Parametric surfaces — tangent planes, surface area: Foundations

4.4.30 · D1 · Maths › Multivariable Calculus › Parametric surfaces — tangent planes, surface area

Is page par yeh assume kiya gaya hai ki aapne Parametric surfaces — tangent planes, surface area mein di gayi koi bhi notation pehle nahi dekhi. Hum har symbol ko ground up se banayenge, ek aisi tartib mein jahan har piece sirf usse pehle ke pieces pe rely kare.

Recall Do rozmarra ke tools jinhe hum use karenge (quick review)

X–y–z axes. Teen number lines, sab ek corner (the origin) pe milti hain aur sab ek doosre se right angle par hain. Pehli, $x$ , right taraf jaati hai; doosri, $y$ , aage jaati hai; teesri, $z$ , upar jaati hai. Space mein koi bhi jagah har axis par kuch amount chalke pahuncha ja sakta hai. Bas yahi "3D coordinates" ka matlab hai. Ek angle $θ$ ka Sine aur cosine. Ek right-angled triangle banao. Uske kisi ek non-right angle $θ$ ke liye: $cos θ$ = ( $θ$ ke paas wali side) ÷ (sabse lambi side), aur $sin θ$ = ( $θ$ ke samne wali side) ÷ (sabse lambi side). Do facts jo hum use karte hain: $cos 0^{\circ} = 1$ aur $sin 0^{\circ} = 0$ (ek bilkul chapti triangle), jabki $cos 9 0^{\circ} = 0$ aur $sin 9 0^{\circ} = 1$ (ek poori tarah "khula hua" right angle). $cos θ$ ko "kitna aligned" do directions hain aur $sin θ$ ko "kitna door-door" — yahi intuition hame neeche chahiye.

0. 3D mein ek point aur ise likhne ka tarika

Sabse pehle: space mein ek point ko teen numbers chahiye — kitna right ( $x$ ), kitna aage ( $y$ ), kitna upar ( $z$ ).

Picture: origin (jahan teen axes milte hain, woh corner) se shuru karo aur pehle axis par $x$ chalo, doosre par $y$ , teesre par $z$ . Tip tumhara point $(x, y, z)$ hai; poora safar-arrow $⟨ x, y, z ⟩$ hai.

Figure — Parametric surfaces — tangent planes, surface area

Figure s01 — 3D mein ek point. Blue segment $x$ -axis ke saath right taraf $x$ steps karta hai, yellow segment $y$ -axis ke saath aage $y$ steps karta hai, green segment $z$ -axis ke saath upar $z$ steps karta hai; red dot finished point $(3, 2, 2.5)$ hai aur origin se ustak ka patla white arrow vector $⟨ 3, 2, 2.5 ⟩$ hai. Alt text: teen colored steps jo ek 3D coordinate box mein ek red point tak pahunchate hain.

Topic ko isko kyun chahiye: ek surface 3D mein rehta hai, isliye uska har point, aur uspar hum jo bhi arrow banate hain, woh in triples mein se ek hai.

1. Parameter plane $(u, v)$ aur map $r$

Yeh ek single most important idea hai jise picture karna hai.

Picture: left par, flat graph paper ka ek tukda — $(u, v)$ plane, ek region jise hum $D$ kehte hain. Right par, 3D space. Map $r$ flat paper ka har dot uthata hai aur use space mein kahin chipka deta hai. Poori sheet, chipakne ke baad, surface ban jaati hai.

Figure s02 — Left: flat parameter region $D$ , blue $u$ -lines aur yellow $v$ -lines se bani grid; har crossing ek input pair $(u, v)$ hai. Right: map $r$ us grid ko 3D mein uthata aur mod-tod karta hai, isliye seedhi blue aur yellow lines surface $S$ par curved lines ban jaati hain. Alt text: left par ek flat colored grid jo right par ek surface par ek wavy colored grid mein badal jaati hai.

Symbol

D

flat parameter paper par woh region jo hum feed kar sakte hain. Symbol

\in

ka matlab hai "belongs to", isliye

(u, v) \in D

ka matlab hai "pair

(u, v)

region

D

ka ek point hai."

2. Grid lines: $u$ -curves aur $v$ -curves

Plain graph paper par do families of lines hoti hain: horizontal wali (fix $v$ , $u$ change karne do) aur vertical wali (fix $u$ , $v$ change karne do). Jab hum paper ko space mein chipkate hain, woh seedhi grid lines surface par curved lines ban jaati hain.

Picture: globe par latitude aur longitude lines. Longitude lines ek family hain, latitude lines doosri; har point ek dono mein se ek ke crossing par baithta hai.

Topic ko isko kyun chahiye: yeh grid curves jis direction mein jaati hain woh do arrows hain jinse poori theory (planes aur area) bani hai. Yahi agla section hai.

3. Tangent arrow — "curve kidhar ja rahi hai?"

Agar tum ek curve par chalo, toh har instant mein tum kisi taraf point kar rahe ho. Woh instantaneous direction-and-speed hi tangent vector hai. Hum yeh notion yahaan, scratch se, build karte hain pehle use karne se.

Definition Tangent / velocity vector — do nearby points se bana

Bead ki position abhi lo, $r (t_{0})$ , aur ek tiny moment baad, $r (t_{0} + h)$ ek chhote step $h$ ke liye. Unke beech ka arrow, $r (t_{0} + h) - r (t_{0})$ , ek chord hai — yeh roughly us taraf point karta hai jis taraf tum ja rahe ho lekin iska length $h$ kitna bada hai uspar depend karta hai. $h$ se divide karo ek fair "per-unit-time" rate paane ke liye, phir imagine karo $h$ zero ki taraf sik ta ja raha hai. Chord ghoomta hai jab tak woh curve ke flat against na ho jaye, seedha aage point karta hua. Woh limiting arrow tangent (velocity) vector hai: $r^{'} (t_{0}) = lim_{h \to 0} \frac{r ( t _{0} + h ) - r ( t _{0} )}{h} .$ Plain words: yeh woh direction hai jisme bead move karta hai, length uski speed ke barabar hai. (Agla section is limiting operation ko derivative naam deta hai.)

Picture: ek bent wire par slide karta bead; woh choti arrow bead par taped, hamesha seedhi aage wire ke saath aimed, kabhi sideways nahi. Woh arrow curve ke flat against leta hai.

Figure s03 — Blue path curve $r (t)$ hai; red dot $t_{0}$ par bead hai; yellow arrow uska tangent (velocity) vector hai, travel ki direction mein path ke saath seedha aimed. Alt text: ek curved blue path par ek bead jiske saath ek yellow arrow curve ke saath point karta hua hai.

Yeh idea Tangent Lines and Velocity Vectors mein aur develop kiya gaya hai, aur aisi ek moving arrow ki length, jodte hue, Arc Length hai — lekin is page ko follow karne ke liye un pages se kuch nahi chahiye. Topic ko isko kyun chahiye: do grid curves mein se har ek ka ek tangent arrow hai, aur woh do arrows tangent plane pin down karenge.

4. Derivative aur partial derivative $\partial$

Woh limit upar — "chord over step, jaise step shrinks to zero" — exactly woh tool hai jise derivative kehte hain.

Ab, hamare map $r (u, v)$ mein do dials hain. Jab hum differentiate karte hain toh batana padta hai hum kaun sa dial nudge kar rahe hain doosre ko hold karte hue. Yahi exactly curly $\partial$ mark karta hai.

$\partial$ kyun aur ordinary $d$ kyun nahi? Plain $\frac{d}{d t}$ ek dial wali function ke liye hai (jaise upar ki curve). Jab do ya zyada hote hain, hume declare karna padta hai kaunsa move karta hai — curly $\partial$ woh declaration hai. Yeh exact sawaal ka jawaab deta hai "sirf $u$ -direction mein change ki rate."

Yahaan ke prerequisites Double Integrals se bhi connect hote hain aur baad mein Change of Variables and the Jacobian se.

5. Do arrows ek plane span karte hain — aur dot product

Do arrows ek hi point se shuru hote hue, jab tak woh ek hi line par na hon, ek flat sheet sweep karte hain: ek plane. $u$ -tangent aur $v$ -tangent exactly yahi karte hain.

Topic ko isko kyun chahiye: "ek point plane mein hai" ko aise phrase kiya jaayega "plane ke andar ka ek arrow normal ke perpendicular hai," aur perpendicular = dot product zero. Woh single equation hi tangent-plane equation hai.

6. Cross product $\times$ — star tool

Yeh woh tool hai jis par poora topic hinge karta hai. Yeh do arrows leta hai aur ek teesra arrow produce karta hai.

Neeche ki picture teen gifts ek saath dikhati hai: flat parallelogram, usse perpendicular uthta arrow, aur $a$ se $b$ tak right-hand twist.

Figure s04 — Blue arrow $a$ aur yellow arrow $b$ green parallelogram fence karte hain; uska area length $∣ a \times b ∣$ ke barabar hai. Red arrow $a \times b$ hai, green sheet ke perpendicular upar uthta hua — iska direction right hand ko $a$ se $b$ ki taraf curl karke set hota hai. Alt text: do arrows ek shaded parallelogram span karte hue jisme ek teesra arrow uspar perpendicular khada hai.

Ise compute karne ke full details Cross Product mein hain. Length bars vs no bars note karo: $r_{u} \times r_{v}$ (no bars) arrow hai — normal ke roop mein use hota hai; $∣ r_{u} \times r_{v} ∣$ (bars ke saath) iska length hai — area factor ke roop mein use hota hai.

7. Catch: jab do arrows collapse ho jaayein (regularity)

Upar sab kuch quietly assume karta tha ki do tangent arrows genuinely ek plane span karte hain. Ho sakta hai woh na karein — aur yeh ek real case hai jiska hume samna karna hai.

8. Flat patch measure karna: $d A = d u d v$ , aur $\iint$ se sum karna

Aakhri symbol: total area paane ke liye hum surface ko countless tiny patches mein katenge, har ek measure karenge, aur sab add karenge. Woh infinite sum ek integral hai.

Topic ko isko kyun chahiye: surface $S$ ka area (§1 ki glued sheet) hai $A (S) = \iint_{D} ∣ r_{u} \times r_{v} ∣ d A = \iint_{D} ∣ r_{u} \times r_{v} ∣ d u d v$ — har symbol ab woh hai jise tum mil chuke ho: $A (S)$ (finished surface $S$ ka area), $\iint$ (patches add karo), $D$ (parameter region), $∣ \times ∣$ (ek stretched patch ka area), $d A = d u d v$ (flat paper par ek tile ka size). Aur §7 ki wajah se integrand wahaan nonzero hai jahan surface regular hai.

Wahi normal arrow $r_{u} \times r_{v}$ , apni sign ke saath rakhte hue, baad mein Surface Integrals and Flux ko power karta hai.

Prerequisite map

Neeche har lettered box is page ki ek idea hai; dash ke baad ka label us section ka naam hai jisme woh rehti hai. Arrows ka matlab hai "feeds into."

Equipment checklist

Khud test karo — right side cover karo aur zor se jawaab do.

Bold

r

kya signal karta hai, ek plain letter ki tulna mein?

Ki object teen numbers ka vector/point hai, ek single number nahi.

Point notation

(x, y, z)

aur arrow notation

⟨ x, y, z ⟩

mein kya fark hai?

Same teen numbers; round brackets ek jagah name karte hain, angle brackets ek displacement arrow name karte hain.

cos θ

aur

sin θ

intuitively kya measure karte hain?

cos θ

= do directions kitne aligned hain (max at

0^{\circ}

);

sin θ

= kitne spread apart (max at

9 0^{\circ}

D

r

, aur

S

kya hain, aur yeh kaise related hain?

D

flat input region hai,

r

gluing rule hai,

S

finished curved surface hai —

S = {r (u, v) : (u, v) \in D}

Surface ko do parameters

u, v

ki kyun zaroorat hai?

Ek surface 2-dimensional hai; tum do independent directions mein ghoom sakte ho, isliye do dials chahiye.

u

-curve kya hai?

Woh path jo tab trace hota hai jab

v

freeze ho aur sirf

u

vary kare.

Ek curve ke liye parameter

t

kya hai?

Single dial (time socho) jo

r (t)

mein feed hota hai; jaise

t

badhta hai point path par chalta hai.

Tangent vector do nearby points se kaise banta hai?

Chord

r (t_{0} + h) - r (t_{0})

lo, step

h

se divide karo, aur

h \to 0

karne do; chord curve ke flat against swing karta hai.

Hum plain

d

ki jagah curly

\partial

kyun use karte hain?

Do dials hain;

\partial

declare karta hai ki hum ek vary kar rahe hain doosre ko freeze karte hue.

r_{u}

geometrically kya hai?

u

-curve ka tangent (velocity) arrow; yeh surface mein flat leta hai.

Dot product kaunsa single number deta hai, aur woh kab zero hota hai?

Alignment measure karne wala ek number; exactly tab zero jab do arrows perpendicular hon.

Cross product

a \times b

ki properties kya hain?

Dono ke perpendicular ek arrow (direction right-hand rule se), jiska length parallelogram ka area hai; aur

a \times b = - b \times a

Regularity condition

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

kya guarantee karta hai, aur kab fail hota hai?

Ek genuine tangent plane aur nonzero area element; fail hota hai jab

r_{u}

r_{v}

zero ho, ya jab woh parallel hon (

sin θ = 0

), jaise globe ke poles.

Cross product ka sign/order baad mein kyun matter karta hai?

Yeh pick karta hai ki surface ka kaunsa side positive normal count hoga — ek orientation — jo flux ke liye essential hai lekin planes aur area ke liye irrelevant hai.

r_{u} \times r_{v}

aur

∣ r_{u} \times r_{v} ∣

mein fark kya hai?

Pehla ek arrow hai (normal); doosra uska length hai (area-stretch factor).

Flat parameter paper par

d A

kya hai, aur use on-surface area mein kya convert karta hai?

d A = d u d v

, ek flat tile; stretch factor

∣ r_{u} \times r_{v} ∣

se multiply karne par on-surface area milti hai.

\iint_{D} d A

tumhe kya karne ka instruction deta hai?

Region

D

ki har tiny tile par sweep karo aur contributions add karo.

Kaunsa tool — dot ya cross — area measure karta hai, aur kyun?

Cross, kyunki area is baare mein hai ki do edges kitni spread hain (

sin θ

), perpendicular hone par sabse bada.