Foundations — Divergence theorem (Gauss's theorem) — statement, flux-divergence connection

4.4.33 · D1 · Maths › Multivariable Calculus › Divergence theorem (Gauss's theorem) — statement, flux-diver

Theorem ko padhne se pehle, humein kuch tools chahiye. Yeh page har ek tool ko zero se banata hai — plain words mein matlab, ek picture, aur wajah ki yeh topic uske bina kaam kyon nahi kar sakta. Hum order mein jaate hain, taaki har idea usse pehle wale par tikha rahe.

1. 3D mein ek point — coordinates $(x,y,z)$

Picture. Teen arrows ek corner par right angles mein milte hain. Room mein koi bhi jagah "3 right, 2 back, 4 up" hai, jise $(3,2,4)$ likhte hain.

Topic ko yeh kyun chahiye. Theorem 3D mein rehta hai. Baaki sab cheezein — fields, surfaces, volumes — sab kuch har ek point $(x,y,z)$ se kuch attach karke banti hain, isliye pehle point ka naam hona zaroori hai.

2. Solid region $V$ aur uski skin $S=\partial V$

Symbol $\partial$ (curly "d") ko "boundary of" padha jaata hai. Toh $\partial V$ ka matlab literally hai "V ki skin."

Picture. Ek aloo jaisi shape (woh $V$ hai). Uski chamakti bahari surface (woh $S$ hai). "Closed" ka matlab hai skin andar ko poori tarah seal karti hai — balloon, bowl nahi.

Topic ko yeh kyun chahiye. Poora theorem do jagahon ke beech ek bridge hai: andar ($V$) aur skin par ($S$). Dono ke liye saaf naam chahiye, aur skin closed honi chahiye taaki "andar" clearly defined rahe.

3. Vectors aur arrow field $\mathbf{F}$

Pehle, ek arrow.

Ab, har point par ek arrow.

Picture. Hawa socho. Har jagah hawa ek speed aur direction mein chalti hai — woh ek arrow hai. Arrows ka poora weather map field $\mathbf{F}$ hai. Teen functions:

• $P(x,y,z)$ = wahan kitni tezi se rightward chalti hai,
• $Q(x,y,z)$ = kitni tezi se backward,
• $R(x,y,z)$ = kitni tezi se upward.

Topic ko yeh kyun chahiye. $\mathbf{F}$ wahi flowing stuff hai — paani ka current, hawa, electric influence. Theorem poora is field ke baare mein hai: kitna leak hota hai, aur kahan ban raha hai. $\mathbf{F}$ ke saath hum jo do cheezein karte hain unke liye Flux integrals aur Divergence and curl dekho.

4. Outward unit normal $\hat{\mathbf{n}}$

Chhota hat $\hat{}$ hamesha "length one" matlab hai — ek pure direction, koi size nahi. "Normal" perpendicular ka purana word hai.

Picture. Ek hedgehog: uski skin ke har point par ek kaanta seedha bahar nikalta hai. Har kaanta wahan $\hat{\mathbf{n}}$ hai. "Outward" matlab andar se door, kabhi aloo ke andar nahi.

Topic ko yeh kyun chahiye. Leaking out measure karne ke liye, humein pata hona chahiye ki har skin point par "bahar" kidhar hai. $\hat{\mathbf{n}}$ woh compass hai. Inward choose karne ki jagah outward choose karna har sign flip kar deta hai — parent note ka pehla mistake box dekho.

5. Dot product $\mathbf{F}\cdot \hat{\mathbf{n}}$ — kitna across flow hota hai

Yeh "kitna $\mathbf{a}$, $\mathbf{b}$ ki direction mein point karta hai" kyun measure karta hai? Kyunki $\cos \theta$ ek dial hai:

• arrows aligned ($\theta =0$): $\cos =1$, full value — maximum agreement,
• arrows perpendicular ($\theta =90^{\circ }$): $\cos =0$, zero — koi agreement nahi,
• arrows opposite ($\theta =180^{\circ }$): $\cos =-1$, full negative.

Yeh tool kyun, koi aur kyun nahi? Humein jawaab chahiye: "$\mathbf{F}$ ka kitna flow actually skin ko pierce kar raha hai, versus uske saath slide kar raha hai?" Flow jo sideways skin ke saath slide karta hai woh kabhi nahi nikalta. Dot product $\mathbf{F}\cdot \hat{\mathbf{n}}$ exactly woh machine hai jo sirf "piercing" part rakhta hai aur sideways part discard karta hai — precisely kyunki $\cos 90^{\circ }=0$ sideways component ko khatam kar deta hai.

Picture. Khidki se takraati baarish. Baarish jo seedhi andar se guzre (aligned with $\hat{\mathbf{n}}$) poori count hoti hai; baarish jo sheeshe par slide kare (perpendicular to $\hat{\mathbf{n}}$) zero count hoti hai.

6. Partial derivatives $\frac{\partial P}{\partial x}$

Curly $\partial$ (straight $d$ nahi) warn karta hai: "aur bhi variables hain, lekin main unhe roka hua hun."

Picture. Ek pahadi par khado. Sirf east mein ek kadam chalo — tumhari height kitni badali? Woh rate east direction mein partial hai. Sirf north chalo — ek alag partial. Wahi landscape, alag sawaal.

Yeh tool kyun, koi aur kyun nahi? Derivation mein, ek tiny box ke right face se nikal rahi flux minus left face wali flux ek difference hai $P(x+\Delta x)-P(x)$. Partial derivative exactly woh tool hai jo ek chhote difference ko ek clean rate mein badalta hai: $P(x+\Delta x)-P(x)\approx \frac{\partial P}{\partial x}\Delta x.$ "Chhote gap ke across difference" ko per-point quantity mein convert karne ka aur koi tarika nahi.

7. Divergence $\nabla \cdot \mathbf{F}$ — har point par creation

Symbol $\nabla$ ("nabla" ya "del") ek bookkeeping gadget hai jo "yeh partial derivatives lo" ke liye khada hai. Dot, dot-product idea recycle karta hai: hum $\nabla =({\partial }_x,{\partial }_y,{\partial }_z)$ ko $\mathbf{F}=(P,Q,R)$ se dot kar rahe hain.

Picture. Ek point ke around ek tiny box. Kuch faces par arrows andar aate hain, kuch par bahar jaate hain. Agar zyada bahar jaaye andar se, toh woh point ek source hai (positive divergence, ek tap). Zyada andar: ek sink (negative, ek drain). Perfectly balanced: divergence $0$.

Topic ko yeh kyun chahiye. Divergence woh "inside" number hai jise theorem add karta hai. Yeh "leaking" ka local, per-point version hai, jise theorem global skin-leaking se connect karega. Fuller treatment: Divergence and curl.

8. Do integrals — ${\iint }_S$ aur ${\iiint }_V$

• ${\iint }_S(\cdots )dS$ — ek surface integral: skin $S$ ko area $dS$ ke tiny patches mein kaato, saare patches par $(\cdots )$ add karo. Do integral signs = ek 2D surface.
• ${\iiint }_V(\cdots )dV$ — ek volume integral: solid $V$ ko volume $dV$ ke tiny cubes mein kaato, saare cubes par $(\cdots )$ add karo. Teen signs = ek 3D solid.

Picture. Aloo ki skin ko tiny stamps ($dS$) se tile karo, versus aloo ke andar tiny sugar cubes ($dV$) pack karo. Har stamp / har cube par likha ek number add karo.

Topic ko yeh kyun chahiye. Theorem do total-ups ke beech ek equals sign hai: skin par total leaking (${\iint }_S$) versus andar total creation (${\iiint }_V$).

Symbols ko ek saath jodna — flux

Ab har piece defined hai, toh finally bina kisi unexplained symbol ke star player padh sakte hain:

Aur theorem ab fully padhne layak hai: $\underset{\text{skin par leaking}}{\underbrace{{\iint }_S\mathbf{F}\cdot \hat{\mathbf{n}}dS}}=\underset{\text{andar creation}}{\underbrace{{\iiint }_V(\nabla \cdot \mathbf{F})dV}}$

Foundations theorem ko kaise feed karti hain

Upar se neeche padho: points se solids aur fields bante hain; field aur outward normal milkar flux dete hain; partial derivatives se divergence banta hai; aur theorem do bottom boxes ko jodne wala bridge hai.

Equipment checklist

Right side cover karo aur khud test karo; parent note padhne se pehle har cheez obvious lagni chahiye.

Connections

• Flux integrals — machine ${\iint }_S\mathbf{F}\cdot \hat{\mathbf{n}}dS$ jo yahan ke dot product se bani hai.
• Divergence and curl — jahan $\nabla \cdot \mathbf{F}$ (aur uska cousin, curl) poori tarah develop kiya gaya hai.
• Parent: the Divergence Theorem — jahan yeh saare symbols finally milte hain.
• Continuity equation — divergence in motion: woh physics jo "creation = leaking" ko ek law banati hai.