Tangent planes and linear approximations to surfaces: Question bank

4.4.5 · D5 · Maths › Multivariable Calculus › Tangent planes and linear approximations to surfaces

Shuru karne se pehle, woh do cheezein yaad karo jinpar sab kuch tika hai:

Recall Formula aur normal (sirf zaroorat padne par dekho)

$(a, b)$ par tangent plane: $z = f (a, b) + f_{x} (a, b) (x - a) + f_{y} (a, b) (y - b)$ , normal vector ke saath $n = ⟨ f_{x}, f_{y}, - 1 ⟩$ . Right-hand side hi linear approximation $L (x, y)$ hai.

Sahi hai ya galat — justify karo

Ek surface jiske kisi point par dono partial derivatives exist karti hain, wahan hamesha tangent plane hoti hai.

Galat. Partials sirf

x

- aur

y

-axes ke saath slope measure karte hain; ek surface dono axes ke saath smooth ho sakti hai lekin diagonally crumpled ho, isliye tumhe full differentiability chahiye, sirf partials ka exist karna kaafi nahi.

Agar

f

(a, b)

par differentiable hai to wahan

f_{x}

aur

f_{y}

exist karte hain.

Sahi. Differentiability zyada strong condition hai; yeh force karti hai ki linear approximation har direction se match kare, jisme khaas taur par woh do axis directions shamil hain jo partials define karti hain.

Agar

f_{x}

aur

f_{y}

(a, b)

ke paas continuous hain, to tangent plane exist karna guaranteed hai.

Sahi. Point ke paas dono partials ki continuity differentiability ke liye ek sufficient condition hai, aur differentiability hi exactly woh cheez hai jo guarantee karti hai ki plane surface se chipki rahe.

Tangent plane aur surface sirf ek point

(a, b, f (a, b))

par touch karte hain.

Saamaanyatah galat. Unhe wahan touch karna aur slope share karna zaroori hai, lekin plane surface ko kahin aur bhi cross ya dobara touch kar sakti hai (jaise ek wavy surface par ek point par tangent plane kahin door wapas use hit kar sakti hai).

Linear approximation

L (x, y)

ek plane hai, isliye yeh sab

(x, y)

ke liye valid hai.

Galat.

L

ek equation ke roop mein har jagah define hai, lekin yeh

f

ko sirf

(a, b)

ke paas hi achhi tarah approximate karti hai; error distance ke square ki tarah badhta hai, isliye door jaane par yeh bekaar ho jaati hai.

Ek smooth surface ke har point par exactly ek tangent plane hoti hai.

Graph $z = f (x, y)$ ke liye sahi jo wahan differentiable ho — dono matched slopes

f_{x}, f_{y}

ek unique non-vertical plane pin karte hain.

Agar

(a, b)

par tangent plane horizontal hai, to wahan dono partials zero hain.

Sahi. Ek horizontal plane ka har direction mein slope zero hota hai, isliye khaas taur par

f_{x} (a, b) = f_{y} (a, b) = 0

— yeh exactly ek critical point hai.

Differential

d z

hamesha true change

Δ z

ke barabar hota hai.

Galat.

d z = f_{x} d x + f_{y} d y

tangent-plane ki prediction hai; actual change

Δ z = f (x, y) - f (a, b)

d z

ke barabar sirf infinitely chhote steps ki limit mein hota hai.

Ek surface par do alag points ek hi tangent plane share kar sakte hain.

Sahi. Jaise ek flat region, ya cylinder, ya ek bump ke symmetric points — sab ke parallel ya identical tangent planes ho sakte hain.

Error dhundho

"Tangent plane:

z = f (a, b) + f_{x} (a, b) x + f_{y} (a, b) y

Slope terms ko base point se displacement se multiply karna chahiye, raw coordinates se nahi: yeh

(x - a)

aur

(y - b)

hona chahiye. Jaise likha hai woh

(a, b, f (a, b))

se bhi nahi guzarta.

z = f (x, y)

ka normal vector

⟨ f_{x}, f_{y}, 1 ⟩

hai."

z

-component

- 1

hai,

+ 1

nahi: surface ko

F = f (x, y) - z = 0

likhne par

\nabla F = ⟨ f_{x}, f_{y}, - 1 ⟩

milta hai. Sign flip orientation ke liye important hai (yeh

z

mein generally neeche ki taraf point karta hai).

"Tangent plane dhundhne ke liye main current point

(x, y)

use karke

A = f_{x} (x, y)

set karta hoon."

Slopes base point par freeze hoti hain:

A = f_{x} (a, b)

, ek number, moving

(x, y)

ka function nahi. Ek plane ke slopes constant hote hain.

f

(a, b)

par continuous hai, isliye wahan differentiable hai aur tangent plane hai."

Continuity bahut weak condition hai; ek cone tip (

x^{2} + y^{2}

at origin) continuous hai lekin uski tangent plane nahi hai. Tumhe partials ka exist karna aur error

f - L

ka distance se tez vanish karna chahiye.

"Kyunki

f = \frac{x y}{x ^{2} + y ^{2}}

(0 se extend kiya gaya) ke dono partials origin par exist karte hain, iski tangent plane

z = 0

hai."

Partials exist karte hain aur zero hain, lekin

f

origin par continuous bhi nahi hai (yeh

y = x

ke saath

\frac{1}{2}

value leta hai), isliye yeh differentiable nahi hai aur iski koi tangent plane nahi — naive

z = 0

ek galat conclusion hai.

x^{2} + y^{2}

ka linear approximation origin par theek hai kyunki function har jagah aur achha behave karta hai."

Origin par yeh surface ek cone point hai — wahan partials exist nahi karte, isliye koi tangent plane aur koi linear approximation us point par exist nahi karta, chahe baaki jagah behavior kaisa bhi ho.

Why questions

Tangent plane mein dono

x

-cross-section tangent line aur

y

-cross-section tangent line kyun honi chahiye?

Kyunki ek common point se guzarne wali do distinct lines ek unique plane determine karti hain; dono partial-slope lines ko match karna exactly woh cheez hai jo plane ko har direction mein surface ka slope share karne par majboor karti hai (directional slopes ki linearity se).

Linear approximation error distance ke square ki tarah kyun badhti hai, linearly kyun nahi?

Differentiability construction se error ke first-order (linear) part ko khatam kar deti hai, isliye remaining leading term second-order hai — curvature se governed — isliye yeh distance squared ki tarah scale karta hai.

Normal paane ke liye hum surface ko level set

F (x, y, z) = f (x, y) - z = 0

ki tarah kyun likhte hain?

Ek level surface par gradient

\nabla F

hamesha uske perpendicular hota hai, isliye

\nabla F = ⟨ f_{x}, f_{y}, - 1 ⟩

seedha normal de deta hai, aur tangent plane woh sab kuch hai jo us normal ke perpendicular ho.

Tangent plane ka idea single-variable tangent line ka 2D twin kyun hai?

Dono ek curved object ko uske best flat local match se replace karte hain: [[Tangent line and linear approximation (single variable)|

y \approx f (a) + f^{'} (a) (x - a)

]] ek slope use karta hai, plane do slopes

f_{x}, f_{y}

use karti hai — same "value plus slope times step" recipe, bas ek extra direction.

Multivariable chain rule tangent-plane/linear-approximation idea par kyun depend karti hai?

Chain rule calculate karti hai ki jab inputs move karte hain to

f

kaise change hota hai; ek point ke paas woh change linear approximation

d z = f_{x} d x + f_{y} d y

se capture hota hai, jo precisely tangent-plane ki predicted change hai.

x

-direction aur

y

-direction mein slope" smoothness guarantee karne ke liye kaafi kyun nahi?

Yeh infinitely many directions mein se sirf do hain; ek function axes ke saath linearly behave kar sakta hai lekin ek diagonal ke saath blow up ho sakta hai, isliye sirf do directions probe karna us failure ko miss kar sakta hai.

Edge cases

Ek surface ka

(a, b)

par sharp cone tip hai (jaise

x^{2} + y^{2}

at origin). Kya tangent plane exist karti hai?

Nahi. Cone tip par slope is baat par depend karta hai ki aap kis direction se approach kar rahe ho, isliye koi single plane sab directions se match nahi kar sakti — wahan partials exist karne mein fail ho jaate hain.

Ek saddle point par dono partials zero hain. Kya tangent plane horizontal hai, aur kya surface uske paas uske neeche rehti hai?

Plane horizontal hai (

f_{x} = f_{y} = 0

), lekin surface ek taraf upar aur doosri taraf neeche jaati hai, isliye yeh ek side par nahi rehti — horizontal tangent plane ka matlab local max ya min nahi hota.

Ek vertical wall jaise

x = c

ko consider karo. Kya ise tangent plane

z = f (a, b) + \dots

ki form mein likha ja sakta hai?

Nahi. Form

z = f (a, b) + A (x - a) + B (y - b)

sirf non-vertical planes produce karta hai (finite slopes); vertical tangent planes is framework se bilkul bahar hain.

Agar

f_{x} (a, b) = f_{y} (a, b) = 0

, kya linear approximation bekaar hai?

Bekaar nahi — yeh correctly predict karti hai

L (x, y) = f (a, b)

, yaani "first order tak flat." Iska bas matlab yeh hai ki interesting behaviour second-order hai (curvature), isliye first-order ek constant estimate deta hai.

Ek genuinely flat surface (khud ek plane) ke liye tangent-plane approximation kitni achhi hai?

Har jagah exact. Ek plane apna khud ka linear approximation hai, error

E

identically zero hai, aur

d z = Δ z

kisi bhi step ke liye, paas ya door.

Base point

(a, b)

par hi linear-approximation error ka kya hota hai?

Yeh exactly zero hai:

L (a, b) = f (a, b)

construction se, kyunki sab displacement terms

(x - a), (y - b)

wahan vanish ho jaate hain.

Connections

Partial derivatives — "sirf do directions mein slope" wale traps ke liye.
Differentiability of multivariable functions — har "partials kaafi nahi" item ki jad.
The gradient vector — normal-vector sign trap.
Directional derivatives — kyun do axis slopes sab directions force karte hain.
Tangent line and linear approximation (single variable) — 1D mirror.
The chain rule (multivariable) — kyun linear-approximation engine aage matter karta hai.