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

4.4.4 · D1 · Maths › Multivariable Calculus › Clairaut's theorem — mixed partials are equal (under conditi

Isse pehle ki tum yeh sentence padh sako " $f_{x y} = f_{y x}$ jab mixed partials continuous hote hain," tumhare paas pehle se lagbhag ek dozen chhote symbols aur pictures hone chahiye. Yeh page har ek cheez ko scratch se build karta hai, ek aisi order mein jahan har cheez pehle wali pe tikti hai. Kuch bhi assume nahi kiya gaya.

0. Starting object: $f (x, y)$

Jo picture tumhare dimaag mein burn ho jaani chahiye: koi curve nahi, balki ek landscape — pahaadon aur ghatiyon ki ek sheet jo flat $x y$ -floor ke upar hover kar rahi hai.

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

Figure dekho. Neeche ka flat grid input pairs $(x, y)$ ka floor hai. Upar floating coloured sheet surface $z = f (x, y)$ hai. Floor pe koi bhi point lo, seedha upar jao, aur wahan sheet ki height output hai. Yeh ek picture hi hai jis se neeche ke har symbol ka secret connection hai.

Related vault reading: Continuity of Multivariable Functions aur Partial Derivatives is surface ko formally describe karte hain.

1. Symbol $\partial$ — "partial", matlab baaki ko still rakho

Hume naya symbol kyun chahiye? Kyunki do inputs ke saath, "slope" ambiguous hai — kis direction mein slope? $\partial$ symbol ka jawab hai "maine ek direction choose ki aur baaki constant rakh raha hoon," toh question unambiguous ho jaata hai.

Figure dekho. $\partial f / \partial x$ measure karne ke liye hum $y$ ko kisi value $b$ par freeze karte hain. Yeh ek vertical slicing plane hai jo surface ko cut karti hai. Plane surface se ek single curve (magenta) pe milti hai. Us curve ke along sirf $x$ change ho raha hai — toh ab hamare paas ek ordinary one-variable curve hai, aur "slope" ka matlab wahi purani cheez hai.

2. $f_{x}$ aur $\frac{\partial f}{\partial x}$ — pehla partial (ek slope)

Picture: yeh figure 2 ki magenta slice-curve ka slope hai — us curve ka apne point par "rise over run".

$f_{y}$ bilkul yehi idea hai roles swap karke: $x$ freeze karo, North chalo.

Poori detail Partial Derivatives mein hai.

3. Step size $h$ (aur $k$ ) aur symbol $lim$

Picture: floor par length $h$ ka ek shrinking horizontal arrow, aur slice-curve par jo chord banta hai woh tilt hota hai jab tak tangent line nahi ban jaati.

Kyun zaroori hai: Clairaut ka poora proof do step sizes $h$ aur $k$ use karke ek second difference se bana hai, phir dono ko zero ki taraf squeeze karo. "Nudge, phir nudge ko shrink karo" ki pakki pakad ke bina proof padhna impossible hai.

4. Mixed partial $f_{x y}$ — ek twist

Ab topic ka star. Hamare paas $f_{x}$ pehle se hai: East-direction mein ek slope. Lekin $f_{x}$ khud $(x, y)$ ki ek nayi function hai — East-slope badalta hai jab tum surface par ghoomte ho. Toh hum ise dobara differentiate kar sakte hain.

Figure dekho. Near edge par (chhota $y$ ) East-slope arrow gently upar point karta hai. Far edge par (bada $y$ ) East-slope arrow steeper hai. Un do arrows ke beech ka change, per unit of North-travel, $f_{x y}$ hai. Agar do arrows identical hote, $f_{x y} = 0$ — koi twist nahi.

$f_{y x}$ mirror hai: pehle North-slope, phir dekho East jaate waqt kaise badalta hai. Clairaut ka claim hai $f_{x y} = f_{y x}$ .

Derivatives stack karne ke baare mein zyada: Higher Order Partial Derivatives.

5. "Continuous" — admission ki price

Yeh topic isse obsess kyun karta hai: proof twist ko kisi unknown nearby point par produce karta hai aur phir us point ko $(a, b)$ ki taraf slide karta hai. Sirf tab jab twist continuous ho, "nearby point par twist" ban sakta hai " $(a, b)$ par twist." Continuity literally woh hinge hai jis par poora theorem swing karta hai. Dekho Continuity of Multivariable Functions.

6. Mean Value Theorem — proof ka engine

Figure dekho. Orange secant do endpoints ko join karta hai; violet tangent kisi interior point $c$ par curve ko touch karta hai aur secant ke exactly parallel hai. MVT guarantee karta hai ki aisa $c$ exist karta hai.

Topic ko isko kyun chahiye: Clairaut ka proof function ke har difference ko derivative times a step mein convert karta hai, chaar baar ( $x$ mein do baar, $y$ mein do baar). Un conversions mein se har ek MVT ka ek use hai. Deep dive: Mean Value Theorem.

7. Second difference $Δ$ — symmetric gadget

Picture: corners $(a, b)$ , $(a + h, b)$ , $(a, b + k)$ , $(a + h, b + k)$ wala ek chhota rectangle; $Δ$ chaar heights ko signs $+, -, -, +$ ke saath combine karta hai. Crucial feature: $x$ aur $y$ ko swap karne se $Δ$ unchanged rehta hai — yeh symmetric hai. Yehi symmetry hai jis ki wajah se ek number $f_{x y}$ bhi padha ja sakta hai aur $f_{y x}$ bhi. Yeh poore theorem ka beej hai; yeh parent note mein poori tarah se appear karta hai.

Prerequisite map

Map padho: surface $f$ partial derivatives ko feed karta hai; partials plus limits twist $f_{x y}$ banate hain; MVT plus $f$ symmetric $Δ$ banate hain; aur theorem twist, symmetric gadget, aur continuity teeno par ek saath khada hai. Aage, yeh ideas Hessian Matrix aur Exact Differential Equations ko power karte hain.

Equipment checklist

Khud test karo — right side cover karo aur reveal karne se pehle zor se jawab do.

\partial

tumhe kiske baare mein warn karta hai?

Ek se zyada input exist karta hai; differentiate karte waqt chosen one ke alawa sab freeze karo.

Ek simple sentence mein,

f_{x}

kya hai?

Surface ki steepness jab tum

+ x

direction mein chalo,

y

held fixed rakhe.

Slope formula mein seedha

h = 0

kyun nahi set kar sakte?

0/0

milega;

lim_{h \to 0}

ki jagah dekhte hain ki step shrink hone par ratio kahan settle hota hai.

Mixed partial

f_{x y}

physically kya measure karta hai?

East-slope

f_{x}

kaise badalta hai jab tum North chalo — surface ka twist.

f_{x y}

mein pehle kaun sa variable differentiate hota hai?

x

(subscripts left to right padhte hain).

MVT kya convert karta hai, aur kisme?

Ek difference

g (a + h) - g (a)

ko ek derivative times a step,

h g^{'} (c)

, mein.

Theorem mixed partials ki continuity kyun maangta hai?

Proof twist ko ek unknown nearby point par yield karta hai; continuity uska limit exactly

(a, b)

par twist ke equal hone deti hai.

Second difference

Δ

mein kya khaas baat hai?

Yeh symmetric hai —

x

aur

y

swap karne se yeh unchanged rehta hai — toh ek number dono

f_{x y}

aur

f_{y x}

ke equal hota hai.