Green's theorem — proof sketch, both forms: Foundations

4.4.29 · D1 · Maths › Multivariable Calculus › Green's theorem — proof sketch, both forms

Is page pe assume kiya gaya hai ki tumne kuch nahi dekha. Hum har woh symbol build karte hain jo parent note use karta hai — $P$ , $Q$ , $d x$ , $d y$ , $\oint$ , $\iint$ , $\partial$ , $n$ , $T$ , $d A$ , $d s$ , $\cdot$ , $\nabla$ — ek blank chalkboard se.

0. Playground: points aur plane

Sab kuch ek flat sheet pe hota hai, jise plane kehte hain. Us par ek jagah ko point kehte hain, likha jaata hai $(x, y)$ .

Figure — Green's theorem — proof sketch, both forms

Kyun zaroori hai: Green's theorem ek region (plane ka ek filled patch) ko uski edge se compare karta hai. Dono sirf points $(x, y)$ ke collections hain, toh yeh baaki sab cheez ka alphabet hai.

1. Vector field: har point pe ek arrow

Har point $(x, y)$ pe hum ek chota arrow chipka dete hain. Yahi vector field hai.

Picture: $F$ ko hawa samjho. Har jagah hawa kisi direction mein kisi strength se chalti hai — wahi arrow hai. $P$ east direction ki hawa hai, $Q$ north direction ki hawa hai.

Topic ko kyun chahiye: Green's theorem ek vector field ke baare mein statement hai. $P$ aur $Q$ parent page ke literally har formula mein aate hain. Inke saath hum kya karte hain yeh dekhne ke liye Line Integrals aur Curl and Divergence dekho.

2. Dot product $a \cdot b$ — "do arrows kitna agree karte hain?"

Koi bhi loop walk karne se pehle, hume arrow-arithmetic ka ek tiny piece chahiye jo flux form use karta hai.

Topic ko kyun chahiye: flux form $F \cdot n$ likhta hai — "hawa $F$ outward direction $n$ mein kitna point karti hai." Woh number exactly $F_{x} n_{x} + F_{y} n_{y}$ hai, yani dot product. Yeh formula §6 ke liye yaad rakho.

3. Curves, closed loops, aur orientation

Topic ko kyun chahiye: parent note ka poora left-hand side hai " $C$ ke around chalo." Direction galat liya toh har equation ka sign flip ho jaata hai — yeh listed Common Mistakes mein se ek hai. Aur agar $C$ piecewise smooth nahi hoti, toh line integral define bhi nahi hota.

4. Region $D$ : filled, aur hole-free

Picture: $D$ ek floor hai; $d A$ tiling mein ek tile hai. Saari tiles pe sum karna double integral hai (agla section).

5. Integrals: teen symbols, ek idea (tiny bits add karo)

Har integral sign ek stretched "S" hai Sum ke liye. Fark sirf yeh hai ki tum kya sum karte ho.

6. Partial derivatives, aur operator $\nabla$

Derivative jawaab deta hai "yeh quantity kitni tezi se change hoti hai?" 2D function jaise $P (x, y)$ pe hume kehna hoga change kis direction mein.

Picture: ek hilly surface $P$ pe khade ho. Due East chalo: tumhari uphill steepness $\partial P / \partial x$ hai. Due North chalo: woh $\partial P / \partial y$ hai. Same hill, do alag slopes.

Do combinations jo parent inhe partials se build karta hai:

Curl (scalar): $Q_{x} - P_{y}$ — local counterclockwise spin measure karta hai.
Divergence: $P_{x} + Q_{y} = \nabla \cdot F$ — local spreading-out measure karta hai.

Dono fully explain kiye gaye hain Curl and Divergence mein; yahan tumhe bas itna jaanna hai ki woh partial derivatives ke combinations hain.

7. Tangent $T$ , normal $n$ , arc length $d s$ , aur "flux"

Flux form ko walk ke saath attached do special arrows chahiye, plus step length ka ek proper measure.

"Out" kyun? Agar region tumhare left pe hai (CCW), toh apne forward-arrow ko right (clockwise) turn karne se woh region se door point karta hai.

Topic ko kyun chahiye: flux form literally $n$ , $d s$ , aur dot product se bana hai. Ab is mein sab kuch define ho gaya hai.

8. Family tree (yeh sab siblings ke roop mein wapas aayenge)

Green's theorem ek bade building mein ek floor hai; Stokes' Theorem aur Divergence Theorem uske 3D relatives hain, aur Conservative Vector Fields woh special case hai jahan loop integral hamesha $0$ hota hai. Tumhe abhi inki zaroorat nahi — bas jaano ki upar wali vocabulary in sab ko unlock karti hai.

Foundations theorem ko kaise feed karti hain

Ise bottom-up padho: har row uske upar wale ideas se bani hai, aur last row theorem hi hai.

Building block	Feeds into
Points $(x, y)$	sab kuch — plane ka alphabet
Vector field $F = (P, Q)$	line integral, curl, divergence, flux
Dot product $\cdot$	divergence $\nabla \cdot F$ aur flux $F \cdot n$
Curve $C$ (simple, closed, piecewise smooth) + orientation	line integral $\oint_{C}$
Region $D$ (filled, simply connected) + tile $d A$	double integral $\iint_{D}$
Partial derivatives + operator $\nabla$	curl $Q_{x} - P_{y}$ aur divergence $P_{x} + Q_{y}$
Tangent $T$ , normal $n$ , $d s$ , flux	flux form $\oint_{C} F \cdot n d s$
Yeh sab milke	Green's theorem (dono forms)

Equipment checklist

Khud ko test karo — agar koi bhi jawaab fuzzy lage, woh section dobara padho.

Point

(x, y)

mein do numbers ka kya matlab hai?

x

= centre se right ki distance,

y

= upar ki distance; dono milke ek dot name karte hain.

Field

F = (P, Q)

mein

P

aur

Q

kya hain?

P

(x, y)

pe arrow ka horizontal (rightward) part;

Q

= vertical (upward) part.

Dot product

a \cdot b

kya hai aur yeh kya measure karta hai?

a_{1} b_{1} + a_{2} b_{2}

— ek single number jo bata hai do arrows direction mein kitna agree karte hain (same way positive, perpendicular zero, opposite negative).

Curve "simple", "closed", aur "piecewise smooth" kab hoti hai?

Simple = kabhi khud ko cross nahi karti; closed = wahin khatam hoti hai jahan se shuru hui; piecewise smooth = finitely many pieces jिनमें से har ek mein well-defined tangent ho.

Positive orientation kaun si hai, aur "on your left" rule kya hai?

Counterclockwise; jab tum chalo toh enclosed region tumhare LEFT mein rehna chahiye.

D

ke liye "simply connected" kya require karta hai?

Ek single filled piece bina kisi hole ke (pancake, doughnut nahi).

\oint

ka matlab kya hai vs

\iint

\oint

= closed loop ke around sum (line integral);

\iint

= filled 2D region ke upar sum (double integral).

Vertical step pe

d x

kya hota hai?

Zero — tum sideways nahi hile, toh

d x = 0

\frac{\partial P}{\partial y}

kya measure karta hai?

P

kitni tezi se change hoti hai jab tum sirf

y

ko nudge karo,

x

ko frozen rakhte hue (

y

-direction mein ek slope).

Operator

\nabla

kya hai aur

\nabla \cdot F

kya hai?

\nabla = (\partial / \partial x, \partial / \partial y)

;

\nabla \cdot F = P_{x} + Q_{y}

, divergence.

Partials ke combinations ke roop mein curl vs divergence?

Curl

= Q_{x} - P_{y}

(spin, ek difference); divergence

= P_{x} + Q_{y}

(spreading, ek sum).

Arc-length element

d s

ka formula kya hai?

d s = d x^{2} + d y^{2}

, hamesha

\geq 0

Tangent

T

se outward normal

n

kaise milta hai?

T

ko clockwise

9 0^{\circ}

rotate karo: agar

T = (d x / d s, d y / d s)

toh

n = (d y / d s, - d x / d s)

Yahan "flux" ka kya matlab hai?

Boundary

C

ke across bahar

F

ka net flow:

\oint_{C} F \cdot n d s

(bahar positive, andar negative).

D

ke andar

P, Q

ke continuous partials kyun hone chahiye?

Taaki field aur uske slopes kabhi blow up na karein; ek singularity (jaise origin pe) hypothesis tod deti hai aur theorem fail ho jaata hai.