Parametric surfaces — tangent planes, surface area

4.4.30 · Maths › Multivariable Calculus

Ek surface 3D mein ek "rubber sheet" ki tarah hoti hai jo space mein rehti hai. Isse cleanly describe karne ke liye hum isko do coordinates $(u, v)$ se paint karte hain. Jab yeh map mil jaata hai, tangent planes aur area cross products se seedhe nikal aate hain.

1. Parametric Surface kya hoti hai?

Do parameters kyun? Ek surface intrinsically 2-dimensional hoti hai — tum uske upar do independent directions mein ghoom sakte ho. Curves ko 1 parameter chahiye ( $r (t)$ ); surfaces ko 2.

Figure — Parametric surfaces — tangent planes, surface area

2. Do Fundamental Tangent Vectors

$r$ ko har ek parameter ke respect mein differentiate karo, doosre ko fixed rakhte hue:

$r_{u} = \frac{\partial r}{\partial u} = ⟨ \frac{\partial x}{\partial u}, \frac{\partial y}{\partial u}, \frac{\partial z}{\partial u} ⟩, r_{v} = \frac{\partial r}{\partial v} = ⟨ \frac{\partial x}{\partial v}, \frac{\partial y}{\partial v}, \frac{\partial z}{\partial v} ⟩$

$r_{u}$ == $u$ -curve ka tangent== hai ( $u$ -direction mein move karne par velocity).
$r_{v}$ == $v$ -curve ka tangent== hai.

Kyun? $r_{u}$ basically curve $u \mapsto r (u, v_{0})$ ki velocity hai, isliye yeh us curve ke along point karta hai — yaani yeh surface ke andar lie karta hai.

3. Tangent Plane — Scratch se Derivation

HUM KYA CHAHTE HAIN: woh flat plane jo surface ko $P_{0} = r (u_{0}, v_{0})$ pe best hug kare.

KAISE: Ek plane ek point aur ek normal vector $n$ se fix hoti hai. Hamare paas point $P_{0}$ hai. Hume ek aisa $n$ chahiye jo surface ke perpendicular ho — yaani dono tangent directions $r_{u}$ aur $r_{v}$ ke perpendicular ho. Cross product exactly wahi tool hai jo do given vectors ke orthogonal ek vector produce karta hai:

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

4. Surface Area — First Principles se Derivation

KYA: Surface ka total area jab $(u, v)$ , $D$ pe range karta hai.

KAISE (chote patches se banao): $D$ ko $Δ u \times Δ v$ size ke chote rectangles mein kaato. Aisa ek rectangle surface pe ek curvy patch mein map hota hai. Uske do edges approximately yeh displacement vectors hain:

$r_{u} Δ u and r_{v} Δ v$

Kyun? $u$ mein $Δ u$ move karne se position $\approx r_{u} Δ u$ shift hoti hai (linear approximation / total differential).

Chota patch approximately ek parallelogram hota hai jo un do edge vectors se span hota hai. Vectors $a, b$ se span hone wale parallelogram ka area $∣ a \times b ∣$ hota hai. Toh:

$Δ S \approx ∣ r_{u} Δ u \times r_{v} Δ v ∣ = ∣ r_{u} \times r_{v} ∣ Δ u Δ v$

Sum karke aur limit lene se ek integral milta hai:

Special case: graph $z = f (x, y)$

$r (x, y) = ⟨ x, y, f (x, y)⟩$ se parametrize karo. Tab $r_{x} = ⟨ 1, 0, f_{x} ⟩$ , $r_{y} = ⟨ 0, 1, f_{y} ⟩$ , aur $r_{x} \times r_{y} = ⟨ - f_{x}, - f_{y}, 1 ⟩ \Rightarrow ∣ r_{x} \times r_{y} ∣ = f_{x}^{2} + f_{y}^{2} + 1$ $A = \iint_{D} f_{x}^{2} + f_{y}^{2} + 1 d A$

5. Worked Examples

6. Common Mistakes

Recall Feynman: ek 12-saal ke bacche ko samjhao

Socho tum ek bumpy hill ko ek stretchy grid blanket se dhak rahe ho jo chote squares se bani hai. Flat zameen pe har square same size ka hota hai, lekin jab tum isse hill pe dabaate ho toh squares khich jaate hain aur tilt ho jaate hain, isliye har ek zyada (ya kam) hill cover karta hai. Yeh pata karne ke liye ki kitni hill hai, tum measure karte ho ki har chota square kitna stretch hua aur sab ko add kar dete ho. Har square ke liye "stretch number" uske do chote arrows leke nikala jaata hai — dekhte hain ki woh kitna area fence karte hain — yahi cross product hai. Woh flat plane jo hill ko ek point pe just touch karti hai, jaise ek kaan ka sheeshe ka tukda uske upar resting ho, woh tangent plane hai, aur uska perpendicular direction usi cross product se milta hai.

7. Flashcards

Ek surface ko kitne parameters describe karte hain, aur kyun?

Do — ek surface intrinsically 2-dimensional hoti hai; tum uske upar do independent directions mein move kar sakte ho.

r_{u}

aur

r_{v}

geometrically kya hain?

u

-curve aur

v

-curve ke tangent vectors; yeh surface mein lie karte hain aur tangent plane span karte hain.

Ek parametric surface ka normal vector kya hota hai?

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

(dono tangent vectors ke perpendicular).

Tangent plane equation batao.

a (x - x_{0}) + b (y - y_{0}) + c (z - z_{0}) = 0

jahan

⟨ a, b, c ⟩ = r_{u} \times r_{v}

us point pe.

Surface area formula aur uske integrand ka matlab batao.

A = \iint_{D} ∣ r_{u} \times r_{v} ∣ d A

; integrand local area-stretch factor hai.

Area ke liye cross product kyun (dot nahi)?

Area = parallelogram spread =

∣ r_{u} ∣∣ r_{v} ∣ sin θ

, jo cross-product magnitude deta hai; dot alignment deta hai.

Graph

z = f (x, y)

ke liye surface area formula?

A = \iint_{D} f_{x}^{2} + f_{y}^{2} + 1 d A

Ek surface ke kisi point pe smooth hone ki condition kya hai?

Waahan

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

hona chahiye.

Parametrization se radius

R

ke sphere ka surface area?

\int_{0}^{2 π} \int_{0}^{π} R^{2} sin ϕ d ϕ d θ = 4 π R^{2}

8. Connections

Cross Product — normal vector aur area factor dono ka source.
Tangent Lines and Velocity Vectors — 1-parameter analogue ( $r^{'} (t)$ ).
Double Integrals — woh machinery jo tiny patch areas ko sum karti hai.
Change of Variables and the Jacobian — $∣ r_{u} \times r_{v} ∣$ surface-version Jacobian hai.
Surface Integrals and Flux — $r_{u} \times r_{v}$ sign ke saath use karta hai (orientation).
Arc Length — 1D cousin: $\int ∣ r^{'} (t) ∣ d t$ vs $\iint ∣ r_{u} \times r_{v} ∣ d A$ .