4.4.31
Ek line integral ek curve (1D) ke saath quantity ko add karta hai. Ek surface integral 3D space mein rehne wali ek surface (2D) ke upar quantity ko add karta hai.
Scalar surface integral ∬ S f d S f density (mass per area) hai, toh uska total mass kya hoga?"Vector surface integral / flux ∬ S F ⋅ d S F andar se per second guzarta hai?"
Poora trick yeh hai: ek curved surface mushkil hai, lekin ek flat parameter rectangle easy hai. Hum ek flat ( u , v ) stretch karta hai.
Definition Parametrized surfaceEk surface ek vector map hai jo uv D r ( u , v ) = ( x ( u , v ) , y ( u , v ) , z ( u , v ) ) , ( u , v ) ∈ D . v u u
Stretch factor kyun chahiye? Jab hum parameters ko tiny amounts d u d v
r u = ∂ u ∂ r , r v = ∂ v ∂ r .
Chhota sa parameter square d u d v parallelogram mein map ho jaata hai jo r u d u r v d v
Intuition Parallelogram ka area = cross product ka magnitudeVectors a b ∥ a × b ∥ ∥ a × b ∥ = ∥ a ∥∥ b ∥ sin θ
Toh real surface area ka tiny patch hai
d S = ∥ r u × r v ∥ d u d v .
Yeh ∥ r u × r v ∥ area scaling factor
Agar f = 1 surface area ∬ S d S
Agar f = total mass milta hai.
Worked example Graph surfacez = g ( x , y ) u = x v = y r ( x , y ) = ( x , y , g ( x , y )) r x = ( 1 , 0 , g x ) , r y = ( 0 , 1 , g y ) . Yeh step kyun? Yeh x y r x × r y = ( − g x , − g y , 1 ) ⇒ ∥ r x × r y ∥ = 1 + g x 2 + g y 2 . Yeh step kyun? Cross product normal deta hai; uski length stretch factor hai.
∬ S f d S = ∬ D f ( x , y , g ) 1 + g x 2 + g y 2 d x d y .
a x 2 + y 2 + z 2 = a 2 z = a 2 − x 2 − y 2 D : x 2 + y 2 ≤ a 2 g x = a 2 − x 2 − y 2 − x g y = a 2 − x 2 − y 2 − y Kyun? g 1 + g x 2 + g y 2 = 1 + a 2 − x 2 − y 2 x 2 + y 2 = a 2 − x 2 − y 2 a 2 . d S = a 2 − x 2 − y 2 a d x d y x 2 + y 2 = ρ 2
=2\pi a\Big[-\sqrt{a^2-\rho^2}\Big]_0^a = 2\pi a^2.$$
*Polar kyun?* Disk aur integrand radially symmetric hain → clean integral. Result $2\pi a^2$ = $4\pi a^2$ ka aadha. ✔
Definition Unit normal aur orientationEk orientable surface mein unit normal ka continuous choice hota hai
n ^ = ∥ r u × r v ∥ r u × r v . n ^ − n ^ orient karta hai: yeh "positive" side pick karta hai. (Möbius strip non-orientable hai — koi consistent choice exist nahi karti.)
Intuition Flux kya measure karta haiSocho fluid velocity field F d S n ^ F n ^ andar se carry karta hai. Sideways flow saath mein slide karta hai, andar se nahi.
Toh patch se flow rate hai ( F ⋅ n ^ ) d S
Intuition Magnitude cancellation itni beautiful kyun haiScalar integrals mein tumhe square root ∥ r u × r v ∥ kabhi woh square root nahi lete — bas F r u × r v easier hota hai!
z = g ( x , y ) r x × r y = ( − g x , − g y , 1 ) upward point karta hai kyunki iska z + 1 > 0 F = ( P , Q , R ) ∬ S F ⋅ d S = ∬ D ( − P g x − Q g y + R ) d x d y .
F = ( x , y , z ) r ( ϕ , θ ) = ( sin ϕ cos θ , sin ϕ sin θ , cos ϕ ) ϕ ∈ [ 0 , π ] θ ∈ [ 0 , 2 π ] Yeh step kyun? Unit sphere pe r F = r r ϕ × r θ = sin ϕ r sin ϕ F ⋅ ( r ϕ × r θ ) = r ⋅ ( sin ϕ r ) = sin ϕ ∥ r ∥ 2 = sin ϕ ∥ r ∥ = 1 ∬ S F ⋅ d S = ∫ 0 2 π ∫ 0 π sin ϕ d ϕ d θ = 2 π ⋅ 2 = 4 π . Divergence Theorem se check karo: ∇ ⋅ F = 3 = 3 4 π = 3 ⋅ 3 4 π = 4 π
Common mistake "Flux ko bhi square root∥ r u × r v ∥ Kyun sahi lagta hai: d S definition ∬ ( F ⋅ n ^ ) d S d S Fix: n ^ d S = ∥ ⋯ ∥ r u × r v ⋅ ∥ ⋯ ∥ d u d v = ( r u × r v ) d u d v F ⋅ ( r u × r v ) no root .
Common mistake Yeh bhool jaana ki flux ka ek SIGN hota hai (orientation).Kyun sahi lagta hai: Scalar integrals ∬ f d S ≥ 0 f ≥ 0 Fix: Flux negative ho sakta hai. r u × r v r v × r u
F ⋅ ( r u × r v ) F ⋅ r Kyun sahi lagta hai: Sphere pe r hai normal direction, toh wahan accidentally kaam kar jaata hai.
Fix: Vector area element tangents ka cross product hai, position vector nahi. Sirf luck se sphere pe yeh align hote hain.
Common mistake Parametrization substitute kiye binaf ( x , y , z ) Kyun sahi lagta hai: Tum f x , y , z Fix: Integrate karne se pehle sab kuch u , v x → x ( u , v )
Recall Active recall — answers cover karo
Parametrization ke terms mein d S ∥ r u × r v ∥ d u d v
Flux scalar integral se zyada easy kyun hota hai? → Norm cancel ho jaata hai; koi square root nahi.
Flux physically kya measure karta hai? → Field ka net flow surface ke andar se .
Agar orientation reverse karo toh kya badalta hai? → Flux sign flip ho jaata hai; scalar integral unchanged rehta hai.
Recall Feynman: 12 saal ke bacche ko explain karoSocho ek bumpy net ek waterfall ke neeche pakdi hui hai. Scalar integral: Agar main poori net paint karna chahta hoon, toh mujhe kitna paint chahiye? Main har chhote square ka area add karta hoon — lekin tilted squares upar se dekhne par bade lagte hain, toh main unhe pehle stretch karta hoon (woh stretch hai ∥ r u × r v ∥ Flux: Agar main jaanna chahta hoon ki kitna paani net ke andar se guzarta hai, toh main sirf woh paani count karta hoon jo seedha holes se guzarta hai, net ke saath slide hone wala nahi. Net se nikla hua arrow (n ^
Mnemonic Dono formulas yaad rakho"Scalar Stretches, Flux Flows."
S calar → S tretch factor ∥ r u × r v ∥ F lux → F ield cross product mein dot ho jaata hai, norm F all away kar jaata hai.
Parametrized surface r ( u , v ) d S d S = ∥ r u × r v ∥ d u d v
Graph z = g ( x , y ) ∥ r x × r y ∥ Scalar surface integral formula state karo. ∬ S f d S = ∬ D f ( r ) ∥ r u × r v ∥ d u d v
Parameters ke terms mein flux integral formula state karo. ∬ S F ⋅ d S = ∬ D F ( r ) ⋅ ( r u × r v ) d u d v
Flux us square root se kyun bachta hai jo scalar integrals ko chahiye? n ^ d S = ∥ ⋯ ∥ r u × r v ∥ ⋯ ∥ d u d v = ( r u × r v ) d u d v
Flux integral kaun si physical quantity compute karta hai? Field F
Agar surface ki orientation reverse karo toh flux ka kya hota hai? Uska sign flip ho jaata hai (normal reverse ho jaata hai)
Vector area element d S d S = n ^ d S = ( r u × r v ) d u d v
Unit sphere se outward F = ( x , y , z ) 4 π 3 ⋅ 3 4 π
Radius a 4 π a 2 2 π a 2
Koi surface non-orientable kyun hoti hai (ek example do)? Unit normal ka koi continuous choice nahi; jaise Möbius strip
Upward normal wale graph ke liye ( P , Q , R ) ∬ D ( − P g x − Q g y + R ) d x d y
magnitude = parallelogram area
Parametrized surface r of u,v
Tangent vectors r_u and r_v
Graph formula sqrt 1+gx^2+gy^2
Line integral over curve 1D
