4.4.32 · D1Stokes' theorem — statement, curl-circulation connection
Intuition Ek core idea (ise do baar padho)Stokes' theorem ek hi sentence kehta hai: kisi surface mein packed tiny spinning ki total amount, us surface ke edge ke around ghoomne wale flow ke barabar hoti hai . Neeche sab kuch — arrows, dots, curl, double integral — sirf wo vocabulary hai jo hume us ek sentence ko precisely likhne ke liye chahiye.
Pehle se ∮ C F ⋅ d r = ∬ S ( ∇ × F ) ⋅ d S har ek mark padhna aana chahiye. Yeh note unhe ek-ek karke earn karta hai, ek aisi order mein jahan har cheez sirf usse pehle wali cheez par depend karti hai.
R 3 R 3 ( x , y , z ) Picture: wo room jisme tum baithe ho — ek corner ko origin ( 0 , 0 , 0 ) x y z Yeh topic kyun chahiye: Stokes 3D mein rehta hai. Iska chhota bhai Green's Theorem flat plane R 2
Ek single point ek aisa triple hai. Position ka function har ek point par ek number (ya ek arrow) assign karta hai. Doosra idea — har point par ek arrow — yahi is show ka star hai.
Ek vector ek arrow hai: iske paas ek length (kitna) aur ek direction (kidhar) hoti hai. Hum ise bold mein likhte hain, jaise F F = ( P , Q , R )
P x Q y R z
field F vector field ek rule hai jo space ke har ek point par ek arrow lagata hai. Picture: weather map par hawa — har jagah ek chhota arrow dikhata hai ki hawa kis taraf aur kitni tez chal rahi hai.
Kyunki arrow jagah-jagah badalta hai, iske components position ke functions hain:
F ( x , y , z ) = ( P ( x , y , z ) , Q ( x , y , z ) , R ( x , y , z ) ) .
Intuition Hume ek field kyun chahiye, ek arrow nahiStokes flow aur swirl ke baare mein hai, aur dono tab hi samajh aate hain jab arrow move karne par change ho. Field woh paani hai tray mein; swirl tab aata hai jab paas-paas ke arrows aapas mein agree nahi karte. Field ko measure karne ke do tarike dekhne ke liye Line Integrals of Vector Fields aur Surface Integrals & Flux dekho.
Do arrows a = ( a 1 , a 2 , a 3 ) b = ( b 1 , b 2 , b 3 ) a ⋅ b = a 1 b 1 + a 2 b 2 + a 3 b 3 = ∣ a ∣ ∣ b ∣ cos θ , ∣ a ∣ a θ
cos 9 0 ∘ = 0 across chalti hai, toh woh tumhari along mein progress ko na help karti hai na hinder. Woh "zero" hi wajah hai kyun sirf F
C ∂ S curve space mein ek rasta hai, jaise ek muda hua taar. Symbol ∂ S S S edge — ek drum ka rim, ek soap film ke aas-paas ka taar. Hum us edge ko aksar C ∂ S = C
d r C d r aage step karte ho. Yeh hamesha curve ke tangent (saath-saath) rehta hai.
Picture: path par pair ke nishan — har chhota step ek d r
∮ ∫ ∮ closed loop hai — tum wahan khatam karte ho jahan se shuru kiya tha. Toh ∮ C C
Inhe milane par, ∮ C F ⋅ d r apne step ke saath kitna push karti hai." Us total ko circulation kehte hain. (Deep dive: Line Integrals of Vector Fields .)
Ek loop ko do taraf se walk kiya ja sakta hai (clockwise / counter-clockwise), aur ek surface ke do sides hain (kaunsa "out" hai). Orientation har ek ke liye ek consistent direction choose karna hai, is tarah linked ki dono agree karein.
Intuition Right-hand rule (kyun sign free nahi hai)Apne right hand ka thumb surface ki chosen "up" direction n ^ curled fingers phir woh direction point karein jis taraf tumhe edge C ek maatra wajah hai kyun Stokes ke dono sides same sign carry karte hain. (Dekho Orientation & the Right-Hand Rule .)
Common mistake "Main sign baad mein fix kar lunga."Magnitudes hamesha match karte hain, isliye ek stray sign aasani se aa sakta hai aur notice karna mushkil hota hai. Orientation ko compute karne se pehle lock karo, right hand ka use karke.
n ^ unit normal length 1 ^ surface se seedha bahar point karta hai, uske perpendicular. Yeh batata hai kaunsi side "up" hai.
Definition Surface, patch, aurd S surface S d A Oriented area element hai
d S = n ^ d A , length patch ka area hai aur jिसकी direction outward normal hai . Yeh kitna bada aur kis taraf face karta hai dono ko ek object mein pack karta hai.
d S through jaati hai, tumhe jaanna hoga ki patch kitni tilted hai. Flow ke edge-on patch mein kuch nahi pakda jaata; face-on mein sabse zyada pakda jaata hai — bilkul dot product ka cos θ ( swirl ) ⋅ d S through pokta hai. Yeh measurement flux hai; dekho Surface Integrals & Flux .
∬ S aur width hoti hai, isliye sweep karne ke liye do directions). ∬ S ( ⋯ ) d A
Definition Partial derivative∂ x ∂ P partial derivative ∂ x P ∂ P / ∂ x P x y z Picture: ek pahaadi par khade ho, ∂ x Topic ko yeh kyun chahiye: swirl paas-paas ke arrows ke beech differences se banta hai, aur derivative exactly woh machine hai jo aisa infinitesimal difference measure karti hai.
Curly ∂ d
∇ ∇ bundle hai: ∇ = ( ∂ x , ∂ y , ∂ z )
∇ × F Curl ek vector hai jo har point par field की spinning measure karta hai — ek chhota paddle-wheel wahan dala jaaye toh kitna rotate karega, aur kis axis ke around. Iska direction spin axis hai (right-hand rule se); iska length spin rate hai.
∇ × F = ( ∂ y R − ∂ z Q , ∂ z P − ∂ x R , ∂ x Q − ∂ y P ) .
× differences kyun
Har component do slopes ka difference hai (jaise ∂ x Q − ∂ y P lopsided turning (ek saccha swirl) bachta hai. Yahi quantity "circulation per unit area" kehlaati hai. (Poori detail: Curl of a Vector Field .)
Common mistake "Curl ek number hai."Flat 2D (Green's Theorem ) mein spin sirf ∂ x Q − ∂ y P kisi bhi axis ke around spin kar sakti hai, isliye curl ek vector hona chahiye jo woh axis carry kare. Stokes, Green ke scalar ko k ^
Ab har mark earn ho gaya hai. Stokes' theorem ko English ki tarah padho:
loop ke around ∮ C har step ke saath push F ⋅ d r = sheet ke upar ∬ S local spin ( ∇ × F ) ⋅ har patch ke through d S
Intuition Pehle se har case cover karo
Curl S → right side 0 0 Conservative Fields & Potential Functions ).Plane mein flat surface, n ^ = k ^ → sirf curl ka z Green's Theorem mein.Do surfaces jo ek edge share karte hain → same C
Dot product agreement meter
Curve C and boundary of S
Orientation right hand rule
Circulation line integral
Khud test karo — har line ka right side cover karo.
Vector field kya hai, ek picture mein? Space ke har point par ek arrow lagaya hua, jaise map par hawa.
Dot product a ⋅ b Do arrows direction mein kitna agree karte hain; yeh ∣ a ∣∣ b ∣ cos θ
a ⋅ b a ⊥ b Zero, kyunki cos 9 0 ∘ = 0
∮ Integration path ek closed loop hai — tum wahan khatam karte ho jahan se shuru kiya tha.
d r Curve ke saath chalte waqt chhota tangent step-arrow.
∂ S = C C S
Right-hand rule kya fix karta hai? Thumb n ^ C
n ^ Outward unit normal; hat ka matlab iska length exactly 1
d S = n ^ d A Ek patch ka area (uski length) aur woh kis taraf face karta hai (uski direction).
Partial derivative ∂ x P P x y , z
Curl scalar hai ya vector, aur iska kya matlab hai? Ek vector; yeh field ki local spinning ki axis aur rate deta hai.
Words mein, Stokes' theorem kya equate karta hai? Edge C S
