4.4.1
Intuition Badi tasveer (YEH KYUN EXIST KARTA HAI)Ek variable ki function y = f ( x ) saare inputs par ek saath depend karta hai: temperature position ( x , y ) ( x , y , z ) aur quantity dono par depend karta hai. Hume ek language chahiye "output kai saare inputs par ek saath depend karta hai" ke liye. Graph hume aisi functions ko dekhne deta hai, aur level sets (curves/surfaces) hume tab bhi dekhne dete hain jab graph itne zyada dimension mein hota hai ki draw nahi kar sakte.
n f : D ⊆ R n → R domain D ( x 1 , … , x n ) z = f ( x 1 , … , x n ) output (ya value) hai.
n = 2 z = f ( x , y ) R 2 n = 3 w = f ( x , y , z ) R 3
Domain kya hota hai: inputs ka wo set jiske liye formula sense deta hai (zero se division nahi, even roots ke andar negative nahi, non-positive numbers ka log nahi).
Worked example Domain dhundhna — har step kyun?Maano f ( x , y ) = 9 − x 2 − y 2
Step: Radicand ≥ 0 9 − x 2 − y 2 ≥ 0 Kyun? Real square roots ke liye andar non-negative hona zaroori hai.Step: Rearrange karke x 2 + y 2 ≤ 9 Kyun? Yeh radius 3 Domain = {( x , y ) : x 2 + y 2 ≤ 9 } Range = [ 0 , 3 ] 9 − x 2 − y 2 0 9 0 3
z = f ( x , y ) {( x , y , z ) : z = f ( x , y ) , ( x , y ) ∈ D } ⊆ R 3 surface hai jo x y
YEH KYUN sirf n ≤ 2 f n + 1 n = 2 n = 3
Intuition Level curves KYUN3D mein ek surface ko flat paper par draw karna mushkil hai. Toh hum wahi karte hain jo cartographers pahadon ke liye karte hain: surface ko constant heights par slice karo aur un slices ko flat x y project karo. Har slice ek curve ban jaati hai jis par uski height ka label hota hai. Curves ki spacing se steepness pata chalta hai — bheed wali curves ka matlab hai steep terrain.
Ek function z = f ( x , y ) c value c yeh hai:
{( x , y ) ∈ D : f ( x , y ) = c } . ek hi output c z = c
Worked example Saddle ki level curves — yeh step kyun?f ( x , y ) = x 2 − y 2 x 2 − y 2 = c
c > 0 x 2 − y 2 = c Kyun? Standard form c x 2 − c y 2 = 1 c < 0 c = 0 x 2 = y 2 ⇒ y = ± x Kyun? Degenerate case jahan hyperbola apne asymptotes mein collapse ho jaata hai.
Crossing lines + nested hyperbolas ka pattern ek saddle ki pehchaan hai.
w = f ( x , y , z ) c value c yeh hai:
{( x , y , z ) : f ( x , y , z ) = c } ⊆ R 3 . zaroor draw kar sakte hain.
f ( x , y , z ) = x 2 + y 2 + z 2
x 2 + y 2 + z 2 = c c > 0 c sphere (origin se constant distance). Kyun? x 2 + y 2 + z 2 c = 0 c < 0 f
Common mistake Common confusions ko steel-man karnaGalti 1: "Level curve f ( x , y ) = c z = c
Kyun sahi lagta hai: aapne sach mein height c Fix: level curve us slice ka projection hai jo x y c label hai, koi coordinate nahi. 3D mein slice ko "trace" kehte hain; projected label level curve hai.
Galti 2: "Equally spaced c
Kyun sahi lagta hai: input mein equal steps aam taur par uniform dikhte hain. Fix: spacing function par depend karti hai. x 2 + y 2 c c ⇔
Galti 3: "f ( x , y , z ) = c
Kyun sahi lagta hai: 2D mein level set ek curve tha. Fix: 3 unknowns mein ek equation generally ek 2D surface define karta hai, curve nahi. Level set ka dimension = − 1
Worked example Pehle forecast karo, phir check karof ( x , y ) = y − x 2 Forecast: calculate karne se pehle, level curves ki kya shape hogi? Woh… parabolas honi chahiye (kyunki y Verify: y − x 2 = c ⇒ y = x 2 + c c
Function f : R n → R Un saare input points ( x 1 , … , x n ) f
z = f ( x , y ) Yeh R 3
f ( x , y ) c Set {( x , y ) : f ( x , y ) = c } c z = c x y
f ( x , y , z ) Iske liye 4 axes chahiye honge (3 inputs + 1 output); hum uski jagah 3D mein level surfaces use karte hain.
f = x 2 + y 2 c > 0 ke liye radius
c ke concentric circles,
c = 0 ke liye ek point,
c < 0 ke liye empty.
f = x 2 + y 2 + z 2 = c Radius
c ki spheres (
c > 0 );
c = 0 par origin point;
c < 0 ke liye empty.
Level curves ka bheed hona kya indicate karta hai? Steepness — wahan function rapidly change karta hai.
n n − 1
Saddle f = x 2 − y 2 c = 0 Do lines y = ± x
Recall Feynman: 12-saal ke bachche ko samjhaoEk pahadi landscape imagine karo. Ek machine (f ( x , y ) Graph asli 3D pahad hai. Lekin flat paper map par 3D nahi dikha sakte, toh rings draw karte ho: har ring un saari jagahon ko connect karta hai jo ek hi height par hain (jaise 100 m, 200 m). Woh rings level curves hain. Jahan rings dur-dur hain wahan chalte ho → gentle slope; rings ek saath dabi hain → cliff! Kisi aisi cheez ke liye jo teen numbers par depend karta hai (jaise poore room mein temperature), pahad bana bhi nahi sakte — toh uski jagah invisible "shells" draw karo jahan value same hai. Woh shells level surfaces hain.
"Same value, same set." Level set = ek hi output share karne wale points. Aur 2D mein C urve, 3D mein S urface — level set hamesha inputs ke space se ek dimension neeche hota hai (socho: "n
example sqrt 9 minus x2 minus y2
Function f of n variables
Level surfaces f equals c
