Foundations — Differentiability in multiple variables

4.4.6 · D1 · Maths › Multivariable Calculus › Differentiability in multiple variables

Parent note padhne se pehle, tumhe uski language ka har piece ground-up se banana hoga. Hum symbol by symbol chalte hain, har ek apni jagah ek plain-words meaning aur ek picture ke saath earn karta hai, ek aisi order mein jahan har step apne pehle wale par tikta hai.

1. Points aur plane: $(x,y)$ aur ${\mathbb{R}}^2$

• Plain words: $\mathbb{R}$ number line hai; ${\mathbb{R}}^2$ do number lines hain jo right angles par cross karti hain.
• Picture: ek floor jo grid se tile ki gayi ho; har crossing ek point $(x,y)$ hai.
• Topic ko yeh kyun chahiye: humaari function is floor ke upar rehti hai. Input hamesha floor par ek point hota hai.

2. Function $f:{\mathbb{R}}^2\to \mathbb{R}$ aur uska graph (ek surface)

• Plain words: floor par ek jagah daalo, wahan ki zameen kitni ounchi hai woh wapas milo.
• Picture: har floor-point ke upar $f(x,y)$ length ka ek vertical pole rakho; saare poles ki tips ek uleha-pulha surface banati hain — $f$ ka graph.
• Topic ko yeh kyun chahiye: "differentiability" is surface ke baare mein ek sawaal hai — kya yeh smooth hai, ya secretly tooti hui hai?

Height ko aksar $z$ kehte hain, toh $z=f(x,y)$.

Dekho Continuity in Multiple Variables ke liye ki "surface mein koi achanak jumps nahi" ka precisely kya matlab hai.

3. Limit arrow $\lim$ aur $h\to 0$

• Plain words: "jaise step tiny hoti jaati hai, yeh kahan ja raha hai?"
• Picture: ek slider $h$ $0$ ki taraf slide kar raha hai; tum output number dekhte ho aur padhte ho ki woh kahan land karta hai.
• Yeh tool kyun, sirf $0$ plug karne ki jagah? Hamare kaafi expressions $\frac{0}{0}$ hote hain jab tum $h=0$ set karte ho — meaningless. Limit humein forbidden division ki jagah trend poochne deti hai.

4. Ek direction mein slope: partial derivative $f_x$, $f_y$

"Kitna steep hai" ki baat karne ke liye hum pehle 1-D slope ki idea yaad karte hain, phir use ek axis ke saath aim karte hain.

Ab $y$ ko fixed rakho aur sirf East chalo ($x$ change karo):

• Picture: surface ko $(a,b)$ se guzarti East–West vertical wall se slice karo. Kati hui edge ek curve hai; $f_x$ uska slope hai.
• $f_y$ same idea hai, $y$ wiggle karte hue (North walk karte hue), North–South wall se slice karte hue.
• Topic ko yeh kyun chahiye: yeh sirf do hi slopes hain jo partials dete hain — aur parent note ki poori warning yahi hai ki do slopes kaafi nahi hain.

Full treatment: Partial Derivatives.

5. Vectors: $\mathbf{h}$, $\mathbf{a}$, aur bold-face convention

• Plain words: $\mathbf{h}$ hai "main base point se kitna door aur kis direction mein gaya".
• Picture: ek amber arrow $\mathbf{a}$ se start hota hai aur naye spot $\mathbf{a}+\mathbf{h}$ ki taraf point karta hai.
• Topic ko yeh kyun chahiye: differentiability har step $\mathbf{h}$ ke liye hold karni chahiye — sirf East aur North nahi, balki har diagonal. Vector hi woh tarika hai jisse hum "ek saath har direction" kehte hain.

6. Ek step ki length: $\Vert \mathbf{h}\Vert$

• Plain words: $\mathbf{a}$ se $\mathbf{a}+\mathbf{h}$ tak ruler-distance.
• Picture: arrow $\mathbf{h}$ ek right triangle ki hypotenuse hai jiske legs $h_1$ aur $h_2$ hain.
• Yeh tool kyun? Differentiability ki definition error ko $\Vert \mathbf{h}\Vert$ se divide karti hai. "Tum kitna chale" se divide karne ke liye pehle woh distance measure karni hogi — exactly yahi norm karta hai.

7. Gradient $\nabla f$ aur dot product $\cdot$

• Plain words: $\nabla f\cdot \mathbf{h}$ = "predicted height-change agar tum step $\mathbf{h}$ lo" = (East-slope × step ka East-part) + (North-slope × step ka North-part).
• Picture: tilted flat board (tangent plane) step $\mathbf{h}$ par exactly itna upar uthti hai.
• Topic ko yeh kyun chahiye: yeh dot product hi tilted plane ka height-change hai. Poora game yahi hai: kya real surface itni hi amount se change hoti hai, plus kuch negligibly small?

Aur: Gradient and Directional Derivatives.

8. "Negligibly small": little-o, $o(\Vert \mathbf{h}\Vert )$

• Plain words: sirf "error zero ho jaati hai" nahi, balki "error itni choti hai jo tum jitna chale uske compare mein kuch bhi nahi".
• Picture: do curves $\Vert \mathbf{h}\Vert \to 0$ ke saath zero ki taraf dive karti hain — error straight line $\Vert \mathbf{h}\Vert$ ke neeche dive karti hai aur waheen rehti hai.
• Yeh tool kyun, sirf $E\to 0$ nahi? Har continuous function mein $E\to 0$ hota hai. Yeh bahut weak hai — yeh ek genuine tangent plane ko point par graze karne wale kisi bhi plane se alag nahi kar sakta. $\Vert \mathbf{h}\Vert$ se divide karna ek sharper test hai jo tangency pin down karta hai.

9. Sentence ko saath mein rakhna

Har symbol earn hone ke baad, parent ka master formula ab plain English mein padha jaata hai:

$f(\mathbf{a}+\mathbf{h})=\underset{\text{start height}}{\underbrace{f(\mathbf{a})}}+\underset{\text{tilted-plane change}}{\underbrace{\nabla f(\mathbf{a})\cdot \mathbf{h}}}+\underset{\text{negligible error}}{\underbrace{o(\Vert \mathbf{h}\Vert )}}$

"Nayi height = purani height + flat tilted board kya predict karta hai + ek leftover itna tiny ki count nahi karta." Woh leftover itna tiny hona — step length se chota — differentiability ki poori definition hai.

Aage yeh Tangent Planes and Linear Approximation, Chain Rule (Multivariable), Total Derivative / Jacobian, aur C1 and Smooth Functions ki notion ko feed karta hai.

Prerequisite map

Equipment checklist