Surface integrals — scalar and vector (flux): Foundations

4.4.31 · D1 · Maths › Multivariable Calculus › Surface integrals — scalar and vector (flux)

Parent page padhne se pehle, tumhe uske har symbol ko khud se jaanna hoga. Neeche, har ek symbol ko milta hai: plain words → picture → topic ko yeh kyun chahiye, ek aisi order mein jahan har cheez strictly pichli pe build hoti hai — koi bhi cheez build hone se pehle use nahi hoti.

1. Points, coordinates, aur 3D space

Hume yeh isliye chahiye kyunki ek surface aisi points ki poori ek sheet hoti hai, aur jo kuch bhi hum compute karte hain woh un points pe rehta hai.

2. Vectors aur unki length $∥ \cdot ∥$

Topic ko yeh kyun chahiye: baad mein, surface ke ek tiny patch ki size ek khaas arrow ki length niklegi. Length woh tool hai jo "yeh arrow kitna bada hai?" ko ek single number mein convert karta hai, isliye patches measure karne se pehle yeh hamare paas hona chahiye.

3. Ek variable ke functions aur derivative

4. Partial derivatives $\frac{\partial}{\partial u}$ aur parameters $u, v$

Curly $\partial$ (straight $d$ ke muqable) ek reminder hai: "doosre variables hain jinhe main rok raha hun."

Topic ko yeh kyun chahiye: partial derivatives woh tool hain jo "ek dial nudge karo" ko surface pe ek actual arrow mein convert karte hain — woh tangent vectors jo hum agley section mein banate hain. Flat-region cousins ke liye Double integrals aur Jacobian and change of variables dekho.

5. Parametrization $r (u, v)$ aur uske tangent vectors

$v$ ko fix rakh ke $u$ slide karna surface pe ek curve paint karta hai (ek "grid line"); grid lines ki do families surface ko ek warped checkerboard jaisa banati hain.

Definition Regularity: do tangents parallel nahi hone chahiye

Patch ki real, nonzero area hone ke liye, $r_{u}$ aur $r_{v}$ linearly independent hone chahiye — matlab unhe genuinely alag directions mein point karna chahiye, koi ek doosre ka stretched copy nahi. Picture: agar do arrows align ho jaayein (parallel), toh woh jo tiny tile span karte hain woh ek line mein squash ho jaata — zero area, ek degenerate patch. Ek parametrization ek point pe regular tab kahlaata hai jab yeh nahi hota. Kyun important hai: yeh surface version hai "Jacobian $\neq = 0$ " ka Jacobian and change of variables se. Sirf regular points pe hum well-defined patch area aur normal direction bana sakte hain. Woh points jahan tangents collapse ho jaate hain (jaise ek sphere parametrization ke poles) unhe special cases ki tarah handle karna padta hai.

Topic ko yeh kyun chahiye: yeh map hi central trick hai. Har mushkil cheez (curved surface) ko har aasaan cheez (flat region $D$ jiske upar tum numbers slide kar sako) se replace kiya jaata hai — jab tak hum regular points pe rahein.

6. Cross product $\times$

Topic ko yeh kyun chahiye — ek saath do kaam:

Iski length $∥ r_{u} \times r_{v} ∥$ = tiny surface patch ki area = stretch factor (likha $d S$ , section 9 mein define kiya).
Iski direction (right-hand rule se fix) = surface kis taraf face karti hai = normal $\hat{n}$ jo flux ke liye use hota hai.

Ek tool dono sawaalon ka jawab deta hai "kitna bada?" aur "kaun si taraf?"

7. Dot product $\cdot$

Topic ko yeh kyun chahiye: $F \cdot \hat{n}$ aur $F \cdot (r_{u} \times r_{v})$ flux integral ka dil hain.

8. Unit normal $\hat{n}$

Exactly do choices hoti hain (pole upar, ya pole neeche $= - \hat{n}$ ). Section 6 se right-hand rule decide karta hai ki $r_{u} \times r_{v}$ kaun sa deta hai; ek side choose karna orienting the surface kahlata hai. Isliye flux ek sign carry karta hai. (Yeh division sirf regular points pe sense deta hai — jahaan $∥ r_{u} \times r_{v} ∥ \neq = 0$ ho, taaki hum zero se divide na kar rahein.)

(x, y, z)

ka kya matlab hai?

Teen instructions jo 3D space mein ek point locate karti hain (

x

y

z

axes ke saath steps).

∥ a ∥

kya compute karta hai, aur kis formula se?

Arrow

a

ki length;

a_{1}^{2} + a_{2}^{2} + a_{3}^{2}

Derivative kaun sa sawal answer karta hai?

"Agar main input ko thoda nudge karun, toh output kitni tezi se change hogi?" — the slope.

\partial

kyun likhte hain

d

ki jagah?

Kai variables exist karte hain;

\partial / \partial u

sirf

u

nudge karta hai aur baaki sab variables freeze karta hai.

Parameters

u, v

aur domain

D

kya hain?

Do dials jo surface pe ek spot pick karte hain;

D

sabhi allowed dial settings ka flat set hai.

r (u, v)

kya hai?

Ek map jo flat

(u, v)

point ko surface pe ek 3D point tak bhejta hai — hard cheez ko flatten karne ki trick.

r_{u}

aur

r_{v}

kya hain?

r

ke partial derivatives; tangent arrows jo surface pe flat lete hain har dial ki direction ke saath.

Regularity condition kya hai aur yeh kyun important hai?

r_{u}, r_{v}

linearly independent hone chahiye (parallel nahi); warna patch zero area mein collapse ho jaata hai — Jacobian

\neq = 0

ka surface version.

Cross product

a \times b

: kaun si length, kaun si direction?

Length = parallelogram ki area (

∥ a ∥∥ b ∥ sin θ

); direction = dono ke perpendicular, right-hand rule se fix.

Order swap karne se cross product ka sign kyun flip hota hai?

Right-hand rule reverse ho jaata hai;

b \times a = - (a \times b)

— yahi flux ke sign ka source hai.

Parallelogram area

sin θ

use karta hai

cos θ

nahi, kyun?

Area = base

\times

height, aur height perpendicular part

∥ b ∥ sin θ

hai.

Dot product

a \cdot b

: yeh kya measure karta hai?

Do arrows kitna same direction mein point karte hain;

∥ a ∥∥ b ∥ cos θ

Flux dot product se kyun bana hai?

Sirf field ka woh component jo normal ke along ho woh surface ke through jaata hai, aur

F \cdot \hat{n}

exactly wahi extract karta hai.

\hat{n}

kya hai aur ise kaise banate hain?

Unit-length normal;

r_{u} \times r_{v}

lo (sirf regular points pe) aur apni length se divide karo.

d S

aur

d S

mein fark?

d S

ek positive patch area hai;

d S = \hat{n} d S

woh patch ek vector ke roop mein hai (area plus facing direction).

\iint_{D} d u d v

kya karta hai?

Flat region

D

ko tiny rectangles mein kaatta hai, unpe koi quantity sum karta hai, shrinking limit leta hai.

Vector field

F

kya hai?

Har point of space pe attached ek arrow — jaise har location pe fluid velocity.

Foundations — Surface integrals — scalar and vector (flux)

1. Points, coordinates, aur 3D space

2. Vectors aur unki length $∥ \cdot ∥$

3. Ek variable ke functions aur derivative

4. Partial derivatives $\frac{\partial}{\partial u}$ aur parameters $u, v$

5. Parametrization $r (u, v)$ aur uske tangent vectors

6. Cross product $\times$

7. Dot product $\cdot$

8. Unit normal $\hat{n}$

9. Double integral $\iint_{D}$ aur area elements $d S$ , $d S$

10. Vector field $F$

Prerequisite map

Equipment checklist