Suppose F=∇f. Take any smooth path r(t), t∈[a,b], from point A to B. The line integral (work) is
∫CF⋅dr=∫ab∇f(r(t))⋅r′(t)dt.
Why this step?dr=r′(t)dt by definition of a line integral.
Now recognize the integrand. By the multivariable chain rule,
dtdf(r(t))=∇f(r(t))⋅r′(t).
Why this step? This is exactly the chain rule for f(x(t),y(t),z(t)) — the rate of change along the curve is the gradient dotted with velocity. This is the whole trick.
So the integrand is a perfect derivative, and the Fundamental Theorem of Calculus gives:
∫C∇f⋅dr=f(r(b))−f(r(a))=f(B)−f(A).
Q: Is F=(y,−x) conservative? Predict, then check.
Forecast: "Rotational-looking, probably not."
Verify:Py=1, Qx=−1. Not equal ⟹ not conservative. Loop integral around unit circle =−2π=0. Forecast confirmed.
Because Py=fxy, Qx=fyx, and Clairaut's theorem gives fxy=fyx.
Why is "simply connected" required?
Holes allow curl-free fields with nonzero loop integrals (e.g. (−y,x)/(x2+y2)).
Method to recover f from F=(P,Q)?
Integrate P in x to get f=∫Pdx+g(y), then set fy=Q to solve for g.
3D condition for conservative?
curlF=0, i.e. Py=Qx,Pz=Rx,Qz=Ry.
Is (y,−x) conservative?
No; Py=1=−1=Qx.
Recall Feynman: explain to a 12-year-old
Imagine a hilly park. At every spot there's an arrow telling you which way a ball would roll — always downhill. That arrow map is a gradient field, and the hill's height map is the potential function. Here's the cool part: if you walk from a bench to the swings, the total "downhill help" you get depends ONLY on the height difference between bench and swings — not on whether you took the long winding path or the short one. And if you walk in a loop back to the same bench, you end up at the same height, so the net help is zero. Fields that work this way are "conservative." Fields like a spinning whirlpool, where you can keep gaining energy by going round and round, are NOT conservative.
Dekho, ek vector field ka matlab hai har point pe ek arrow — jaise wind ka map ya force field. Kuch fields special hote hain: woh kisi scalar functionf ke gradient hote hain. Yeh f ek "pahaadi ki height map" jaisa samjho, aur arrows hamesha downhill (ya uphill) point karte hain. Aise fields ko conservative bolte hain, aur woh hidden scalar f ko potential function kehte hain.
Kyun important hai? Kyunki conservative field me line integral (yaani work) sirf start aur end point pe depend karta hai — beech ka raasta matter hi nahi karta. Formula: ∫C∇f⋅dr=f(B)−f(A). Isliye agar tum loop maar ke wapas same point pe aa jaao, total work zero ho jaata hai. Yeh "conservation of energy" ka maths version hai.
Check kaise karein? 2D me simple test: agar ∂P/∂y=∂Q/∂x ho (aur domain me koi hole na ho, yaani simply connected ho), to field conservative hai. Yeh test actually Clairaut's theorem se aata hai — mixed partials equal hote hain. Lekin dhyaan rakho: agar domain me hole ho (jaise origin missing), to test fail kar sakta hai — woh famous (−y,x)/(x2+y2) wala example.
Potential f nikalne ka recipe yaad rakho: pehle P ko x me integrate karo, g(y) leftover aayega; phir uss ko y me differentiate kar ke Q ke saath match karo, g(y) fix ho jaayega. Mnemonic: "Curl zero, find the hill; integrate P, then fix the frill." Practice karo do-teen examples, concept bilkul clear ho jaayega.
