Vector fields — definition, visualization: Question bank

4.4.23 · D5 · Maths › Multivariable Calculus › Vector fields — definition, visualization

Shuru karne se pehle, ek vocabulary reminder taaki koi symbol unseen na rahe:

True or false — justify

A vector field is the same object as a single vector.

False. Ek single vector ek arrow hai; ek field ek rule hai jo har point pe alag arrow de sakta hai. Single vector sirf constant field ka special case hai.

F (x, y) = (3, 3)

ek vector field hai.

True. Constant fields bhi fields hoti hain — rule bas har jagah same arrow

(3, 3)

output karta hai. Input aur output ki same dimension bhi hold hoti hai.

Har function

R^{2} \to R^{3}

ko unke points pe attached arrows ki tarah draw kiya ja sakta hai.

False. Yeh ek valid function hai, lekin "arrow rooted at its point" wali picture tab hi cleanly kaam karti hai jab input aur output ka dimension same ho, kyunki 2D plane ke point mein 3D arrow fit nahi hota.

Agar do fields ka

∣ F ∣

har point pe same ho, toh woh same field hain.

False. Magnitude direction ko ignore karta hai; jaise

(x, y)

aur

(- y, x)

dono ka magnitude

r

har jagah same hai, phir bhi ek radial hai aur doosra rotational.

Field

F (x, y) = (- y, x)

position vector

(x, y)

ki same direction mein point karta hai.

False. Inka dot product

- y x + x y = 0

hai, isliye

F

radius ke perpendicular hai — circle ke tangent pe, uske saath nahi (dekho Divergence and Curl).

Koi bhi radial-looking field automatically ek gradient (conservative) field hoti hai.

False.

(x, y) = \nabla (\frac{1}{2} (x^{2} + y^{2}))

conservative hai, lekin sirf "outward point karna" guarantee nahi karta; conservativeness ek condition hai components pe, jo Conservative Fields and Potential Functions ke zariye test hoti hai.

Agar

F = \nabla f

ho, toh saare arrows reverse karne ke baad bhi gradient field milti hai.

True.

- F = \nabla (- f)

, toh reversed field

- f

ka gradient hai — phir bhi conservative.

Ek constant nonzero field ek gradient field ho sakti hai.

True.

(a, b) = \nabla (a x + b y)

, toh uniform fields linear scalar functions ke gradients hoti hain.

Drawing mein lambe arrows ka matlab hamesha yeh hota hai ki wahan field "zyada important" hai.

False. Length magnitude (strength) encode karta hai, importance nahi; aur same-length plots mein hum deliberately length drop karke color use karte hain, toh length meaningless ho sakti hai.

Spot the error

F (x, y) = (x, y)

ka magnitude

x + y

hai, kyunki components add karte hain."

Galat: magnitude

x^{2} + y^{2} = r

hai,

x + y

nahi. Tum squares ke sum ka square root lete ho, kabhi bhi raw sum nahi (jo negative bhi ho sakta hai).

"Origin pe rotation field

(- y, x)

phir bhi kisi direction mein point karta hai, jaise upar."

(0, 0)

pe field

(- 0, 0) = (0, 0)

hai — zero vector, jiska koi direction nahi hota. Origin ek degenerate point hai jahan arrow vanish ho jaata hai.

"Inverse-square field

c r /∣ r ∣^{3}

space mein har jagah defined hai."

Yeh

r = 0

pe blow up karta hai jahan

∣ r ∣ = 0

division undefined banata hai. Origin ko domain

D

se exclude karna padega.

F /∣ F ∣

hamesha direction extract karne ka safe tarika hai."

Yeh wahan fail karta hai jahan

∣ F ∣ = 0

ho (division by zero), jaise radial ya rotational field ka center. Zero-magnitude points ka koi unit direction nahi hota.

"Kyunki

\nabla (x^{2} + y^{2}) = (2 x, 2 y)

aur

(x, y)

same direction mein point karte hain, toh woh literally equal fields hain."

Har point pe same direction, lekin alag magnitude:

(2 x, 2 y)

do guna lamba hai. Equal direction ka matlab equal vector nahi hota.

"Jis field ke saare arrows same length ke hon, woh necessarily constant field hai."

Nahi — same-length style mein draw ki gayi rotation field ke drawn arrows barabar hote hain lekin woh har point pe alag directions mein point karte hain. Equal magnitude

\neq =

equal vector.

"Kyunki gravity 'andar kheenchti hai', isliye inverse-square field mein

c > 0

hona chahiye."

Andar kheenchne ke liye arrow ko

r

ke opposite hona chahiye, toh

c < 0

; positive

c

outward push karega (jaise ek repulsive charge Flux and the Divergence Theorem mein).

Why questions

F

ke output dimension ka input dimension ke barabar hona arrow picture ke liye kyun zaroori hai?

Taaki output arrow ko input point pe same space mein attach kiya ja sake — plane-point ko plane-arrow chahiye, warna "arrow rooted here" meaningless hai.

Hum aksar saare arrows same length draw kyun karte hain aur color use kyun karte hain?

Kyunki strong regions ke paas lambe arrows overlap karke pattern chupa dete hain; length fix rakhne se direction readable rehti hai aur magnitude color pe shift ho jaata hai.

Zero dot product yeh prove karne ke liye kaise kaafi hai ki rotation field "pure swirl" hai?

F \cdot (x, y) = 0

ka matlab hai ki

F

radius ke perpendicular hai, isliye us point se guzarne wale circle ka tangent hai, toh flow center ki taraf aaye ya jaaye bina circulate karta hai.

Kisi moving curve ke saath velocity naturally vector field kyun hoti hai?

Path ke har point pe ek location bhi hai aur motion ki ek direction+speed bhi — exactly "arrow at every point" wala data — dekho Parametric Curves and Velocity.

Inverse-square form mein

r /∣ r ∣^{2}

ki jagah

r /∣ r ∣^{3}

kyun use kiya jaata hai?

r /∣ r ∣

unit outward direction hai; use

∣ r ∣^{2}

se divide karne se magnitude

\propto 1/ r^{2}

milti hai, aur dono divisions combine karne se denominator mein

∣ r ∣^{3}

aata hai.

Do alag scalar functions kabhi genuinely alag gradient fields kyun nahi de sakte agar woh sirf ek constant se differ karein?

\nabla (f + C) = \nabla f

kyunki constant ka derivative zero hota hai; constant add karna "height" shift karta hai lekin slope nahi, toh arrow field identical rehti hai.

∣ F ∣ = P^{2} + Q^{2}

hai na ki

∣ P ∣ + ∣ Q ∣

— kyun?

Yeh legs

P

aur

Q

wale arrow ki Pythagorean length hai; straight-line arrow length squares aur square root use karti hai, absolute values ka sum nahi (jo ek alag, "taxicab" distance measure karta hai).

Edge cases

Jahan

F = 0

ho, wahan drawn arrow kaisa dikhta hai?

Woh ek point tak simat jaata hai — zero length, undefined direction. Yeh field ke critical/equilibrium points hote hain (radial aur rotational fields ke centers).

Origin se bahut door jaane pe radial field

(x, y)

ka kya hota hai?

Iska magnitude

r = x^{2} + y^{2}

bina bound ke badhta hai, toh arrows lambe se lambe hote jaate hain — field infinity pe unbounded hai.

Inverse-square field

∣ F ∣ = ∣ c ∣/ r^{2}

ka kya hoga jab

r \to 0^{+}

aur jab

r \to \infty

Jab

r \to 0^{+}

toh yeh infinity tak diverge karta hai (source pe singular); jab

r \to \infty

toh yeh zero tak decay karta hai — pass mein strong, door mein negligible.

Kya ek aisi domain pe defined field jo mein ek hole ho (jaise origin minus

R^{2}

) phir bhi valid vector field hai?

Haan —

D

koi bhi region ho sakta hai; bad points ko exclude karna (jahan components undefined hain) exactly waisa hi hai jaise hum inverse-square jaise singular fields handle karte hain.

Kya gradient field mein koi zero vector ho sakta hai?

Haan — jahan bhi

\nabla f = 0

ho, yaani

f

ke maximum, minimum, ya saddle pe; jaise

\nabla (x^{2} + y^{2}) = (2 x, 2 y)

origin pe vanish karta hai, jo minimum hai.

Positive

x

-axis pe jahan

y = 0

ho, rotation field

(- y, x)

kya output karta hai?

(- 0, x) = (0, x)

, ek purely vertical arrow (upar jab

x > 0

, neeche jab

x < 0

) — axis ke perpendicular, jo axes pe bhi tangent-to-circle behaviour confirm karta hai.

Recall Ek-line summary

Ek field arrows ka rule hai, na ki ek arrow; magnitude Pythagorean hai; zero-vector points pe direction khatam ho jaati hai; aur singular fields simply bad points apne domain se drop kar deti hain.

Connections

Gradient and Directional Derivatives — kaun si fields gradients hain.
Divergence and Curl — traps ke peeche swirl vs. spread diagnostics.
Conservative Fields and Potential Functions — "kya yeh $\nabla f$ hai?" ka real test.
Flux and the Divergence Theorem — $c$ ka sign aur outward/inward flow.
Parametric Curves and Velocity — path ke saath velocity as a field.
Line Integrals — arrows reverse karna work ka sign flip karta hai.