Divergence — definition, physical meaning (flux density)

4.4.24 · Maths › Multivariable Calculus

KYA hai divergence?

Gehri baat yeh hai: yeh dono definitions ek hi cheez hain. Geometric wali batati hai kyun tumhe farak padta hai; partial-derivative wali batati hai kaise compute karo. Chalo doosri ko pehli se derive karte hain.

KAISE: First principles se Derivation

Hum ek tiny box se bahar flux compute karte hain aur usse uske volume se divide karte hain.

Setup. Ek tiny box rakho point $(x,y,z)$ pe jiske sides $\Delta x,\Delta y,\Delta z$ hain, toh $V=\Delta x\Delta y\Delta z$. Box ke 6 faces hain; hum har ek se outward flux add karte hain.

$x$-axis ke perpendicular dono faces se flux. Right face ($x+\Delta x$) ka outward normal $+\hat{x}$ hai, left face ($x$) ka $-\hat{x}$ hai. Sirf $F_1$ contribute karta hai (yeh $x$-component hai jo $\pm \hat{x}$ ke saath dot hota hai). Har face ka flux approximate karo (field value)×(area) ke roop mein:

${\text{Flux}}_x\approx \underset{\text{right se bahar}}{\underbrace{F_1(x+\Delta x,y,z)\Delta y\Delta z}}-\underset{\text{left se bahar}}{\underbrace{F_1(x,y,z)\Delta y\Delta z}}$

Minus kyun? Left face pe outward normal $-\hat{x}$ direction mein point karta hai, toh $\vec{F}\cdot \hat{n}=-F_1$.

Area $\Delta y\Delta z$ bahar nikalo: ${\text{Flux}}_x\approx [F_1(x+\Delta x,y,z)-F_1(x,y,z)]\Delta y\Delta z$

Yeh step kyun? Bracket exactly $F_1$ ka difference hai step $\Delta x$ ke upar — yeh ek derivative banne ki taiyari mein hai. $\Delta x$ se multiply aur divide karo:

${\text{Flux}}_x\approx \frac{F_1(x+\Delta x,y,z)-F_1(x,y,z)}{\Delta x}\Delta x\Delta y\Delta z\stackrel{\Delta x\to 0}{\to }\frac{\partial F_1}{\partial x}V$

Bilkul same reasoning se $y$- aur $z$-faces ke liye: ${\text{Flux}}_y\approx \frac{\partial F_2}{\partial y}V,{\text{Flux}}_z\approx \frac{\partial F_3}{\partial z}V$

Add karo aur $V$ se divide karo. Total flux $\approx \left(\frac{\partial F_1}{\partial x}+\frac{\partial F_2}{\partial y}+\frac{\partial F_3}{\partial z}\right)V$, toh

$\frac{\text{flux}}{V}\to \frac{\partial F_1}{\partial x}+\frac{\partial F_2}{\partial y}+\frac{\partial F_3}{\partial z}=\mathrm{div}\vec{F}.■$

Bas yahi hai — flux per unit volume = partial derivatives ka sum. Geometric definition ne formula ko exist karne par majboor kar diya.

KYUN formula "sahi lagte hai"

Worked examples

Common mistakes (steel-manned)

Flashcards

Connections

• Curl — rotation density (cross-product sibling; same $\nabla$, alag operation)
• Divergence Theorem (Gauss) (local flux-density ko global flux mein convert karta hai: $\iiint \mathrm{div}\vec{F}dV=\iint \vec{F}\cdot \hat{n}dS$)
• Gradient and the del operator (jahan se $\nabla$ aata hai)
• Flux through a surface (surface integral jo average ho raha hai)
• Continuity equation (physics: $\partial \rho /\partial t+\nabla \cdot (\rho \vec{v})=0$)
• Laplacian (${\nabla }^{2}f=\mathrm{div}(\mathrm{grad}f)$)

Concept Map

Deep Dive

• D1 · Foundations — har symbol zero se
• D2 · Visual walkthrough — derivation pictures mein
• D3 · Worked examples — har scenario
• D4 · Exercises — graded, full solutions
• D5 · Question bank — concept traps