Functions of several variables — graphs, level curves, level surfaces: Visual walkthrough

4.4.1 · D2 · Maths › Multivariable Calculus › Functions of several variables — graphs, level curves, level

Hum ek central result derive karte hain:

Level curves of f (x, y) = x^{2} + y^{2} are concentric circles of radius c .

Neeche sab kuch us sentence ko, symbol by symbol, earn karta hai.

Step 1 — Two-input machine asal mein hoti kya hai

KYA HAI. Ek function $f (x, y)$ ek aisi machine hai jisme do dials hain aur ek readout hai. Aap flat floor par ek jagah choose karte ho — ek pair of numbers $(x, y)$ — aur machine ek single number $z$ wapas deti hai, jo height hai.

YE YAHAN SE KYUN SHURU KAREN. Surface ko cut karne se pehle hume agree karna hoga ki surface exist karti hai. Har wo symbol jo hum use karenge yahan se janam leta hai: $x$ = kitna east, $y$ = kitna north, $z$ = kitna upar.

PICTURE. Figure dekho. Flat grid floor hai (the $x y$ -plane). Ek chosen point $(x, y)$ marked hai; usse ek vertical stick $z = f (x, y)$ lambai ka utha hua hai. Us stick ki tip surface ka ek point hai. (Hum puri note mein plain letters $x, y, z$ use karte hain — marked point bas koi bhi choice hai.)

Figure — Functions of several variables — graphs, level curves, level surfaces

height, the readout z = the machine f (east dial x, north dial y)

Step 2 — Poori surface ugao: the paraboloid

KYA HAI. Step 1 ko har floor point par ek saath karo. Saari stick-tips collect karo. Woh tips ka cloud hi graph hai — ek surface floor ke upar float karti hui.

KYUN. Ek akela stick ek number hai. Surface poori function hai ek hi nazar mein dekhi hui. Bowl ko poora kaat'ne se pehle humein chahiye.

PICTURE. Orange surface bowl $z = x^{2} + y^{2}$ hai. Dhyan do ye floor ko exactly ek point par touch karti hai (origin, jahan dono dials $0$ hain) aur upar aur bahar ki taraf flare karti hai. Steepness badhti hai jaise tum charhte ho — yaad rakho, ye Step 6 mein wapas aata hai.

z = x^{2} + y^{2} ⟹ z \geq 0, z = 0 ⟺ (x, y) = (0, 0)

Yahan $x^{2} \geq 0$ aur $y^{2} \geq 0$ hamesha, toh unka sum kabhi zero se neeche nahi ja sakta — isliye bowl kabhi floor ke neeche nahi jaata.

Step 3 — Height $c$ par horizontal slice kato

KYA HAI. Ek height choose karo, use $c$ kaho (bas ek chosen number, jaise " $c = 4$ metres"). Sheesha ka ek bilkul flat piece bowl mein us height par push karo. Jahan glass surface se milti hai, hume ek curve milta hai jo 3D mein float kar raha hai — is curve ko height $c$ par trace kehte hain.

Letter $c$ KYUN? $c$ ka matlab hai constant: is poori curve par height kabhi nahi badlti, wo $c$ par frozen hai. Hum ek symbol use karte hain taki poori family (saari heights) ek formula share kare.

Sirf horizontal cut kyun, koi aur nahi? Kyunki hum equal value waale jagah chahte hain. Equal height ka matlab hai equal readout $z = c$ . Tilted cut alag-alag heights ko mix kar deta aur wo meaning khatam ho jaati.

PICTURE. Violet ring wahi hai jahan glass sheet $z = c$ orange bowl ko choomti hai. Ye hawa mein height $c$ par rehti hai — ye abhi floor par nahi hai.

the surface’s height z = frozen height of the glass c on the trace

Step 4 — Dono heights ko agree karvao (key algebra)

KYA HAI. Trace par, do baatein ek saath sach hain: point surface par hai ( $z = x^{2} + y^{2}$ ) aur glass height par hai ( $z = c$ ). Dono ko equal set karo.

KYUN. Ye akela step hai jo geometry ko equation mein badalta hai. "Dono par hona" ka matlab hai dono descriptions se same $z$ , toh hum ek ko doosre mein substitute karte hain aur $z$ gayab ho jaata hai — sirf $x$ aur $y$ ke beech ek relation reh jaata hai.

PICTURE. Figure mein dono coloured height-labels ( $z$ bowl se, $z$ glass se) milte hain aur cancel hote hain, sirf $x, y$ mein ek flat equation reh jaati hai.

height from the bowl x^{2} + y^{2} = height from the glass c

Ab har symbol: $x^{2} + y^{2}$ machine ka output hai floor point par; $c$ wo constant hai jo humne choose kiya. Equation kehti hai "kaunse floor points machine ko exactly $c$ read karvate hain?" — ye sawaal exactly level curve hai jo page ke upar define ki gayi hai.

Step 5 — Shape pehchano: radius $c$ ka circle

KYA HAI. $x^{2} + y^{2} = c$ ko us circle se compare karo jo tum pehle se jaante ho, $x^{2} + y^{2} = r^{2}$ (origin se fixed distance $r$ par saare points).

YE COMPARISON KYUN, KOI AUR NAHI? Kyunki $x^{2} + y^{2}$ literally squared distance hai floor point $(x, y)$ se origin tak (floor par Pythagoras: do legs $x$ aur $y$ hain). Toh " $x^{2} + y^{2} = c$ " ka matlab hai "squared distance $= c$ ", yani "distance $= c$ " — ek circle.

PICTURE. Floor par right triangle ke legs $x$ aur $y$ hain aur hypotenuse origin tak seedhi line hai; uski length squared $x^{2} + y^{2}$ hai. Us length ko freeze karne par circle milta hai.

x^{2} + y^{2} = c ⟹ match r^{2} = c r = radius; needs c \geq 0 c

Hum positive root lete hain kyunki radius ek length hai, kabhi negative nahi. Ye ek quiet choice hai jo degenerate cases rule out karti hai — jinhe hum ab seedha face karte hain.

Step 6 — Har case: $c > 0$ , $c = 0$ , $c < 0$ , aur crowding

KYA HAI. Ek achhi derivation ko $c$ ki har value ke liye survive karna chahiye. Teeno run karo.

$c > 0$ : $r = c$ ek genuine positive radius hai → ek sahi circle. Kyun: ek real positive length radius ho sakti hai.
$c = 0$ : $x^{2} + y^{2} = 0$ force karta hai $x = 0$ aur $y = 0$ (do squares tabhi zero sum kar sakte hain jab dono zero hon) → level "curve" ek single point, origin, mein simat jaati hai. Kyun: glass bowl ko sirf uske lowest point par touch karti hai.
$c < 0$ : $x^{2} + y^{2}$ kabhi negative nahi ho sakta, toh koi bhi floor point kaam nahi karta → level set empty hai. Kyun: glass sheet bowl ke neeche hai aur kuch touch nahi karti.

Rings outward kyun crowd karti hain. Equal spaced heights $c = 1, 2, 3, 4$ lo. Unke radii hain $1, 2, 3, 4 = 1, 1.41, 1.73, 2$ . Radii ke beech gaps shrink karte hain: $0.41, 0.32, 0.27$ . Toh height mein equal steps se circles milte hain jo bahar jaate jaate bunch karte hain — aur bunched contours ka matlab hai steep zameen. Ye exactly bowl ka steep hona hai jaisa tum charh'te ho, ab flat map par visible.

PICTURE. Floor map mein nested circles $c = 1, 2, 3, 4$ labelled hain; shrinking gaps dekho. Ek tiny dot $c = 0$ mark karta hai; ek faint crossed-out ring remind karati hai ki $c < 0$ empty hai.

case c > 0 c = 0 c < 0 level set circle, r = c point (0, 0) empty why positive length is a valid radius two squares sum to 0 only if both are 0 x^{2} + y^{2} \geq 0 can never equal a negative

Step 7 — Map ko ulta padho (sanity check)

KYA HAI. Sirf flat contour map par khade ho (koi bowl draw nahi) aur kisi bhi point par height rebuild karo. Agar tum $(x, y)$ par ho, dekho kaun sa circle tumhare se guzarta hai — uska label tumhari height hai.

KYUN. Ek derivation tabhi trustworthy hoti hai jab wo reversible ho. Map → surface ko surface → map undo karna chahiye.

PICTURE. Ek hiker-dot $(1.5, 0)$ par ring $c = 2.25$ par baitha hai, kyunki $1. 5^{2} + 0^{2} = 2.25$ . Vertical arrow Step 1 ki stick ko reconstruct karta hai.

floor point (x, y) ⟼ label of its ring = x^{2} + y^{2} = z

Humne Step 1 ka readout recover kar liya — loop close hota hai. ✅

Ek-picture summary

Sab ek saath: orange bowl, teen glass sheets jo violet traces cut karti hain $c = 1, 2, 3$ par, aur unke magenta shadows floor par nested circles ki tarah land karte hain — gaps visibly shrinking outward.

Recall Feynman retelling — poora walk plain shabdon mein

Socho ek aisi machine jो flat floor ke kisi bhi spot par khadi hokar ek height batati hai — kitna east ho usse square karke, kitna north ho usse square karke, aur dono add karke. Ye har jagah karo aur heights ek orange bowl banate hain floor par baitha hua (Steps 1–2). Ab ek flat glass sheet bowl mein kisi fixed height $c$ par ghusao; jahan glass bowl se milti hai tumhe hawa mein ek ring milti hai — ek trace (Step 3). Poocho "kaunse floor spots ne woh height di?" — bowl ki height ko glass ki height ke equal set karo, height cancel ho jaata hai, aur tumhare paas $x^{2} + y^{2} = c$ reh jaata hai (Step 4). Kyunki $x^{2} + y^{2}$ middle se sirf squared distance hai (Pythagoras), iska matlab hai "sab spots centre se same distance $c$ par" — ek circle (Step 5). Har height try karo: positive se real circle milta hai, zero se sirf centre dot milta hai, negative se kuch nahi milta kyunki do squares add karke negative nahi aa sakta (Step 6). Saari air-rings ko seedha floor par gira do aur tumhe ek contour map milta hai jiske circles bahar jaate jaate bunch karte hain — bunched circles matlab bowl steep ho rahi hai (Step 6 again). Aakhir mein, map ulta padho: kahi bhi khade ho, dekho tum kis ring par ho, aur uska label tumhari height hai — machine, rebuild ho gayi (Step 7).

Recall

Hum level curves ke liye horizontally kyun cut karte hain?

Horizontal ka matlab equal height

z = c

hai, yani equal output — exactly wo points jo ek value share karte hain.

x^{2} + y^{2} = c

mein kya disappear hua aur kyun?

Height

z

— kyunki point surface par bhi hai (

z = x^{2} + y^{2}

) aur glass par bhi (

z = c

), toh humne dono equal set kiye.

Radius

c

kyun hai,

c

kyun nahi?

x^{2} + y^{2}

origin se squared distance hai, toh distance uska square root hai.

c = 0

aur

c < 0

par kya hota hai?

c = 0

se single point

(0, 0)

milta hai;

c < 0

se empty set milta hai (squares ka sum kabhi negative nahi hota).

Trace vs level curve?

Trace 3D mein height

c

par float karti hai; level curve uska floor par shadow hai jisme

c

sirf ek label hai.

Circles outward kyun crowd karti hain?

Equal height-steps se radii

1, 2, 3, \dots

milte hain jinke gaps shrink karte hain — crowding steepness signal karta hai.

Connections

Parent topic — ye walkthrough uska central "slice-and-project" claim zero se build karta hai.
Gradient vector — rings ke across steepest ascent ki taraf point karta hai; yahan crowding uski size preview karti hai.
Partial derivatives — bowl ke slopes $x$ - aur $y$ -directions ke along.
Quadric surfaces — bowl $z = x^{2} + y^{2}$ paraboloid hai, ek named quadric.
Directional derivatives — contour map par chalte waqt height kitni change hoti hai.