Divergence — definition, physical meaning (flux density): Visual walkthrough

4.4.24 · D2 · Maths › Multivariable Calculus › Divergence — definition, physical meaning (flux density)

Shuru karne se pehle, un do words pe agree karte hain jinpar sab kuch tika hai.

Step 1 — Point pe ek tiny box daalo

KYA. Jis point ki hume parwah hai use pick karo, uski coordinates $(x, y, z)$ rakho. Uske around, ek tiny rectangular box banao jiske side-lengths hain $Δ x$ (width), $Δ y$ (depth), $Δ z$ (height). Chhota symbol $Δ$ (Greek "delta") bas matlab hai "ek chhota step."

KYUN. Divergence hai flux per unit volume. Flux ke liye hume ek closed surface chahiye jisme arrows cross karte hue gine ja sakein; sabse simple closed surface hai 6 flat faces wala box. Box easy hai kyunki har face exactly ek axis ke along point karta hai, isliye har face pe $F$ ka sirf ek component matter karta hai.

PICTURE. Box ka volume hai $V = width Δ x \cdot depth Δ y \cdot height Δ z .$ Hum saare 6 walls se nikalne wale arrows count karenge, unhe add karenge, phir $V$ se divide karenge.

Step 2 — Outward normal: "bahar" kaunsi taraf hai?

KYA. Har face pe hum ek chhota arrow $\overset{n}{^}$ (padho "n-hat") draw karte hain jo box ke seedha bahar point karta ho, us face ke perpendicular, exactly $1$ length ke saath. Hat ka matlab hai "length one" — yeh sirf ek direction carry karta hai, koi size nahi.

KYUN. Flux fluid ko box se nikalta hua count karta hai. "Nikalna" matlab hai outward direction mein ek wall cross karna. Isliye har face pe hume pata hona chahiye ki bahar kaunsi taraf hai — exactly yahi $\overset{n}{^}$ record karta hai. $\overset{n}{^}$ ke saath move karta fluid "nikalna" count hota hai (positive); $\overset{n}{^}$ ke against move karta fluid enter ho raha hai (negative).

PICTURE. $x$ -axis ke perpendicular do faces ke liye:

Right face (at $x + Δ x$ ): $\overset{n}{^} = + \overset{x}{^}$ (right point karta hai, box se bahar).
Left face (at $x$ ): $\overset{n}{^} = - \overset{x}{^}$ (left point karta hai, phir bhi box se bahar).

Woh tool jo measure karta hai " $F$ kitna $\overset{n}{^}$ ke along point karta hai" woh hai dot product $F \cdot \overset{n}{^}$ . Dot product kyun aur kuch kyun nahi? Kyunki kisi vector ka ek unit direction ke saath dot product exactly us vector ki us direction pe chhaya (projection) return karta hai — precisely woh "wall ke through nikla hua part" jisse humne flux define kiya tha.

Step 3 — Do $x$ -faces se bahar flux

KYA. Sirf $F_{1}$ (arrow ka rightward part) $x$ -faces cross karta hai, kyunki un faces pe $\overset{n}{^}$ hai $\pm \overset{x}{^}$ aur $F \cdot (\pm \overset{x}{^}) = \pm F_{1}$ . Har face ke flux ko (wahan field value) $\times$ (face area) se approximate karo. $x$ -faces ka area $Δ y Δ z$ hai.

Flux_{x} \approx right face se bahar (\overset{n}{^} = + \overset{x}{^}) F_{1} (x + Δ x, y, z) Δ y Δ z - left face se bahar (\overset{n}{^} = - \overset{x}{^}) F_{1} (x, y, z) Δ y Δ z

$F_{1} (x + Δ x, y, z)$ — rightward push right wall pe measure kiya gaya.
$F_{1} (x, y, z)$ — rightward push left wall pe measure kiya gaya.
$Δ y Δ z$ — har $x$ -face ka area.
Minus isliye aata hai kyunki left wall pe $\overset{n}{^} = - \overset{x}{^}$ hai, isliye $F \cdot \overset{n}{^} = - F_{1}$ : wahan rightward fluid enter ho raha hai, jo subtract hota hai.

KYUN. Hum $x$ ke along net "bahar minus andar" chahte hain. Fluid right se nikalta hai aur (agar $F_{1} > 0$ ho) left se enter karta hai. Subtract karne se sach mein bacha hua milta hai.

PICTURE. Do parallel walls, same area, arrows dono ko pierce karte hue. Jo right pierce karta hai woh nikalta hai; jo left pierce karta hai woh enter karta hai aur subtract hota hai.

Step 4 — Ek step pe difference ek derivative banne ki raah pe hai

KYA. Common area $Δ y Δ z$ factor out karo:

Flux_{x} \approx [F_{1} (x + Δ x, y, z) - F_{1} (x, y, z)] Δ y Δ z .

Bracket mein hai $F_{1}$ mein badlaav jab tum $Δ x$ distance right step karte ho.

KYUN yeh tool — the derivative. Ek raw difference jaise $F_{1} (x + Δ x) - F_{1} (x)$ depend karta hai $Δ x$ kitna bada hai; yeh awkward hai. Derivative ek sharper sawaal ka jawab deta hai: " $F_{1}$ yahan $x$ ke per unit mein kitni rate se change hota hai?" Hum difference ko rate mein convert karte hain $Δ x$ se divide aur re-multiply karke:

Flux_{x} \approx rate of change of F_{1} in x \frac{F _{1} ( x + Δ x , y , z ) - F _{1} ( x , y , z )}{Δ x} = V Δ x Δ y Δ z .

PICTURE. $F_{1}$ ko $x$ ke against plot karo: bracket hai rise, $Δ x$ hai run, aur rise/run hai slope of the line joining the two wall-values. Box ko chhota karne se woh line curve ko kiss karti hai — slope tangent slope ban jaata hai.

Step 5 — Box shrink karo: slope ban jaata hai $\partial F_{1} / \partial x$

KYA. Box ko shrink hone do, $Δ x \to 0$ . Us connecting line ka slope exact tangent slope ban jaata hai — partial derivative:

Δ x \to 0 lim \frac{F _{1} ( x + Δ x , y , z ) - F _{1} ( x , y , z )}{Δ x} = \frac{\partial F _{1}}{\partial x} .

Curly $\partial$ (padho "partial-dee") matlab: $y$ aur $z$ ko frozen rakho, sirf $x$ ko wiggle karo. Hum doosron ko freeze karte hain kyunki humare do $x$ -walls same $y, z$ pe hain — sirf $x$ unke beech differ karta hai.

Toh flux mein $x$ -contribution hai

Flux_{x} \approx \frac{\partial F _{1}}{\partial x} V .

KYUN. Limit " $V \to 0$ " mein box picture ek exact statement ban jaati hai point ke baare mein, na ki ek fat box ke liye approximation.

PICTURE. Step 4 jaisa hi two-point plot, lekin do dots slide karke saath aa jaate hain; secant line rotate hokar tangent pe aa jaati hai, jiski steepness hai $\partial F_{1} / \partial x$ .

Step 6 — Baaki chaar faces, same reasoning se

KYA. $y$ -faces (top aur bottom) ka area $Δ x Δ z$ hai aur sirf $F_{2}$ feel karte hain; $z$ -faces (front aur back) ka area $Δ x Δ y$ hai aur sirf $F_{3}$ feel karte hain. Steps 3–5 ko word for word copy karo, sirf labels rotate karke:

Flux_{y} \approx \frac{\partial F _{2}}{\partial y} V, Flux_{z} \approx \frac{\partial F _{3}}{\partial z} V .

Notice karo matching indices: $F_{1}$ pairs karta hai $x$ ke saath, $F_{2}$ pairs karta hai $y$ ke saath, $F_{3}$ pairs karta hai $z$ ke saath. Har component ko apni walls ki apni pair ke across compare kiya jaata hai, jo apni axis ke along hain. Isliye cross-terms jaise $\partial F_{1} / \partial y$ kabhi nahi aate — left/right walls sirf $x$ mein differ karte hain, isliye sirf $F_{1}$ ka $x$ ke saath change matter karta hai.

KYUN. Space teeno axes ko same treat karta hai; koi privileged direction nahi hai, isliye $x$ wali baat $y$ aur $z$ pe bhi same lagti hai.

PICTURE. Wahi box, ab top/bottom pair aur front/back pair highlight hai, har ek ke apne arrows aur apne matching partial derivative ke saath.

Step 7 — Teeno add karo, $V$ se divide karo

KYA. Total outward flux saare chhe faces ka sum hai:

Flux_{total} \approx (\frac{\partial F _{1}}{\partial x} + \frac{\partial F _{2}}{\partial y} + \frac{\partial F _{3}}{\partial z}) V .

$V$ se divide karo aur $V \to 0$ le jao (divergence ki definition):

div F = V \to 0 lim \frac{Flux _{total}}{V} = \frac{\partial F _{1}}{\partial x} + \frac{\partial F _{2}}{\partial y} + \frac{\partial F _{3}}{\partial z} . ■

Har term = net outflow ek axis ke along, per unit volume.
Sum = point pe total spreading rate.
$V$ cleanly cancel ho jaata hai kyunki har face-flux mein $V$ ka ek factor tha.

KYUN. "Flux per unit volume" literally matlab hai "bacha hua fluid box ke size se divide karo." Aisa karne se box ka koi mention nahi bcha — answer sirf point ka hai.

Step 8 — Degenerate case: symmetric flow, zero net divergence

KYA. $F = (x^{2}, 0, 0)$ ko origin pe test karo. Iska true divergence hai $\partial (x^{2}) / \partial x = 2 x$ , jo $x = 0$ pe $0$ hai. Chalte hain box ko confirm karte dekhte hain.

Origin pe centred box $[- h, h]^{3}$ ke liye:

Right wall at $x = + h$ : $F_{1} = (+ h)^{2} = h^{2}$ , $\overset{n}{^} = + \overset{x}{^}$ , area $(2 h)^{2}$ → flux $+ h^{2} (2 h)^{2}$ .
Left wall at $x = - h$ : $F_{1} = (- h)^{2} = h^{2}$ , $\overset{n}{^} = - \overset{x}{^}$ , area $(2 h)^{2}$ → flux $- h^{2} (2 h)^{2}$ .

Woh exactly cancel ho jaate hain: net flux $= 0$ , toh divergence $= 0$ . ✓

KYUN yeh dikhaya. Yeh tumhe us trap se bachata hai "bade arrows $\Rightarrow$ bada divergence." Yahan dono walls pe arrows equally strong hain ( $h^{2}$ each) phir bhi equally dono sides se nikal rahe hain — bacha hua kuch nahi. Divergence walls ke across imbalance measure karta hai, arrows ka size nahi. (Dekho parent mistake list.)

PICTURE. Left aur right walls ke saath equal-length arrows dono outward point karte hue — mirror images. Equal out, equal out, lekin $x = 0$ pe woh balance hokar koi net source nahi bacha.

Ek-picture summary

Sab kuch compressed: box → six face-fluxes → teen matching differences → teen partial derivatives → unka sum. Coloured path trace karo "walls pe arrows count karo" se "teen slopes add karo" tak.

Recall Walkthrough ki Feynman retelling

Apne point ke around ek tiny imaginary box rakho. Uske chhe walls mein se har ek ek chhota gate hai; count karo kitna fluid har gate se bahar jaata hai. Gates ko pairs mein group karo — left/right, top/bottom, front/back. Left/right pair ke liye, sirf flow ka rightward part ( $F_{1}$ ) cross kar sakta hai, aur bacha hua hai "kitna right cross kiya minus kitna left cross kiya." Woh difference, tiny gap $Δ x$ pe spread karke, sirf slope hai $F_{1}$ ka jab tum right chalte ho — partial derivative $\partial F_{1} / \partial x$ . Top/bottom pair deta hai $\partial F_{2} / \partial y$ , front/back pair deta hai $\partial F_{3} / \partial z$ . Teen slopes add karo aur tumhare paas hai total bacha hua fluid per box-volume — the divergence. Agar walls balance karein (dono sides se equal out), toh bacha hua zero hai chahe arrows kitne bhi bade hon. Yahi poora idea hai: teen matching slopes, add karke, woh rate hai jis par fluid us point pe born hota hai.

Quick self-check

div (x, y, z)

kitna hoga?

1 + 1 + 1 = 3

(har jagah ek source).

\partial F_{1} / \partial y

kabhi kyun nahi aata?

Left/right walls sirf

x

mein differ karte hain, isliye sirf

F_{1}

x

ke saath change unke flux ko affect karta hai; cross-changes belong karte hain Curl — rotation density ko.

F = (x^{2}, 0, 0)

ke liye

x = 0

pe, centred box se net flux kitna hai?

Zero — dono

x

-walls pe

F_{1} = h^{2}

hai aur unke fluxes cancel ho jaate hain; matches karta hai

2 x ∣_{0} = 0

V

se divide karne par kya cancel hota hai?

Har face-flux mein

V

ka ek factor tha, isliye box ka size disappear ho jaata hai aur answer sirf point ka reh jaata hai.

Connections

Divergence — definition, physical meaning (flux density) — woh parent jise yeh page derive karta hai.
Flux through a surface — woh surface integral jo humne face-by-face approximate kiya.
Divergence Theorem (Gauss) — infinitely saare tiny boxes ko ek global flux law mein glue karta hai.
Gradient and the del operator — jahan $\nabla$ aur $\partial$ notation janm leta hai.
Curl — rotation density — woh sibling jo un cross-terms se banta hai jo humne discard kiye.
Continuity equation — jahan yeh "fluid born per volume" idea physics ban jaata hai.
Laplacian — gradient ka divergence.