Parametric surfaces — tangent planes, surface area: Question bank

4.4.30 · D5 · Maths › Multivariable Calculus › Parametric surfaces — tangent planes, surface area

Shuru karne se pehle, wo words aur symbols jo baar baar aate hain, simple language mein:

Definition Quick vocabulary anchor

Parametric map $r (u, v) = ⟨ x (u, v), y (u, v), z (u, v)⟩$ : ek rule jo do numbers $(u, v)$ leta hai aur 3D space mein ek point output karta hai. Jaise jaise $(u, v)$ apni allowed values mein ghoomte hain, output point poori surface trace karta hai — $r$ ko socho jaise wo rubber sheet ko paint kar raha ho, $(u, v)$ uske do colours hain.
Parameter region $D$ : allowed $(u, v)$ values ka flat, 2D region (flat $uv$ -plane ka ek patch). Ye wo "flat blanket" hai pehle jab tak use surface par press nahi kiya jaata; surface ka har point exactly $D$ ke ek point se aata hai.
Tangent vector $r_{u}$ : wo "velocity arrow" jo tumhe milta hai jab tum surface par chalte ho aur sirf parameter $u$ change hota hai. Ye surface ke andar flat rehta hai, jaise rubber sheet par kheencha hua ek arrow.
Normal vector $n = r_{u} \times r_{v}$ : ek arrow jo surface se seedha bahar nikalta hai, surface par bane har arrow ke perpendicular.
Cross product $a \times b$ : ek machine jo do arrows khata hai aur ek teesra arrow ugalta hai, dono ke perpendicular, jiska length un do arrows dwara bane parallelogram ka area hota hai. Dekho Cross Product.
Dot product $a \cdot b = ∣ a ∣∣ b ∣ cos θ$ : ek machine jo do arrows khata hai aur ek single number output karta hai jo batata hai ki wo kitna ek hi direction mein point karte hain (unka alignment). Ye exactly tab zero hota hai jab arrows perpendicular hote hain.

True ya false — justify karo

Map

r (u, v)

sirf batata hai ki surface points kahan hain, isliye surface ki shape hi matter karti hai.

False — surface points ka same set alag alag

r

's se paint kiya ja sakta hai, aur

r_{u}, r_{v}

painting par depend karte hain, lekin surface area

\iint_{D} ∣ r_{u} \times r_{v} ∣ d A

same nikalta hai kyunki wo true geometric spreading measure karta hai.

Agar kisi point par

r_{u} \times r_{v} = 0

ho, to surface wahan flat hai.

False — iska matlab hai parametrization degenerate hai (do grid directions collapse ho gayi ya ek line mein aa gayi), isliye wahan tangent plane defined nahi hai. Ek perfectly smooth sphere par ye poles par hota hai purely coordinates ki wajah se, geometry ki wajah se nahi.

∣ r_{u} \times r_{v} ∣ = ∣ r_{u} ∣ ∣ r_{v} ∣

hamesha hota hai.

False — ye sirf tab hota hai jab do tangent arrows perpendicular hon. Generally

∣ r_{u} \times r_{v} ∣ = ∣ r_{u} ∣∣ r_{v} ∣ sin θ

hota hai, jo chhota hota hai jab bhi grid lines oblique angle par milti hain.

Order badal kar

r_{v} \times r_{u}

karne se compute kiya gaya tangent plane badal jaata hai.

False — isse normal doosri taraf point karta hai, lekin

P_{0}

se guzarney wala plane jo us line ke perpendicular hai, identical hai. Sign sirf orientation/flux ke liye matter karta hai (dekho Surface Integrals and Flux).

Tangent plane ka normal

r_{u} \times r_{v}

aur surface-area factor

∣ r_{u} \times r_{v} ∣

alag alag quantities hain.

False — ye same cross product hain; normal vector hai, area factor uski length hai. Yahi hai "ek cross se dono kaam" wala idea.

Graph

z = f (x, y)

ke liye, surface area

x y

-plane mein uski shadow

D

ke area ke barabar hoti hai.

False — shadow

\iint_{D} d A

hai, lekin tilted surface lambi hoti hai, jo

\iint_{D} f_{x}^{2} + f_{y}^{2} + 1 d A \geq area (D)

deti hai, equality sirf tab jab surface horizontal ho (

f_{x} = f_{y} = 0

Agar surface har jagah smooth hai, to uski parametrization bhi har jagah smooth hai.

False — shape ki smoothness geometric hai, lekin

r_{u} \times r_{v}

phir bhi "coordinate singularities" par vanish ho sakti hai (jaise sphere ke poles par) jahan map degenerate hota hai, jabki surface locally utni hi smooth hai jitni baaki jagah.

Area factor

∣ r_{u} \times r_{v} ∣

ek constant hai jise integral se bahar nikal sakte ho.

Generally False — ye usually

(u, v)

par depend karta hai (e.g. sphere ke liye

R^{2} sin ϕ

). Ye sirf tab bahar nikalta hai jab wo constant ho, jaise flat plane

z = 2 x + 2 y

ke liye.

Error dhundho

"Area

= \iint_{D} r_{u} \cdot r_{v} d A

Dot product alignment measure karta hai (

cos θ

), aur ye exactly tab zero hota hai jab edges perpendicular hon — precisely maximum area ka case. Area ke liye

∣ r_{u} \times r_{v} ∣

chahiye, jiska factor

sin θ

9 0^{\circ}

par peak karta hai.

"Area

= \iint_{D} d A

parameter region par, kyunki

d A

already area hai."

Ye flat

(u, v)

plane

D

mein area hai, curved surface par nahi. Map har tiny square ko stretch karta hai, isliye tumhe

d A

ko local stretch

∣ r_{u} \times r_{v} ∣

se weight karna hoga (Jacobian-type factor, cf. Change of Variables and the Jacobian).

z = f (x, y)

ka normal

⟨ f_{x}, f_{y}, 1 ⟩

hai."

r_{x} \times r_{y}

se sahi normal

⟨ - f_{x}, - f_{y}, 1 ⟩

hai — horizontal components mein minus sign hota hai kyunki surface steepest increase ki direction ke against uthti hai. (Plane equation ke liye koi bhi sign kaam karta hai, lekin vector

⟨ - f_{x}, - f_{y}, 1 ⟩

hai.)

"Sphere ke liye,

∣ r_{ϕ} \times r_{θ} ∣ = R^{2}

, isliye

A = R^{2} \cdot (π) (2 π) = 2 π^{2} R^{2}

Factor

R^{2} sin ϕ

hai,

R^{2}

nahi — ye poles par vanish hota hai aur equator par peak karta hai.

R^{2} sin ϕ

integrate karne par sahi

4 π R^{2}

milta hai.

r_{u}

surface ke perpendicular hai kyunki ye ek derivative hai."

Ulta —

r_{u}

tangent hai: ye ek aisi curve ki velocity hai jo surface par rehti hai (freeze

v

, move

u

), isliye ye surface ke andar flat rehta hai. Dekho Tangent Lines and Velocity Vectors.

"Tangent plane dhundhne ke liye mujhe sirf point

P_{0}

chahiye."

Ek plane ko ek point aur face karne ki ek direction chahiye — normal

n = r_{u} \times r_{v}

n

ke bina infinitely many planes

P_{0}

se guzar sakti hain.

"Ek parametric surface ko ek parameter chahiye, jaise ek parametric curve."

Surface intrinsically two-dimensional hai — tum do independent directions mein ghoom sakte ho — isliye ise do parameters

(u, v)

chahiye. Ek parameter sirf ek curve trace karta hai.

Why questions

Area factor ke liye cross product kyun, dot product kyun nahi?

Area measure karta hai ki do edge-arrows kitna spread apart hain, jo

∣ r_{u} ∣∣ r_{v} ∣ sin θ

hai — exactly cross-product magnitude. Dot product measure karta hai ki wo kitna align karte hain (

cos θ

), jo galat quantity hai.

Δ u

move karne par surface point approximately

r_{u} Δ u

kyun shift hota hai?

Ye linear (tangent-line) approximation hai: ek point ke paas curve

u \mapsto r (u, v_{0})

velocity

r_{u}

se move karta hai, isliye ek chhota step

Δ u

displacement

\approx r_{u} Δ u

produce karta hai, bilkul jaise distance

\approx

speed

\times

time.

Chhota curved patch flat parallelogram ki tarah kyun behave karta hai?

Infinitesimally small

Δ u \times Δ v

par curvature negligible hai aur do edges straight arrows

r_{u} Δ u

r_{v} Δ v

hain. Do arrows ek parallelogram fence off karte hain, jiska area ek cross-product magnitude hai — isliye

∣ r_{u} \times r_{v} ∣

appear karta hai.

Area ke liye cross product ki magnitude kyun lete hain lekin tangent plane ke liye vector kyun rakhte hain?

Area ek size hai (ek non-negative number), isliye sirf length chahiye; plane ko perpendicular hone ki ek direction chahiye, isliye pura vector rakhte hain. Same cross product, do alag pieces of information use hoti hain.

"Smooth" (regular) point ke liye

r_{u} \times r_{v} \neq = 0

kyun zaroori hai?

Agar ye zero hai, to do tangent arrows parallel hain (ya ek vanish ho gaya), isliye wo genuinely 2D plane span nahi karte — wahan koi well-defined tangent plane nahi hai aur koi reliable area factor nahi hai.

Surface-area factor Jacobian wale idea jaisa kyun hai change of variables mein?

Dono answer karte hain "ye map tiny areas ko kitna stretch karta hai?" Jacobian

∣ det J ∣

flat regions ke beech maps ke liye karta hai;

∣ r_{u} \times r_{v} ∣

flat parameter region

D

se curved surface par map ke liye karta hai. Dekho Change of Variables and the Jacobian.

Edge cases

Sphere ke poles

ϕ = 0, π

par

∣ r_{ϕ} \times r_{θ} ∣ = R^{2} sin ϕ

ka kya hota hai?

0

ho jaata hai kyunki

sin (0) = sin (π) = 0

— wahan

θ

-circle ek point mein shrink ho jaata hai, isliye wo meridians collapse ho jaate hain. Ye ek coordinate singularity hai, sphere mein koi hole nahi, aur ye integral mein harmlessly zero contribute karta hai.

Agar do grid curves

0^{\circ}

angle par milti hain (tangent arrows parallel), to patch area kya hai?

Zero, kyunki

sin 0 = 0

: parallel arrows ek degenerate parallelogram fence off karte hain jisme koi area nahi. Parametrization wahan do independent directions mein spread out karne mein fail ho gayi hai.

Perfectly flat surface jaise

z = 2 x + 2 y

ke liye area factor constant kyun hai?

f_{x} = f_{y} = 2

constants hain, isliye

f_{x}^{2} + f_{y}^{2} + 1 = 9 = 3

har jagah; tilt kabhi nahi badlti. Ek constant factor shadow ke area ko sirf multiply karta hai,

3 \cdot area (D)

deta hai.

Area factor kya hota hai jab surface exactly flat

x y

-plane,

z = 0

, ho?

f_{x} = f_{y} = 0

0 + 0 + 1 = 1

milta hai, isliye surface area exactly shadow area ke barabar hoti hai — koi stretching nahi, jo ye sanity check hai jo formula ko pass karna chahiye.

Agar tum parameter direction reverse karo (

u \to - u

), kya computed area badlti hai?

Nahi —

r_{u}

sign flip karta hai, lekin

∣ r_{u} \times r_{v} ∣

magnitude use karta hai, isliye area unchanged rehti hai. Sirf orientation-sensitive quantities jaise flux flip feel karenge.

Ek akela tangent vector surface direction ke baare mein kya batata hai — kya wo plane banane ke liye kaafi hai?

Nahi — ek arrow surface mein sirf ek direction deta hai. Tumhe do non-parallel tangent arrows chahiye jo full 2D tangent plane span kare aur unke cross product se normal define kare.

Recall Poore trap set ka ek-line summary

Tangent arrows surface ke andar rehte hain; unka cross product normal (vector) bhi hai aur area stretch (uski length) bhi; area ke liye magnitude use karo, planes ke liye vector rakho, aur coordinate singularities se savdhan raho jahan cross product harmlessly vanish ho jaata hai.