame mein: skin ko tiny swirls (curl × area) mein kaato, dekho interior arrows cancel ho jaate hain, aur sirf rim-lap (circulation) bachta hai.

Recall Feynman: plain words mein bolo

Ek wire loop ko paani ki ek shallow, swirling tray mein daalo. Main jaanna chahta hun ki paani loop ke edge ke around kitni tez daudta hai. Poori edge follow karne ki bajaye, main inside ko tiny squares ki grid mein kaatta hun aur measure karta hun ki har chhote square ke around paani kitna spin karta hai — wahi "curl" hai. Jab do squares touch karte hain, ek unki shared wall ke saath clockwise paani dhakelta hai aur doosra counter-clockwise: perfect cancellation. Har inside wall apne padosi se cancel ho jaati hai. Sirf outside walls — loop ka actual rim — ke paas koi nahi hota unhe cancel karne ke liye. Isliye andar ke saare tiny spins add karne se mujhe exactly rim ke around flow milta hai. Woh sentence, exact banaya gaya, Stokes' theorem hai: skin ke through curl-flux rim ke around circulation ke barabar hai.

Recall

Tiny-loop law (2D swirl) ::: ${\oint }_{\text{tiny}}\mathbf{F}\cdot d\mathbf{r}\approx ({\partial }_{x}Q-{\partial }_{y}P)\Delta A$ Why do interior edges vanish when tiling? ::: Har interior edge do neighbours ke zariye ek baar har direction mein chali jaati hai, isliye dono contributions exactly cancel ho jaate hain. Which tool measures "how $P$ changes as $y$ moves"? ::: Partial derivative $\partial P/\partial y$. What does $\hat{\mathbf{n}}$ do in the 3D tiny-loop law? ::: Yeh dot product ke zariye curl vector ka woh part select karta hai jo loop ki axis ke saath aligned hai. Flat horizontal $S$ reduces Stokes to which theorem? ::: Green's Theorem.