Tangent planes and linear approximations to surfaces: Visual walkthrough

4.4.5 · D2 · Maths › Multivariable Calculus › Tangent planes and linear approximations to surfaces

Shuru karne se pehle: ek chhoti si dictionary, taaki kuch bhi mystery na rahe.

Step 1 — Zoom in karo jab tak pahaad flat na lage

KYA. Koi bhi smooth pahaad lo aur apna camera ek jagah mein zoom karo. Jaise-jaise tum zoom karte ho, curviness pighal jaati hai aur tumhare neeche ka patch ek flat, tilted board jaise dikhne lagta hai.

KYUN. Yahi flat board tangent plane ka poora idea hai. Agar hum is board ki equation dhundh sakein, toh hum ise us jagah ke paas asli pahaad ki sasti substitute ki tarah use kar sakte hain. Board dhundhna hi is page ka poora goal hai — neeche sab kuch sirf iske exact tilt ko pin karne ke baare mein hai.

PICTURE. Left mein surface bulge karti hai. Right mein (zoomed in), wahi patch almost ek plane hai. Red dot hamara base camp $(a, b, f (a, b))$ hai.

Figure — Tangent planes and linear approximations to surfaces

Step 2 — Pahaad ko do taraf se slice karo taaki uske do slopes pata chalein

KYA. Pahaad ko base camp se do vertical walls se kaato:

Wall A: $y = b$ freeze karo, $x$ ko move karne do. Is wall ke saath pahaad ka edge $x$ -direction mein ek curve hai.
Wall B: $x = a$ freeze karo, $y$ ko move karne do. Woh edge $y$ -direction mein ek curve hai.

KYUN. Ek tilted board ke exactly do independent tilts hote hain: left–right aur front–back. Har slice unme se ek reveal karti hai. Slicing ek dara dene wale 3D sawaal ("pahaad kaise tilt karta hai?") ko do friendly 1D sawaalon mein badal deti hai ("is curve ka slope kya hai?"), jinka jawab hum ordinary derivative se pehle se jaante hain.

PICTURE. Blue curve Wall A mein rehti hai ( $x$ mein move hoti hai); green curve Wall B mein rehti hai ( $y$ mein move hoti hai). Dono red base-camp point se gujarti hain.

Step 3 — Har slice ke andar tangent line kheencho

KYA. Har slice mein, slice-curve ko base camp par uski tangent line se replace karo. Blue slice → blue tangent line; green slice → green tangent line.

KYUN. Yeh 1D linear approximation hai jis par hum trust karte hain: $predicted height y \approx height at base f (a) + slope f^{'} (a) step taken (x - a) .$ Har symbol: $f (a)$ woh hai jahan se shuru karte ho, $f^{'} (a)$ woh hai jitni tezi se height per unit step change hoti hai, aur $(x - a)$ base se kitna step liya woh hai. Slope ko step se multiply karo, start mein add karo — tumhe predicted height milti hai. Hum yahi kaam do baar karne wale hain aur results ko jodne wale hain.

PICTURE. Do straight tangent lines, ek per slice, har curve ko exactly red dot par touch karti hain.

Step 4 — Sabse general non-vertical plane likho

KYA. Point $(a, b, z_{0})$ se gujarne wala har non-vertical plane likha ja sakta hai $z = z_{0} + A (x - a) + B (y - b) .$

KYUN. Hum abhi tilts nahi jaante, toh unhe unknown letters $A$ aur $B$ chhod dete hain aur do slices use karke unhe solve karte hain. Har symbol padho:

$z_{0}$ ::: base camp ke seedha upar plane ki height. Jab $x = a$ aur $y = b$ , aakhri do terms zero ho jaate hain aur $z = z_{0}$ .
$A$ ::: plane ka $x$ -direction mein tilt (uska apna $x$ -slope).
$(x - a)$ ::: tum base camp se $x$ mein kitna step le chuke ho.
$B$ ::: plane ka $y$ -direction mein tilt.
$(y - b)$ ::: tum base camp se $y$ mein kitna step le chuke ho.

PICTURE. Floor ke upar float karta ek flat tilted sheet, uska height-above-base $z_{0}$ marked hai, uske do step-arrows $(x - a)$ aur $(y - b)$ floor par drawn hain.

Step 5 — Plane ko hill se touch karaao (height match karo)

KYA. Maango ki plane aur hill base camp par agree karein: $z_{0} = f (a, b)$ set karo.

KYUN. Ek tangent plane ko surface ko touch karna chahiye, uske upar ya neeche float nahi karna chahiye. Exact spot $(a, b)$ par plane ki height $z_{0}$ hai aur hill ki height $f (a, b)$ hai; touch karne ka matlab hai woh equal hain. Toh unknown $z_{0}$ ab known hai.

Ab plane yeh padhti hai $z = f (a, b) + A (x - a) + B (y - b) .$

PICTURE. Floating sheet neeche slide karti hai jab tak uska base-camp point exactly red surface point par nahi baith jaata — contact ban gaya.

Step 6 — $x$ -tilt match karo: $A$ solve karo

KYA. Sirf Wall A ke saath chalo ( $y = b$ freeze karo). Plane par yeh collapse ho jaata hai $z = f (a, b) + A (x - a),$ slope $A$ ki ek straight line. Plane ke liye hill ko hug karne ke liye, yeh blue slice ki tangent ke equal hona chahiye, jiska slope $f_{x} (a, b)$ hai. Slopes match karne par $A = f_{x} (a, b)$ milta hai.

KYUN. Wall A mein $(y - b)$ term zero hai (kyunki $y = b$ ), toh $B$ gayab ho jaata hai aur sirf $A$ bachta hai. Yeh $A$ ko perfectly isolate karta hai — walls hamare liye algebra karte hain. Plane ka $x$ -slope hill ke $x$ -slope ke barabar hona chahiye nahi toh sheet blue curve se peel away ho jaayegi.

PICTURE. Plane, Wall A se slice hoke, ek straight line dikhata hai jo blue tangent line ke upar hai — same steepness $A = f_{x}$ .

Step 7 — $y$ -tilt match karo: $B$ solve karo

KYA. Step 6 ko Wall B ke saath mirror karo ( $x = a$ freeze karo). Plane $z = f (a, b) + B (y - b)$ tak collapse ho jaata hai, slope $B$ , jo green slice ke slope $f_{y} (a, b)$ ke barabar hona chahiye. Toh $B = f_{y} (a, b)$ .

KYUN. Same trick, doosri direction: $x = a$ freeze karna $A$ term ko khatam karta hai, $B$ isolate hota hai. Plane ka front–back tilt hill ke front–back tilt ke barabar hona chahiye.

Dono solved letters substitute karke:

PICTURE. Dono walls ek saath: plane blue tangent (slope $f_{x}$ ) aur green tangent (slope $f_{y}$ ) par hai — poori tarah pin ho gayi.

Slopes $f_{x}, f_{y}$ Partial derivatives se aate hain; vector $⟨ f_{x}, f_{y}, - 1 ⟩$ jo is plane ke perpendicular khada hai woh The gradient vector view hai.

Step 8 — Degenerate aur edge cases (kabhi surprise mat ho)

KYA. Teen "kya agar" situations jinse formula ko survive karna chahiye:

Flat spot (dono tilts zero). Agar $f_{x} (a, b) = f_{y} (a, b) = 0$ — ek summit, ek valley floor, ya ek saddle — toh formula $z = f (a, b)$ ban jaata hai: ek bilkul horizontal plane. Sahi hai: pahaad ki choti par flat sticker level rahta hai.
Vertical tangent (formula exist karne se mana karta hai). Cone $f = x^{2} + y^{2}$ ke liye tip $(0, 0)$ par, slopes blow up karte hain (point sharp hai, smooth nahi). Zoom karne se needle-tip kabhi flat nahi hoti, toh koi tangent plane exist nahi karta — aur hamaara "non-vertical plane" form $z = z_{0} + A (x - a) + B (y - b)$ honestly ek vertical wall represent nahi kar sakta. Method tumhe fail ho kar warn karta hai.
Walls ke saath smooth lekin diagonally wild. Partials exist kar sakte hain phir bhi surface har direction mein flatten nahi hogi (e.g. $f = \frac{x y}{x ^{2} + y ^{2}}$ origin ke paas). Hamaari slices ne sirf do directions check kiye; ek rogue diagonal humein betray kar sakta hai. Isliye plane tab hi genuinely fit hota hai jab $f$ differentiable ho, sirf sliceable nahi.

YEH KYUN DIKHAO. Kyunki ek reader jo sirf happy case dekha ho woh formula ko har jagah trust karega. Har edge case validity ki ek boundary mark karta hai — exact error-shrinks-faster-than-distance test ke liye Differentiability of multivariable functions dekho, aur walls se miss ki gayi diagonal directions ke slopes ke liye Directional derivatives dekho.

PICTURE. Teen mini-panels: ek dome par level cap, spike jisme koi plane nahi, aur ek twisty saddle-ish surface jahan axis-slices theek hain lekin diagonal nahi.

Ek-picture summary

Sab ek saath: hill, base camp, do slices apne tilts $f_{x}$ aur $f_{y}$ ke saath, aur tangent plane dono tangent lines par resting. Kisi bhi nearby floor point ke upar plane ki height linear approximation $L (x, y)$ hai.

Recall Feynman retelling — poora walk plain words mein

Main ek chocolate hill par khada tha aur apni aankhein zoom karta raha jab tak zameen ek flat tilted board jaisi nahi lagi (Step 1). Yeh jaanne ke liye ki board ko kaise tilt karna chahiye, maine hill ko do glass walls se slice kiya: ek front-to-back, ek left-to-right (Step 2). Har slice ne ek curve dikhaya, aur maine har curve ko mere paon par touch karti straight tangent line kheenchi (Step 3). Do crossing lines ek flat sheet fix karti hain. Toh maine sabse general flat sheet likhi, uske do tilts ko blanks $A$ aur $B$ chhod diya (Step 4). Maine sheet ko neeche slide kiya jab tak woh mere paon ke neeche hill ko touch nahi kar li — usne height set kar di (Step 5). Phir maine demand ki sheet bilkul left-right slice jaisi lean kare: usne $A = f_{x}$ fill in kiya (Step 6). Front-back ke liye bhi same: $B = f_{y}$ (Step 7). Jod ke: height, plus har tilt times har step — tangent plane. Finally maine weird spots check kiye: summit par sheet level hai, needle-tip par koi sheet hi nahi hai, aur ek sneaky twisty surface par do slices mujhse jhooth bol sakti hain — yahi exactly woh reason hai kyun "differentiable" (sirf "has slopes" nahi) password hai (Step 8).

Connections

Parent topic — Tangent planes
Partial derivatives — tilts $f_{x}, f_{y}$ yahan se aate hain.
The gradient vector — normal $⟨ f_{x}, f_{y}, - 1 ⟩$ jo hamare plane ke perpendicular hai.
Differentiability of multivariable functions — Step 8 ke peeche ka password.
Tangent line and linear approximation (single variable) — 1D line jise humne do baar joda.
The chain rule (multivariable) — is linear-approximation engine par chalta hai.
Directional derivatives — diagonal directions ke slopes jo hamaari do walls se miss ho gayi.