Divergence — definition, physical meaning (flux density): Question bank

4.4.24 · D5 · Maths › Multivariable Calculus › Divergence — definition, physical meaning (flux density)

Traps se pehle, ek chhota sa visual refresher taaki neeche har symbol samajh mein aaye.

Recall Divergence ke do roop (tap karke reveal karo)

Geometric: $div F (P) = V \to 0 lim \frac{1}{V} \iint_{\partial V} F \cdot \overset{n}{^} d S$ — outward flux per tiny volume. Yahan $V$ point $P$ ke aas-paas ek chhota sa blob hai, $\partial V$ uski skin hai, aur $\overset{n}{^}$ (unit-length arrow, "the normal") seedha skin se bahar ki taraf point karta hai. Computational: $div F = \nabla \cdot F = \frac{\partial F _{1}}{\partial x} + \frac{\partial F _{2}}{\partial y} + \frac{\partial F _{3}}{\partial z}$ — sirf diagonal partial derivatives ka sum, ek single number.

Definition ke peeche ki picture

Neeche wale tiny cube ko dekho. Yeh $(x, y, z)$ par baitha hai, sides $Δ x, Δ y, Δ z$ hain, isliye iska volume $V = Δ x Δ y Δ z$ hai. Iske chhon faces mein se har ek ka outward normal $\overset{n}{^}$ (red arrows) seedha bahar ki taraf poking karta hai. Ek face se flux $F \cdot \overset{n}{^}$ times face ka area hota hai — flow ka sirf woh part jo normal ke saath hai, bahar nikalta hai.

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

Formula ko cube se derive karna ("why", step by step)

Sirf $x$ -axis ke perpendicular do faces par focus karo aur neeche wala figure dekho. Right face (at $x + Δ x$ ) ka normal $+ \overset{x}{^}$ hai; left face (at $x$ ) ka normal $- \overset{x}{^}$ hai.

Step 1 — har face se kya nikalti hai. Right face par $F \cdot \overset{n}{^} = + F_{1} (x + Δ x, y, z)$ ; left face par $F \cdot \overset{n}{^} = - F_{1} (x, y, z)$ (woh minus pura trap hai — outward normal ulti direction mein point karta hai). Har ek ko face area $Δ y Δ z$ se multiply karo aur add karo: $Flux_{x} \approx [F_{1} (x + Δ x, y, z) - F_{1} (x, y, z)] Δ y Δ z .$

Step 2 — yeh disguise mein derivative kyun hai. Bracket ek step $Δ x$ par $F_{1}$ ka difference hai. $Δ x$ se multiply aur divide karo: $Flux_{x} \approx \frac{F _{1} ( x + Δ x , y , z ) - F _{1} ( x , y , z )}{Δ x} = V Δ x Δ y Δ z Δ x \to 0 \frac{\partial F _{1}}{\partial x} V .$

Step 3 — baaki do axis-pairs ke liye bhi same karo. Bilkul isi reasoning se $y$ -faces $\frac{\partial F _{2}}{\partial y} V$ dete hain aur $z$ -faces $\frac{\partial F _{3}}{\partial z} V$ dete hain.

Step 4 — chhhon faces ko sum karo, $V$ se divide karo. Total flux $\approx (\frac{\partial F _{1}}{\partial x} + \frac{\partial F _{2}}{\partial y} + \frac{\partial F _{3}}{\partial z}) V$ , isliye $\frac{flux}{V} \to \frac{\partial F _{1}}{\partial x} + \frac{\partial F _{2}}{\partial y} + \frac{\partial F _{3}}{\partial z} = div F .$ Geometric definition ne diagonal-partials formula ko force karke banaya — aur upar wali shape-independence ki wajah se, woh formula divergence hai chahe tumne koi bhi blob imagine kiya ho.

True or false — justify karo

Har answer mein ek reason hona chahiye, sirf T/F nahi.

F = (5, 0, 0)

(ek uniform field, saare arrows equal) ki divergence zero hai.

True. Utna hi fluid left face se andar aata hai jitna right face se bahar jaata hai; kuch bhi create nahi hota.

\partial (5) / \partial x = 0

, isliye div

= 0

— bade arrows, koi spreading nahi.

Agar

F

ka magnitude har jagah large hai, toh uski divergence bhi large honi chahiye.

False. Divergence flow ke change ko measure karta hai ek point par, flow ki size ko nahi. Swirl

(- y, x, 0)

ke arrows door tak bahut bade hain phir bhi divergence exactly

0

hai.

Divergence ek vector hai kyunki

\nabla

aur

F

dono vectors hain.

False. Operation dot product

\nabla \cdot F

hai, jo do vectors ko collapse karke ek number banata hai. Vector-in, scalar-out. (Curl cross product use karta hai aur woh ek vector return karta hai.)

Ek field ka ek point par positive divergence aur doosre point par negative divergence ho sakta hai.

True. Divergence ek point-by-point scalar field hai.

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

ke liye,

div = 2 x

x < 0

ke liye negative hai (wahan ek sink) aur

x > 0

ke liye positive hai (wahan ek source).

Term

\partial F_{2} / \partial x

divergence mein contribute karta hai.

False. Sirf matching-index (diagonal) partials appear karte hain:

\partial F_{1} / \partial x

\partial F_{2} / \partial y

\partial F_{3} / \partial z

. Cross-terms jaise

\partial F_{2} / \partial x

rotation describe karte hain aur curl mein rehte hain, divergence mein nahi.

Agar ek fluid incompressible hai (uski density unchanged carry hoti hai), toh

div v = 0

har jagah hoga.

True — incompressible flow ke liye general mein, steady ya unsteady. Incompressibility ka matlab hai har fluid parcel apna volume rakhta hai, yaani har point par har instant "in equals out": exactly

div v = 0

. Yeh divergence-free condition hai jo Continuity equation ke peeche hai.

Divergence aur curl ek field ki usi measurement ke do naam hain.

False. Divergence (dot product) net outflow per volume measure karta hai; curl (cross product) rotation per area measure karta hai. Swirl

(- y, x, 0)

ki divergence zero hai lekin Curl — rotation density nonzero hai.

div F

ki units

F

ki units se alag hoti hain.

True. Divergence flux-per-volume hai: tum field ki units ko ek length se divide karte ho (

\partial / \partial x

se). Agar

F

ek velocity (m/s) hai, toh div per-second (1/s) hai.

Ek field jo har jagah radially inward point karti hai uski divergence hamesha negative hoti hai.

Zyaadaatar true, lekin field check karo. Inward-pointing

(- x, - y, - z)

- 3 < 0

deta hai, ek sink. Lekin inverse-square inward field origin se door zero ho sakti hai — sirf "inward pointing" sign fix nahi karta; rate of change karta hai.

div F = 0

ka matlab koi fluid move nahi kar raha.

False. Iska matlab hai us point par koi fluid create ya destroy nahi ho raha — flow vigorous ho sakta hai. Zero divergence "jitna bahar jaata hai utna andar aata hai" hai, "kuch nahi hota" nahi.

Error dhundo

Har line mein ek flawed statement ya step hai. Correction reveal karo.

div (x, y, z)

ek vector

(1, 1, 1)

hai."

Error: add karne se pehle ruk gaye. Divergence diagonal partials ko ek number mein sum karta hai:

1 + 1 + 1 = 3

, ek scalar. Tumne terms list ki lekin dot product kabhi liya hi nahi.

div F = \frac{\partial F _{1}}{\partial x} + \frac{\partial F _{1}}{\partial y} + \frac{\partial F _{1}}{\partial z}

Error: teeno terms

F_{1}

use karti hain. Yeh hona chahiye

\partial F_{1} / \partial x + \partial F_{2} / \partial y + \partial F_{3} / \partial z

— component

i

variable

i

se differentiated, us face se match karta hua jis par flux cross karta hai.

"Swirl

(- y, x, 0)

ke liye:

\partial (- y) / \partial x = - 1

, isliye divergence negative hai."

Error:

- y

mein koi

x

nahi hai, isliye

\partial (- y) / \partial x = 0

- 1

nahi. Isi tarah

\partial (x) / \partial y = 0

. Divergence

0

hai;

- 1

aur

+ 1

jo actually aate hain woh curl mein belong karte hain.

"Left face par flux

+ F_{1} Δ y Δ z

hai."

Error: left face par outward normal

- \overset{x}{^}

direction mein point karta hai, isliye

F \cdot \overset{n}{^} = - F_{1}

. Wahan flux

- F_{1} Δ y Δ z

hai; is sign ko bhool jaana puri derivation ko kharab kar deta hai (yeh upar wala Step 1 minus sign hai).

"Divergence Theorem kehta hai

\iint_{\partial V} F \cdot \overset{n}{^} d S = div F

Error: right side volume par divergence ka integral hona chahiye:

∭_{V} div F d V

. Divergence Theorem (Gauss) dekho — ek global flux summed local flux densities ke barabar hai, kisi single value ke nahi.

"Kyunki flux ek surface integral hai, divergence bhi ek surface par measure honi chahiye."

Error: divergence flux per unit volume zero volume ki limit mein hai — ek point quantity. Surface integral sirf numerator ke roop mein appear karta hai

V

se divide karne aur shrink karne se pehle.

"Laplacian

\nabla^{2} f

ek vector hai kyunki yeh divergence se aata hai."

Error:

\nabla^{2} f = div (grad f)

ek vector field ki divergence leta hai, isliye output ek scalar hai. Laplacian dekho.

Why questions

Divergence dot product kyun use karta hai plain multiplication ki jagah?

Kyunki ek face se flux

F \cdot \overset{n}{^}

hai — sirf outward normal ke saath field ka component fluid ko push karta hai bahar. Dot product exactly woh projection extract karta hai; doosri directions face ke saath slide ho jaati hain aur kuch bahar nahi le jaatein.

Sirf ek axis ke saath derivative (across nahi) us axis ke faces ke liye kyun count karti hai?

x

-faces flow ke

x

-component se cross hoti hain; wahan net outflow hai "right par

x

-flow left se kitni tezi hai" — yeh

\partial F_{1} / \partial x

hai.

y

mein move karte waqt

F_{1}

mein change flow ko face ke saath sideways shift karta hai, bahar nikalne mein kuch add nahi karta.

Divergence scalar kyun hai jabki curl vector hai?

Net spreading ki koi direction nahi hoti — yeh "out minus in" ka ek single balance hai, isliye ek number kaafi hai. Rotation ek axis ke baare mein hoti hai, isliye tumhe ek direction (axis) aur ek rate dono chahiye — isliye Curl — rotation density ke liye ek vector.

Definition mein volume ko zero tak kyun shrink karte hain?

Ek single point

P

par ek local rate paane ke liye, kisi chunk par total ki jagah.

V

se divide karna total flux ko flux density mein badalta hai;

V \to 0

lena koi bhi averaging hata deta hai taaki number point ka ho, blob ka nahi.

Formula derive karne ke liye cube use kyun kar sakte hain jab definition koi bhi shape allow karti hai?

Kyunki smooth fields ke liye limit shape-independent hai: koi bhi chhota region size

V

ka deta hai (wohi local rate)

\times V

plus corrections jo

V

se tezi se khatam ho jaate hain. Isliye cube ka answer ball ke answer ke barabar hai — hum sirf sabse aasaan faces wali shape pick karte hain.

\nabla

gradient se aata hai phir bhi divergence kaise banaata hai?

\nabla = (\partial / \partial x, \partial / \partial y, \partial / \partial z)

ek "derivative slots ka vector" hai. Ek scalar par apply karne par yeh slots bharta hai (gradient); ek vector field ke saath dot karne par yeh har slot ko matching component ke saath pair karta hai (divergence). Same operator, alag multiplication — Gradient and the del operator dekho.

Do fields jinke arrows ek point par identical hain phir bhi alag divergences kyun ho sakti hain?

Divergence neighbourhood par depend karta hai, single arrow par nahi. Yeh padhta hai ki field point ke aas-paas kaise change ho rahi hai, isliye do fields jo

P

par agree karti hain lekin nearby differ karti hain, divergence mein disagree karengi.

Positive

\partial F_{1} / \partial x

x

-direction mein net outflow kyun mean karta hai?

Iska matlab hai outgoing (right-face) flow

F_{1} (x + Δ x)

incoming (left-face) flow

F_{1} (x)

se zyada hai.

x

ke saath zyada bahar jaata hai andar aane se, isliye us direction mein fluid sourced ho raha hai.

Edge cases

Zero field

F = (0, 0, 0)

ki divergence kya hai?

Exactly

0

. Saare partials

0

hain; kuch flow nahi hota, isliye kuch create ya destroy nahi hota — trivial "no source" case.

Kya divergence us point par define ho sakti hai jahan field blow up karti hai, jaise

F = (x, y, z) / r^{3}

ka origin?

Ordinary formula se nahi — field origin par undefined hai. Origin se door uski divergence exactly

0

hai, phir bhi saara source origin par concentrated hai: distributional sense mein

\nabla \cdot (r / r^{3}) = 4 π δ (r)

(ek Dirac delta spike), jo smooth pointwise formula nahi dekh sakta. Yeh ek point charge ke field ke peeche ki mathematics hai.

Agar ek box ke right aur left faces se equal outward flow hai, toh divergence mein

x

-contribution kya hai?

Zero. Example:

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

x = 0

— dono faces equally leak karte hain, isliye

\partial F_{1} / \partial x = 2 x = 0

wahan. Symmetric outflow koi net source nahi deta.

Har point par zero divergence tumhe kisi bhi closed surface se total flux ke baare mein kya batata hai?

Woh zero hai, Divergence Theorem (Gauss) se:

\iint F \cdot \overset{n}{^} d S = ∭ 0 d V = 0

. Jo kuch bhi ek closed region mein andar aata hai use bahar bhi jaana chahiye.

Ek source region (div

> 0

) aur sink region (div

< 0

) ke beech boundary par divergence ka kya hota hai?

Yeh smoothly

0

se guzarta hai (ek continuous field ke liye): ek surface hoti hai jahan creation exactly destruction ko balance karta hai. E.g.

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

zero cross karta hai

x = 0

par.

Kya divergence 2D field ke liye define hoti hai, aur kya box argument abhi bhi kaam karta hai?

Haan. 2D mein ek tiny rectangle use karo:

div F = \partial F_{1} / \partial x + \partial F_{2} / \partial y

, aur "flux per volume" rectangle ke chaar edges se "flux per area" ban jaata hai — wohi derivation ek kam axis ke saath.

Shrinking volume

V

ki shape divergence ki value badal deti hai?

Nahi. Ek smooth field ke liye limit shape-independent hai — ek cube, ball, ya ellipsoid sab wohi number dete hain — jo exactly isliye hai ki definition "

V \to 0

" shape mein unspecified chhod sakti hai.

Connections

Divergence — definition, physical meaning (flux density) (woh parent jise yeh bank drill karta hai)
Curl — rotation density (cross-product sibling jo vector-vs-scalar traps pakadta hai)
Divergence Theorem (Gauss) (har "total flux" edge case ke peeche)
Gradient and the del operator (kyun $\nabla$ grad aur div dono banata hai)
Flux through a surface (woh integral jo definition mein average ho raha hai)
Continuity equation (incompressibility = zero divergence)
Laplacian (gradient ki divergence — ek scalar)