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

4.4.4 · D2 · Maths › Multivariable Calculus › Clairaut's theorem — mixed partials are equal (under conditi

Step 0 — Woh picture jis par sab kuch tikа hai

Figure dekho. Surface woh pastel sheet hai. Marked point par, coral arrow hai "East chalo" direction aur uski steepness hai $f_x$; lavender arrow hai "North chalo" aur uski steepness hai $f_y$. Slope bas rise over run hai ek tiny step ke liye — ramp ke tilt se zyada mysterious kuch nahi.

Ab show ka star:

Poora theorem yeh claim hai: East-then-North measure kiya gaya twist = North-then-East measure kiya gaya twist. Ab hum ise prove karte hain — algebra par tab tak bharosa nahi karte jab tak picture force na kare.

Step 1 — Ek tiny rectangle banao, charon corner ki heights padho

KAISA DIKHTA HAI: base plane par chaar coloured dots corners hain; surface tak jaate vertical sticks chaar heights hain. Inhe naam do:

$\underset{\text{far corner: East \& North}}{\underbrace{f(a+h,b+k)}}\underset{\text{East only}}{\underbrace{f(a+h,b)}}\underset{\text{North only}}{\underbrace{f(a,b+k)}}\underset{\text{base}}{\underbrace{f(a,b)}}$

Step 2 — Corners ko ek symmetric number $\Delta$ mein combine karo

KAISA DIKHTA HAI: figure mein green $(+)$ corners aur coral $(-)$ corners. Magic property notice karo: East aur North ki roles swap karo (swap $h\leftrightarrow k$, axes swap karo) aur chaar terms sirf reorder ho jaate hain — $\Delta$ unchanged rehta hai. Use pata nahi aur parwah bhi nahi ki hum kaunsi direction ko "pehla" kehte hain. Yahi chupi symmetry proof ka poora engine hai.

Step 3 — Ise ek North-cheez ke East-difference ke roop mein todo

KAISA DIKHTA HAI: figure mein do lavender vertical bars $g(a)$ (left) aur $g(a+h)$ (right) hain. $\Delta$ unki do lengths ka difference hai.

Step 4 — Pehla Mean Value Theorem (East direction mein)

KAISA DIKHTA HAI: figure $c_1$ ko East interval ke andar kahin mark karti hai (hum nahi jaante kahan — ek floating tick ke roop mein "?" ke saath dikhaya gaya hai). Us column par, ab hum $x=c_1$ hold karte hain aur dekhte hain ki East-slope $f_x$ North-height $b$ aur $b+k$ ke beech kaise badli. Woh bracket $f_x$ ka $y$ mein difference hai — dusre MVT ke liye begging kar raha hai.

Step 5 — Doosra Mean Value Theorem (North direction mein) → $f_{xy}$

KAISA DIKHTA HAI: mystery point $(c_1,d_1)$ pastel rectangle ke andar kahin baith a hai — sirf "andar" tak pinned down, dashed box aur ek floating dot se dikhaya gaya. Ye yaad rakho: $\Delta =hk\cdot (\text{wahan East-then-North twist})$.

Step 6 — Usi $\Delta$ ko doosre taraf se todo → $f_{yx}$

KAISA DIKHTA HAI: wahi rectangle, ab doosre taraf se slice ki gayi (horizontal-gap-first). Ek doosra mystery point $(c_2,d_2)$ — generally $(c_1,d_1)$ se alag interior point, lekin box ke andar bhi trapped.

Step 7 — Inhe equal karo aur rectangle shrink karo

KAISA DIKHTA HAI: nested pastel rectangles ka sequence single point $(a,b)$ ki taraf shrinking, dono mystery dots andar ki taraf funnel karte hue. Continuity guarantee karti hai ki twist-values smoothly centre tak ride karti hain bजाय leap karne ke.

Step 8 — Degenerate aur edge cases (kabhi koi scenario skip mat karo)

• Negative steps ($h<0$ ya $k<0$): rectangle West ya South ki taraf extend ho jaata hai. MVT $[a+h,a]$ par equally well kaam karta hai; factor $hk$ correct sign carry karta hai aur identically cancel ho jaata hai. Result unchanged. Picture: box $(a,b)$ ke doosri taraf flip ho jaata hai — same story.
• $h=0$ ya $k=0$ (ek degenerate rectangle): toh $\Delta =0$ trivially hoga (corners ke do pairs coincide ho jaate hain) — koi information nahi, toh hum kabhi zero se divide nahi karte kyunki hum limit $h,k\to 0$ lete hain nonzero values ke through, kabhi zero par nahi.
• Continuity FAIL hoti hai — crease case: agar $f_{xy}$ ya $f_{yx}$ $(a,b)$ par continuous nahi hai, toh Step 7 toot jaata hai. Do hidden points $(a,b)$ ke paas alag paths se aa sakte hain aur alag limiting twist-values par land kar sakte hain. Ye hypothetical nahi hai: $f(x,y)=\{\begin{array}{ll}\frac{xy(x^2-y^2)}{x^2+y^2},& (x,y)\ne (0,0)\\ 0,& (x,y)=(0,0)\end{array}\Rightarrow f_{xy}(0,0)=-1\ne f_{yx}(0,0)=1.$

KAISA DIKHTA HAI: do panels. Left — smooth surface: dono mystery dots same twist value par slide karte hain (curves meet karti hain). Right — creased counterexample surface: do dots alag ridges ke saath aate hain, twist-values $-1$ aur $+1$ par split ho jaati hain. Crease culprit hai; smoothness use forbid karti hai.

Ek-picture summary

Sab kuch ek canvas par: chaar signed corners wali tiny rectangle single symmetric number $\Delta$ feed karti hai. Ise East→North todo aur nikalta hai $hk{f}_{xy}$; ise North→East todo aur nikalta hai $hk{f}_{yx}$. Kyunki ye usi same $\Delta$ hai, $hk$ se divide karo, box shrink karo, aur — provided twist continuous hai — dono $(a,b)$ par identical value par funnel karte hain.

Active recall