4.4.5 · D1Tangent planes and linear approximations to surfaces
Ek smooth surface, paas se dekho toh ek flat tilted sheet jaisi lagti hai — aur woh sheet bilkul describe ho jaati hai sirf ek point par uski height se aur do perpendicular directions mein kitni steeply chadhti hai. Yeh poora topic uss flat sheet ko equation mein likhne aur usse nearby heights guess karne ke baare mein hai.
Yeh page kuch bhi assume nahi karta. Parent note Tangent Planes padhne se pehle, usmein use hone wala har symbol yahan ground up se build kiya gaya hai, ek aisi order mein jahan har ek sirf pehle waale par depend karta hai.
Sab kuch three-dimensional space mein hota hai. Hum ek point ko teen numbers se label karte hain.
( x , y , z )
x right (ya left agar negative),y forward (ya back),z up (ya down).
Picture yeh hai : teen number-lines (axes) right angles par origin ( 0 , 0 , 0 ) x y ground banate hain; z
Topic ko yeh kyun chahiye: ek surface iss space mein float karti ek shape hai, aur tangent plane usi space mein ek aur shape hai. Axes ke bina hum nahi keh sakte kuch bhi kahan hai.
f ( x , y ) f do inputs x y ek number return karta hai — ek height. Hum "f ( x , y ) f x y
Picture yeh hai : ground par spot ( x , y ) f ( x , y )
Worked example Notation padhnaAgar f ( x , y ) = x 2 + y 2 f ( 1 , 2 ) x = 1 y = 2 1 2 + 2 2 = 5
Topic ko yeh kyun chahiye: surface hai hi woh function. Poora game ek spot ke paas f
Intuition Ek rule se ek shape takAgar hum har ground spot ( x , y ) ( x , y , f ( x , y )) surface banate hain — ek curved sheet ground ke upar hover karti hui. z = f ( x , y ) z
Figure mein bowl z = x 2 + y 2 ( a , b ) surface point ( a , b , f ( a , b ))
( a , b ) a b fixed numbers hain: woh particular ground spot jahan hum apni flat sheet banate hain. x y variable rehte hain — woh ( a , b )
x − a y − b x − a y − b step hai ( a , b ) ( x , y )
Picture yeh hai : ground par fixed dot ( a , b ) ( x , y ) x − a y − b
x x − a step x − a x x − a = 0 f ( a , b )
Topic ko yeh kyun chahiye: "linear approximation" matlab base par height plus slope × step . Step precisely ( x − a , y − b )
Do dimensions se pehle, Tangent line and linear approximation (single variable) se ek-dimensional idea yaad karo: curve y = f ( x ) slope uski tangent line ka "rise over run" hai — jaise tum right step karte ho height kitni tezi se badalti hai.
Intuition Kyun chahiye humein surface par slope ke liye ektool
Surface par tum infinitely many directions mein chal sakte ho, har ek ki apni steepness hoti hai. Sabse clever trick yeh hai ki ek variable ko freeze karo taaki surface ek plain 1D curve ban jaaye — phir ordinary slope use karo. Woh frozen-slice slope ek partial derivative hai.
Definition Partial derivativesf x f y
f x ( a , b ) y = b fixed rakho, sirf x "surface kitni steeply chadhti hai jab main right step karta hoon?" f y ( a , b ) x = a y "Kitni steeply jab main forward step karta hoon?"
Picture yeh hai : surface ko ek vertical wall se slice karo. x f x y f y
Worked example Partial compute karnaf ( x , y ) = x 2 + y 2 f x y constant maan kar sirf x f x = 2 x x f y = 2 y ( 1 , 2 ) f x = 2 f y = 4
Topic ko yeh kyun chahiye: tangent plane ko bilkul surface jaisi tilt karni chahiye. Uske do tilts hain hi f x f y Partial derivatives mein hai.
Definition Ek non-vertical planeEk flat, tilted sheet jismein koi fold nahi. Aisi koi bhi sheet likhi ja sakti hai
z = z 0 + A ( x − a ) + B ( y − b ) z 0 A B
Picture yeh hai : cardboard ki ek stiff sheet ( a , b ) z 0 A B
Symbols padhna: base point par x − a = 0 y − b = 0 z = z 0 z 0 x − a = 1 z A A hai hi rightward slope.
Topic ko yeh kyun chahiye: yeh jawab ki raw shape hai. Parent note z 0 = f ( a , b ) A = f x B = f y woh tangent plane banane ke liye.
Jab hum section 5 ke plane mein z 0 = f ( a , b ) A = f x ( a , b ) B = f y ( a , b )
L ( x , y ) L ( x , y ) ( x , y ) flat sheet ki height ka naam hai:
L ( x , y ) = f ( a , b ) + f x ( a , b ) ( x − a ) + f y ( a , b ) ( y − b ) . L linear ke liye khada hai — formula mein steps sirf first power tak hain, koi squares nahi, isliye iski graph ek plane hai. Yeh bilkul wahi right-hand side hai jaise z = z 0 + A ( x − a ) + B ( y − b )
Picture yeh hai : kisi bhi ground spot ( x , y ) L ( x , y ) flat sheet (curved surface nahi) se padhte ho.
Topic ko yeh kyun chahiye: baad ke har formula — ≈ L
≈ f ( x , y ) ≈ L ( x , y ) lagbhag section 6 ki flat-sheet height L ( x , y )
Topic ko yeh kyun chahiye: linear approximation ka poora point yeh hai ki yeh ek accha guess hai, exact truth nahi. Woh gap error hai, Differentiability of multivariable functions ke zariye study kiya gaya.
Δ z Δ Δ z = f ( x , y ) − f ( a , b ) real surface ki height badli jab tum ( a , b ) ( x , y )
d x d y d z d x = x − a d y = y − b d z = f x d x + f y d y flat sheet predict karti hai. Chhote steps ke liye, Δ z ≈ d z
d z L ( x , y ) = f ( a , b ) + f x ( x − a ) + f y ( y − b ) change hai
L ( x , y ) − L ( a , b ) = [ f ( a , b ) + f x ( x − a ) + f y ( y − b ) ] − f ( a , b ) = f x ( x − a ) + f y ( y − b ) . x − a = d x y − b = d y d z = f x d x + f y d y d z f ( x , y ) ≈ L ( x , y ) Δ z d z
Picture yeh hai : tumhare step ke upar do heights uth rahi hain — curved surface ki real rise Δ z d z
Topic ko yeh kyun chahiye: differentials d z error estimates karta hai (jaise rectangle-area example).
( x , y ) → ( a , b ) lim kya value kisi expression mein aati hai jab ( x , y ) kisi bhi direction se ( a , b )
Intuition Kyun limit, aur sirf "small error" nahiTangent plane tabhi honest hai jab uska error tumse approach karne se tezi se khatam ho jaaye. Yahan L ( x , y ) f ( x , y ) − L ( x , y ) ( x − a ) 2 + ( y − b ) 2 lim ( x , y ) → ( a , b ) ( x − a ) 2 + ( y − b ) 2 f ( x , y ) − L ( x , y ) = 0 tumne kitna step kiya uski tulna mein bhi tiny hai . Yahi differentiable ka exact matlab hai.
Topic ko yeh kyun chahiye: yeh woh fine print hai jo guarantee karta hai ki plane sach mein surface ko "hug" karta hai — Differentiability of multivariable functions ka subject.
⟨ p , q , r ⟩ p q r ⟨ ⟩
Definition Dot product aur perpendicularityDo vectors ⟨ p , q , r ⟩ ⟨ u , v , w ⟩ dot product single number p u + q v + r w perpendicular (right angles par) hote hain exactly jab unka dot product 0
n perpendicular — surface se seedha bahar nikalta hua jaise pahadi par ek flagpole. Tangent plane woh sab kuch hai jo n .
Picture yeh hai : flat sheet surface par leti hui, ek arrow n
n = ⟨ f x , f y , − 1 ⟩
1 f x t x = ⟨ 1 , 0 , f x ⟩ 1 f y t y = ⟨ 0 , 1 , f y ⟩
Plane ka normal vector dono ke perpendicular hona chahiye. n n ⋅ t x = f x ( 1 ) + f y ( 0 ) + ( − 1 ) f x = f x − f x = 0 , n ⋅ t y = f x ( 0 ) + f y ( 1 ) + ( − 1 ) f y = f y − f y = 0. n = ⟨ f x , f y , − 1 ⟩ − 1
Topic ko yeh kyun chahiye: yeh plane likhne ka ek doosra, elegant tarika deta hai, aur yeh The gradient vector aur Directional derivatives se connect hota hai.
base point a b and steps x-a y-b
plane form height plus tilts
partial derivatives fx fy
approx symbol and deltas dz
TANGENT PLANE TOPIC 4.4.5
Kya tum bina dekhe har ek ka jawaab de sakte ho? Check karne ke liye reveal karo.
( x , y , z ) Ek point origin se kitna right, forward, aur up hai.
f ( x , y ) Ground spot ( x , y )
z = f ( x , y ) Ek curved surface, ground ke upar float karte saare points ( x , y , f ( x , y ))
x − a y − b Fixed base point ( a , b )
Seedhe shabdon mein, f x ( a , b ) Surface ka slope jab tum x y b
f x y
Kyun plane z = z 0 + A ( x − a ) + B ( y − b ) Yeh automatically ( a , b ) z 0 A , B
L ( x , y ) Flat-sheet height ka naam: L ( x , y ) = f ( a , b ) + f x ( a , b ) ( x − a ) + f y ( a , b ) ( y − b )
Δ z d z Δ z d z L
d z = f x d x + f y d y Yeh L ( x , y ) − L ( a , b ) d x = x − a d y = y − b
Differentiability limit words mein kya kehta hai? Error f − L
Kyun n = ⟨ f x , f y , − 1 ⟩ Dono in-plane arrows ⟨ 1 , 0 , f x ⟩ ⟨ 0 , 1 , f y ⟩ 0
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