Clairaut's theorem — mixed partials are equal (under conditions)

4.4.4 · Maths › Multivariable Calculus

Intuition Badi picture (WHY yeh matter karta hai)

Jab aap kisi function $f (x, y)$ ke do partial derivatives lete hain — ek baar $x$ ke respect mein, ek baar $y$ ke respect mein — to aap jiس order mein differentiate karte hain, woh zyaadatar matter nahi karta. Matlab, $f_{x y} = f_{y x}$ . WHY hume yeh suspect karna chahiye? Kyunki ek partial derivative ek local slope measure karta hai, aur ek second mixed partial measure karta hai ki ek direction ka slope doosri direction mein move karne par kitna change hota hai. Geometrically, "jaise main $y$ mein move karta hoon, $x$ -slope kitna twist hota hai" aur "jaise main $x$ mein move karta hoon, $y$ -slope kitna twist hota hai" — yeh dono surface ki same warping measure kar rahe hain. Theorem kehta hai: agar surface kaafi smooth hai, toh yeh dono measurements exactly agree karti hain.

Mixed partial kya hota hai?

Ise first principles se derive karna

Step 1 — Ek symmetric second difference banao. Step sizes $h, k > 0$ fix karo. Define karo $Δ (h, k) = f (a + h, b + k) - f (a + h, b) - f (a, b + k) + f (a, b) .$ Yeh step kyun? Yeh combination mixed second derivative ka discrete analogue hai. Notice karo ki yeh perfectly symmetric hai: $h \leftrightarrow k$ aur $a \leftrightarrow b$ swap karne par (yaani $x$ aur $y$ ke roles swap karne par) $Δ$ unchanged rehta hai. Yahi hidden symmetry poora trick hai.

Step 2 — Ise $y$ -difference ka $x$ -difference ke roop mein group karo. Maano $g (x) = f (x, b + k) - f (x, b)$ . Tab $Δ (h, k) = g (a + h) - g (a) .$ Yeh step kyun? Humne $Δ$ ko $x$ -direction mein $g$ ka difference bana diya, isliye hum $x$ mein Mean Value Theorem apply kar sakte hain.

Step 3 — $x$ mein MVT apply karo. MVT se koi $c_{1}$ hoga $a$ aur $a + h$ ke beech jo satisfy kare $g (a + h) - g (a) = h g^{'} (c_{1}) = h [f_{x} (c_{1}, b + k) - f_{x} (c_{1}, b)] .$ Yeh step kyun? $g^{'} (x) = f_{x} (x, b + k) - f_{x} (x, b)$ , aur MVT ek difference ko derivative times step $h$ mein convert karta hai.

Step 4 — Phir MVT apply karo, ab $y$ mein. Bracket $f_{x} (c_{1}, \cdot)$ ka $y$ -difference hai. MVT se koi $d_{1}$ hoga $b$ aur $b + k$ ke beech jo satisfy kare $f_{x} (c_{1}, b + k) - f_{x} (c_{1}, b) = k f_{x y} (c_{1}, d_{1}) .$ Isliye $Δ (h, k) = h k f_{x y} (c_{1}, d_{1}) .$ Yeh step kyun? $f_{x}$ ko $y$ ke respect mein differentiate karne par $f_{x y}$ milta hai — exactly wahi mixed partial jo hume chahiye.

Step 5 — Steps 2–4 doosre order mein karo. Ab maano $ϕ (y) = f (a + h, y) - f (a, y)$ , toh $Δ (h, k) = ϕ (b + k) - ϕ (b)$ . Same $Δ$ hai. Do MVTs (pehle $y$ mein, phir $x$ mein) dete hain, kisi $c_{2}, d_{2}$ ke liye, $Δ (h, k) = h k f_{y x} (c_{2}, d_{2}) .$ Yeh step kyun? $Δ$ ki symmetry ka matlab hai ki hum ise kisi bhi order mein khol sakte hain.

Step 6 — Equate karo aur limit lo. Dono expressions same $Δ$ ke equal hain, isliye $hk$ se divide karne par: $f_{x y} (c_{1}, d_{1}) = f_{y x} (c_{2}, d_{2}) .$ Jaise $h, k \to 0$ , saare chaar points $(c_{1}, d_{1}), (c_{2}, d_{2}) \to (a, b)$ . Kyunki $f_{x y}$ aur $f_{y x}$ continuous hain, dono sides $(a, b)$ par apni values par converge karti hain: $f_{x y} (a, b) = f_{y x} (a, b) .$ Continuity kyun zaroori hai: points $c_{i}, d_{i}$ unknown hain; hum sirf jaante hain ki woh $(a, b)$ ki taraf squeeze hote hain. Continuity hi "kisi nearby point par value" ko " $(a, b)$ par value" banne deti hai. Iske bina, limit fail ho sakti hai — neeche diya gaya famous counterexample dekho.

Figure — Clairaut's theorem — mixed partials are equal (under conditions)

Worked examples

Counterexample (jab yeh FAIL karta hai)

Recall Feynman: 12-saal ke bacche ko explain karo

Socho ek stretchy trampoline hai. Uspar khade hokar poochho: "Agar main East chalta hoon, toh ground kitni tezi se jhukti hai?" Yeh ek $x$ -slope hai. Ab poochho "aur jaise main North shift karta hoon, woh East-tilt kaise change hoti hai?" Yeh $f_{x y}$ hai — ek twist. Clairaut kehta hai ki jo twist tum measure karte ho (East-then-North) wahi twist hogi jo tum North-then-East measure karte. Yeh aise hai ki trampoline ka corner kitna dented hai yeh same hoga chahे tum kisi bhi taraf se aao — jab tak trampoline smooth ho aur koi sharp creases na hon. Agar koi pinch ya crease hai (ek point jahan slopes jump karte hain), toh dono tarikay agree nahi kar sakte.

Active recall

Clairaut's theorem kya conclude karta hai?

f_{x y} = f_{y x}

— mixed second partials equal hote hain.

Clairaut's theorem ki key HYPOTHESIS kya hai?

Mixed partials point ke aas-paas ek open region mein continuous hote hain.

f_{x y}

mein pehle kaun sa variable differentiate hota hai?

x

(subscripts left to right padhe jaate hain).

Leibniz form

\partial^{2} f / \partial y \partial x

mein pehle kaun sa hota hai?

x

(

f

ke sabse paas wala operator pehle act karta hai).

Clairaut's theorem prove karne ke liye kaun si single symmetric quantity use hoti hai?

Second difference

Δ = f (a + h, b + k) - f (a + h, b) - f (a, b + k) + f (a, b)

Proof ko mixed partials ki continuity kyun chahiye?

MVT points unknown hote hain; continuity unki limit values ko

(a, b)

par value banne deti hai.

Ek aisa function batao jahan origin par mixed partials differ karte hain.

f = x y (x^{2} - y^{2}) / (x^{2} + y^{2})

(aur

0

origin par):

f_{x y} (0, 0) = - 1 \neq = f_{y x} (0, 0) = 1

f_{xx y}

jaisi higher derivatives ke liye value kya determine karta hai?

Sirf har variable ki count (yahan do

x

's, ek

y

), order nahi — smoothness ke under.

f = x^{2} y^{3}

ke liye

f_{x y}

compute karo.

\partial_{y} (2 x y^{3}) = 6 x y^{2}

Proof ko konsa tool do baar drive karta hai?

Mean Value Theorem (ek baar

x

mein, ek baar

y

mein).

Connections

Partial Derivatives — building blocks $f_{x}, f_{y}$ .
Mean Value Theorem — proof ka engine.
Hessian Matrix — Clairaut Hessian ko symmetric banata hai, second-derivative test ke liye key hai.
Exact Differential Equations — exactness condition $\partial_{y} M = \partial_{x} N$ equal mixed partials par rely karti hai.
Continuity of Multivariable Functions — woh hypothesis jo sab kuch work karati hai.
Higher Order Partial Derivatives — jahan iterated Clairaut hume freely reorder karne deta hai.