Divergence — definition, physical meaning (flux density): Worked examples

4.4.24 · D3 · Maths › Multivariable Calculus › Divergence — definition, physical meaning (flux density)

Yeh page ek drill hai. Divergence — definition, physical meaning (flux density) parent note ne idea (flux per tiny volume) aur formula build kiya tha. Yahaan hum har tarah ki problem cover karte hain jo aa sakti hai, taaki koi bhi case aisa na ho jo tumne pehle dekha na ho.

Kuch bhi compute karne se pehle, ek reminder plain words mein. Vector field ka matlab sirf itna hai: space ke har point par ek arrow. Arrow $F = (F_{1}, F_{2}, F_{3})$ mein teen numbers hote hain — kitna east ( $F_{1}$ ), kitna north ( $F_{2}$ ), kitna upar ( $F_{3}$ ) point karta hai. Divergence poochta hai: is point ke aas-paas, kya arrow-stuff net bahar ja raha hai (ek tap) ya net aa raha hai (ek drain)? Hum ise measure karte hain

$div F = \frac{\partial F _{1}}{\partial x} + \frac{\partial F _{2}}{\partial y} + \frac{\partial F _{3}}{\partial z} .$

Symbol $\frac{\partial F _{1}}{\partial x}$ (padho "partial of $F_{1}$ with respect to $x$ ") ka matlab hai: east-arrow $F_{1}$ kitni tezi se badhta hai jab tum east ki taraf step karo, $y, z$ ko frozen rakhte hue. Hum sirf matching partial lete hain — $F_{1}$ ke saath $x$ , $F_{2}$ ke saath $y$ , $F_{3}$ ke saath $z$ . Yeh "diagonal" rule hi poora khel hai.

The scenario matrix

Yeh un sabhi case-classes ki poori list hai jo yeh topic tumpar throw kar sakta hai. Neeche har cell ko aage ek worked example se cover kiya gaya hai.

#	Case class	Kya tricky hai	Example
A	Positive constant divergence (source)	ek uniform "tap" padhna	Ex 1
B	Negative constant divergence (sink)	ek "drain" ka sign	Ex 2
C	Zero divergence, bade arrows (rotation)	size ≠ spreading	Ex 3
D	Sign-varying divergence (yahaan positive, wahaan negative)	ek field, kai behaviours	Ex 4
E	Degenerate: uniform field	sabse bade arrows, div $= 0$	Ex 5
F	Cross-term trap ( $\partial F_{2} / \partial x$ )	kaun se partials count karte hain	Ex 6
G	Limiting check: ek real box shrink karo, $V \to 0$ lo	definition = formula	Ex 7
H	Real-world word problem (fluid tap)	units, interpretation	Ex 8
I	Exam twist: woh jagah dhundho jahan $div = 0$	solve karo, sirf compute nahi	Ex 9
J	2D radial with a singularity at the origin	zero everywhere phir bhi ek source	Ex 10

Cell A — positive constant divergence (source)

Cell B — negative constant divergence (sink)

Cell C & E — zero divergence with big arrows

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

Cell D — sign-varying divergence

Cell F — the cross-term trap

Worked example Example 6: partials jo relevant lagte hain par hain nahi

$F = (3 y, 5 x, 7 z)$ . $div F$ compute karo.

Forecast: pehle slot mein $y$ hai aur doosre mein $x$ — unhe differentiate karne ka man karta hai. Answer guess karo, phir check karo ki woh andar jaate hain ya nahi.

$F_{1} = 3 y$ : matching partial hai $\partial (3 y) / \partial x = 0$ ( $3 y$ ke andar koi $x$ nahi). Yeh step kyun? Divergence sirf $\partial F_{1} / \partial x$ use karta hai — $\partial F_{1} / \partial y$ nahi. Cross-partial $\partial (3 y) / \partial y = 3$ curl ka kaam hai, yahaan nahi.
$F_{2} = 5 x$ : matching partial $\partial (5 x) / \partial y = 0$ .
$F_{3} = 7 z$ : matching partial $\partial (7 z) / \partial z = 7$ .
Diagonal sum karo: $0 + 0 + 7 = 7$ .

Matlab: bahut saari cross-coupling ke bawajood, net spreading poori tarah $z$ -part se aati hai.

Verify: sirf $F_{3}$ apne axis ke saath badhta hai; $div = 7$ . (Agar tumne galti se cross-terms $3$ aur $5$ use kiye hote, toh $15$ aata — woh ek curl-flavoured mistake hai.) ✓

Cell G — the limiting definition, checked by hand

Worked example Example 7: ek real box shrink karo aur formula recover karo

$F = (x, 0, 0)$ . Formula kehta hai $div = 1$ . Ise geometric definition (flux per volume) se prove karo, origin par centred side $2 h$ ke cube par.

Forecast: kya actual flux ÷ volume exactly $1$ deta hai, $h$ se independent?

Sirf do $x$ -faces matter karte hain. Top/bottom/front/back faces par outward normal $\pm \overset{y}{^}$ ya $\pm \overset{z}{^}$ hai, aur $F = (x, 0, 0)$ mein zero $y, z$ parts hain, toh $F \cdot \overset{n}{^} = 0$ wahaan. Yeh step kyun? Flux hai $F \cdot \overset{n}{^}$ ; ek field jisme koi $y, z$ components nahi hain woh $y, z$ ki taraf face karne wale faces se koi flux pass nahi karta.
Right face $x = h$ : $F_{1} = h$ , outward normal $+ \overset{x}{^}$ , area $(2 h)^{2} = 4 h^{2}$ . Flux out $= h \cdot 4 h^{2} = 4 h^{3}$ .
Left face $x = - h$ : $F_{1} = - h$ , outward normal $- \overset{x}{^}$ , toh $F \cdot \overset{n}{^} = - (- h) = + h$ . Flux out $= h \cdot 4 h^{2} = 4 h^{3}$ . Yeh step kyun? Left par, field value aur normal dono negative hain — do minus signs outflow ko positive banate hain. Fluid sachchi mein dono faces se bahar jaata hai.
Total flux $= 4 h^{3} + 4 h^{3} = 8 h^{3}$ . Volume $= (2 h)^{3} = 8 h^{3}$ .
Flux per volume $= \frac{8 h ^{3}}{8 h ^{3}} = 1$ , har $h$ ke liye. $h \to 0$ lo: phir bhi $1$ .

Matlab: geometric definition aur formula exactly agree karte hain — figure mein balloon ko shrink hote dekho.

Verify: $8 h^{3} /8 h^{3} = 1 = \partial x / \partial x$ . $h$ se independent, jaise limit demand karti hai. ✓

Cell H — real-world word problem (with units)

Worked example Example 8: a leaky pipe field

Ek tank mein water velocity $v = (0.4 x, 0.4 y, - 0.8 z)$ hai, metres per second mein measure ki gayi, position metres mein. Kya origin par paani create ho raha hai ya destroy, aur per unit volume kitni rate par?

Forecast: $x, y$ mein do positive spreads aur $z$ mein ek squeeze — kya woh cancel karenge?

$\frac{\partial ( 0.4 x )}{\partial x} = 0.4 s^{- 1}$ . Yeh step kyun? Units: velocity (m/s) ko length (m) se differentiate karne par per-second ( $s^{- 1}$ ) milta hai — ek rate, exactly wahi jo "per unit volume per second" ko chahiye.
$\frac{\partial ( 0.4 y )}{\partial y} = 0.4 s^{- 1}$ ; $\frac{\partial ( - 0.8 z )}{\partial z} = - 0.8 s^{- 1}$ .
Sum: $div v = 0.4 + 0.4 - 0.8 = 0 s^{- 1}$ .

Matlab: divergence $0$ har jagah → flow incompressible hai; kuch create ya destroy nahi hota. Sideways spread karna downward squeeze se exactly compensate hota hai. Yeh constant density ke saath Continuity equation hai: $\nabla \cdot v = 0$ .

Verify: $0.4 + 0.4 - 0.8 = 0$ ; units sabhi $s^{- 1}$ , consistent. ✓

Cell I — exam twist: find where divergence vanishes

Cell J — 2D radial with a singularity

Worked example Example 10: woh field jo har jagah source-free hai phir bhi tap ki tarah behave karta hai

Plane mein, $F = (\frac{x}{x ^{2} + y ^{2}}, \frac{y}{x ^{2} + y ^{2}})$ lo (arrows bahar point karte hain, par distance ke saath shrink hote hain — figure dekho). Iska divergence origin se door compute karo, aur origin explain karo.

Forecast: arrows bahar point karte hain, toh tum positive source guess karte. Par woh baahri taraf kamzor hote hain — shayad woh kamzori spreading ko exactly cancel kar deti hai?

$r^{2} = x^{2} + y^{2}$ likho, toh $F_{1} = x / r^{2}$ . Quotient rule use karo $\partial r^{2} / \partial x = 2 x$ ke saath: $\frac{\partial}{\partial x} (\frac{x}{r ^{2}}) = \frac{1 \cdot r ^{2} - x \cdot 2 x}{r ^{4}} = \frac{r ^{2} - 2 x ^{2}}{r ^{4}} = \frac{y ^{2} - x ^{2}}{r ^{4}} .$ Yeh step kyun? $F_{1}$ ek fraction hai jiska top aur bottom dono $x$ rakhte hain; quotient rule handle karta hai "ek ratio kitni tezi se change hota hai."
Symmetry se ( $x \leftrightarrow y$ ): $\frac{\partial}{\partial y} (\frac{y}{r ^{2}}) = \frac{x ^{2} - y ^{2}}{r ^{4}}$ .
Add karo: $div F = \frac{y ^{2} - x ^{2}}{r ^{4}} + \frac{x ^{2} - y ^{2}}{r ^{4}} = 0$ ( $r \neq = 0$ ke liye).

Matlab: origin se door outward spreading exactly arrows ke shrink hone se undo ho jaati hai — divergence $0$ har jagah except ek point. Par origin ke aas-paas kisi bhi circle se total flux ek fixed positive number hai: saara "source" singular origin par concentrated hai, jahan formula break down karta hai (zero se division). Yeh point sources aur Divergence Theorem (Gauss) ke peeche ka flavor hai.

Verify: do fractions exact negatives hain, $r \neq = 0$ ke liye $0$ tak sum hote hain; $(1, 0)$ par check karo: $(0 - 1) /1 + (1 - 0) /1 = 0$ ✓. $(1, 2)$ par: $(4 - 1) /25 + (1 - 4) /25 = 3/25 - 3/25 = 0$ ✓.

Recall Kaun sa cell tumhare liye sabse mushkil tha?

Zero-divergence-with-big-arrows (C/E) aur singular source (J) sabse zyada logon ko trip karte hain. Dono ek hi baat kehte hain: divergence net leaving measure karta hai, arrow size nahi, aur appearance nahi.

Recall

Big arrows wala field phir bhi zero divergence rakh sakta hai — true ya false?

True; e.g.

(100, 0, 0)

ya swirl

(- y, x, 0)

, dono div

= 0

div (2 x, 2 y, 2 z) = ?

2 + 2 + 2 = 6

Kis surface par

(x^{2}, y^{2}, z^{2})

ka zero divergence hai?

Plane

x + y + z = 0

(x / r^{2}, y / r^{2})

ka origin se door zero divergence kyun hai?

Outward spreading exactly arrows ke

1/ r

se shrink hone se cancel ho jaati hai; source singular origin par concentrated hai.

F = (3 y, 5 x, 7 z)

ke liye divergence mein kaun se partials jaate hain?

Sirf diagonal wale

\partial (3 y) / \partial x, \partial (5 x) / \partial y, \partial (7 z) / \partial z = 0 + 0 + 7 = 7

Connections

Divergence — definition, physical meaning (flux density) (parent — idea aur derivation)
Curl — rotation density (jahan Ex 3 & 6 ke cross-terms actually rehte hain)
Divergence Theorem (Gauss) (Ex 7 ka local flux/volume global flux mein turn karta hai; Ex 10 ka origin explain karta hai)
Continuity equation (Ex 8: $\nabla \cdot v = 0$ incompressibility hai)
Flux through a surface (Ex 7 mein hand se compute ki gayi surface integral)
Gradient and the del operator (jahan se $\nabla$ aata hai)
Laplacian ( $\nabla^{2} f = div (grad f)$ )