Foundations — Divergence — definition, physical meaning (flux density)

4.4.24 · D1 · Maths › Multivariable Calculus › Divergence — definition, physical meaning (flux density)

Is page par assume kiya gaya hai ki tumne pehle kuch nahi dekha. Hum parent note ke har symbol ko — $\vec{F}$, $F_1$, $\partial /\partial x$, $\nabla$, $\hat{n}$, $dS$, $\iint$, $\lim$, dot $\cdot$ — picture se shuru karke build karte hain, ek aisi order mein jahan har ek symbol sirf unhi par lean karta hai jo pehle aa chuke hain.

0. Arrow kya hota hai, aur unka ek field kya hota hai?

Ek akela arrow boring hota hai. Interesting object tab banta hai jab space ke har point ko apna arrow milta hai.

Figure dekho: wohi field $\vec{F}$ arrows ke ek poore carpet ki tarah dikhaya gaya hai. Wo carpet — koi ek arrow nahi — wohi cheez hai jis par divergence kaam karta hai.

Topic ko ye kyun chahiye? Kyunki divergence ek sawaal hai ki arrows spot to spot kaise change hote hain. Bina ek field (varying arrows) ke, measure karne ke liye kuch bhi nahi hai.

1. Ek arrow ko pieces mein todna: components $F_1,F_2,F_3$

Kisi bhi arrow ko ek "kitna right", ek "kitna upar", aur ek "kitna aage" instruction se build kiya ja sakta hai.

Figure mein, tircha arrow apni horizontal shadow $F_1$ aur vertical shadow $F_2$ mein split hota hai. Arrow bilkul "$F_1$ steps right, phir $F_2$ steps up" hai.

Topic ko ye kyun chahiye: divergence formula har component ko alag-alag treat karta hai — $x$-piece $F_1$, $y$-piece $F_2$, $z$-piece $F_3$. "$x$-flow" ke baare mein baat karne ke liye pehle tumhe use naam dena hoga: wo naam $F_1$ hai.

2. EK direction mein change ki rate: partial derivative $\frac{\partial F_1}{\partial x}$

Ab measuring ka key tool. Pehle us idea ko yaad karo jis se ye aata hai.

Lekin ek field mein, position ke teen knobs hain ($x$, $y$, $z$). Hum chahte hain wo change jo sirf unhe mein se ek ko nudge karne se hota hai.

Figure mein: $y$ ko freeze karo, dashed line ke saath right slide karo, aur sirf dekho ki rightward-arrows kaise lambe hote hain. Wo growth rate $\partial F_1/\partial x$ hai.

Topic ko ye kyun chahiye: "zyada fluid right face se leave karta hai left se enter karne se" bilkul wohi hai "$F_1$ right par zyada bada hai left se jab tum $x$ mein step karte ho" — wohi hai $\partial F_1/\partial x$. Ye ek symbol poora physical meaning carry karta hai.

3. Teen partials ko ek symbol mein pack karna: $\nabla$ (del)

Hum baar baar "$\partial /\partial x$, $\partial /\partial y$, $\partial /\partial z$" likhte rehte hain. Mathematicians in teen operations ko ek symbol mein bundle karte hain taaki formulas chhote rahein.

Topic ko ye kyun chahiye: ye hume poora divergence formula ek tidy expression $\nabla \cdot \vec{F}$ ke roop mein likhne deta hai bajaaye ek three-term sum ke.

4. Dot product $\cdot$: matching parts multiply karo, phir add karo

$\nabla$ ko $\vec{F}$ ke saath combine karne ke liye hum dot product use karte hain. Yaad karo ye do ordinary vectors ke saath kya karta hai.

Ab ise $\nabla$ ke pehle slot mein apply karo. $\partial /\partial x$ se "multiply karna" ka matlab hai "partial derivative lo": $\nabla \cdot \vec{F}=\frac{\partial }{\partial x}{F}_1+\frac{\partial }{\partial y}{F}_2+\frac{\partial }{\partial z}{F}_3=\frac{\partial F_1}{\partial x}+\frac{\partial F_2}{\partial y}+\frac{\partial F_3}{\partial z}$

Topic ko ye kyun chahiye: dot exactly wohi machine hai jo ek vector-input ($\vec{F}$) ko scalar-output (har point par ek number) mein badalta hai. "Vector in, scalar out" divergence ki signature hai.

5. Normal $\hat{n}$, area $dS$, aur flux $\iint \vec{F}\cdot \hat{n}dS$

Parent ka geometric definition teen aur pieces maangta hai. Ek point ke around ek closed bubble imagine karo.

Figure mein: arrows jo bubble ko outward cross karte hain ($\hat{n}$ ke saath aligned) positive count karte hain; inward cross karne wale negative count karte hain. Flux running total hai.

Topic ko ye kyun chahiye: divergence define hoti hai shrinking-bubble limit mein flux per unit volume ke roop mein. Flux raw quantity hai; divergence flux hai jo ek single point tak squeeze ho gayi hai.

6. Limit ${\lim }_{V\to 0}$ — bubble ko ek point tak shrink karna

Degenerate check: agar field bilkul balanced hai (har scale par same in jitna out, jaise uniform wind), to flux $0$ hai har bubble ke liye, to limit $0$ hai. Divergence $=0$ ka matlab hai "yahan koi net creation nahi."

Symbols ko saath rakhna

$\underset{\text{P par ek number}}{\underbrace{\mathrm{div}\vec{F}(P)}}=\underset{\S 6}{\underbrace{\underset{V\to 0}{\lim }}}\frac{1}{V}\underset{\S 5\text{ flux}}{\underbrace{\iint \vec{F}\cdot \hat{n}dS}}=\underset{\S 3,\S 4}{\underbrace{\nabla \cdot \vec{F}}}=\underset{\S 1,\S 2}{\underbrace{\frac{\partial F_1}{\partial x}+\frac{\partial F_2}{\partial y}+\frac{\partial F_3}{\partial z}}}$

Upar ka har symbol un sections mein earn kiya gaya tha jo neeche mark hain. Left half hai kyun tumhe parwah hai (geometry); right half hai kaise compute karte hain (partials). Parent note prove karta hai ki ye dono same object hain.

Foundations topic ko kaise feed karte hain

Equipment checklist

Apne aap ko test karo — parent note padhne se pehle tumhe har cheez ka jawaab dena aana chahiye.

Connections

• Gradient and the del operator — $\nabla$ ka birthplace jo §3 mein use hua.
• Flux through a surface — §5 ka surface integral poori detail mein.
• Curl — rotation density — wohi $\nabla$ se cross product kya build karta hai.
• Divergence Theorem (Gauss) — local density ko global flux se link karta hai.
• Continuity equation — physics jo ye symbols use karti hai.
• Laplacian — gradient ka divergence, ${\nabla }^{2}f$.