Conservative vector fields — potential functions

4.4.26 · Maths › Multivariable Calculus

WHY does path-independence follow? (Scratch se derive karo)

Maano $F = \nabla f$ . Koi bhi smooth path $r (t)$ , $t \in [a, b]$ , point $A$ se $B$ tak lo. Line integral (work) hai $\int_{C} F \cdot d r = \int_{a}^{b} \nabla f (r (t)) \cdot r^{'} (t) d t .$

Yeh step kyun? $d r = r^{'} (t) d t$ line integral ki definition se.

Ab integrand ko pehchano. Multivariable chain rule se, $\frac{d}{d t} f (r (t)) = \nabla f (r (t)) \cdot r^{'} (t) .$

Yeh step kyun? Yeh exactly $f (x (t), y (t), z (t))$ ke liye chain rule hai — curve ke saath change ki rate gradient aur velocity ka dot product hai. Yahi poora trick hai.

Toh integrand ek perfect derivative hai, aur Fundamental Theorem of Calculus deta hai: $\int_{C} \nabla f \cdot d r = f (r (b)) - f (r (a)) = f (B) - f (A) .$

Figure — Conservative vector fields — potential functions

HOW to test if a field is conservative

HOW to recover the potential $f$ (worked method)

Strategy: $P$ ko $x$ mein integrate karo, phir differentiate karke bachi hui $y$ ki function ko fix karo.

Forecast-then-Verify

Recall Compute karne se pehle forecast karo

Q: Kya $F = (y, - x)$ conservative hai? Pehle predict karo, phir check karo. Forecast: "Rotational-looking lag raha hai, shayad nahi." Verify: $P_{y} = 1$ , $Q_{x} = - 1$ . Equal nahi ⟹ conservative nahi. Unit circle ke around loop integral $= - 2 π \neq = 0$ . Forecast confirm ho gaya.

Common mistakes (Steel-manned)

Flashcards

F

ke conservative hone ka matlab kya hai?

Ek scalar potential

f

exist karta hai jaise ki

F = \nabla f

Fundamental Theorem of Line Integrals ka statement?

\int_{C} \nabla f \cdot d r = f (B) - f (A)

— sirf endpoints pe depend karta hai.

Conservative field ka loop integral?

\oint_{C} F \cdot d r = 0

Conservativeness ke liye 2D test?

\partial P / \partial y = \partial Q / \partial x

(simply connected domain pe).

Cross-partial test kaam kyun karta hai?

Kyunki

P_{y} = f_{x y}

Q_{x} = f_{y x}

, aur Clairaut's theorem deta hai

f_{x y} = f_{y x}

"Simply connected" kyun required hai?

Holes curl-free fields ko nonzero loop integrals ke saath allow karte hain (jaise

(- y, x) / (x^{2} + y^{2})

F = (P, Q)

f

recover karne ka method?

P

x

mein integrate karo

f = \int P d x + g (y)

pane ke liye, phir

f_{y} = Q

set karo

g

solve karne ke liye.

3D mein conservative hone ki condition?

curl F = 0

, yaani

P_{y} = Q_{x}, P_{z} = R_{x}, Q_{z} = R_{y}

Kya

(y, - x)

conservative hai?

Nahi;

P_{y} = 1 \neq = - 1 = Q_{x}

Recall Feynman: ek 12-saal ke bachche ko samjhao

Ek pahadi park imagine karo. Har jagah ek arrow hai jo bata raha hai ki ball kidhar roll karegi — hamesha neeche ki taraf. Woh arrow map ek gradient field hai, aur pahadi ka height map potential function hai. Yahan cool part hai: agar tum bench se swings tak jaate ho, toh tumhe jo total "downhill help" milti hai woh sirf bench aur swings ke height difference pe depend karti hai — is baat pe nahi ki tumne lamba winding raasta liya ya chhota wala. Aur agar tum loop mein chalo aur usi bench pe wapas aao, toh tum same height pe ho, isliye net help zero hai. Aisi tarah kaam karne wale fields "conservative" hain. Aisi fields jaise ek spinning whirlpool, jahan tum chakkar lagaate raho aur energy lete raho, woh NOT conservative hain.

Connections

Gradient and directional derivative — potential ka gradient hi field hai.
Line integrals of vector fields — jo FTLI evaluate karta hai.
Curl of a vector field — $0$ -curl criterion.
Clairaut's theorem (equality of mixed partials) — test ka engine.
Green's theorem — simply-connected subtlety explain karta hai.
Conservation of energy in physics — physical meaning.

Conservative vector fields — potential functions

WHAT is a conservative field?

WHY does path-independence follow? (Scratch se derive karo)

HOW to test if a field is conservative

HOW to recover the potential $f$ (worked method)

Equivalences ki poori chain

Forecast-then-Verify

Common mistakes (Steel-manned)

Flashcards

Connections

Concept Map

Deep Dive