Functions of several variables — graphs, level curves, level surfaces: Question bank

4.4.1 · D5 · Maths › Multivariable Calculus › Functions of several variables — graphs, level curves, level

Pehle "trace" vs "level curve" ko pin down karo (visual anchor)

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

Amber ring (trace) ko dekho jo bowl par height $c$ par hover kar raha hai, aur cyan ring (uski level curve) ko dekho jo seedha neeche ground par flatten ho gayi hai. Yeh bhi notice karo ki ground rings baahri taraf bheed baar rahi hain — woh crowding steepness cue hai jo neeche baar baar test hoti hai.

Sach ya jhooth — justify karo

A level curve

f (x, y) = c

lives in

R^{3}

at height

z = c

Jhooth. Trace 3D mein height

c

par rehta hai, lekin level curve us trace ki shadow hai jo

x y

-plane par project hoti hai, isliye woh 2D mein rehti hai —

c

sirf ek label hai.

Equally spaced values of

c

always give equally spaced level curves.

Jhooth. Spacing function par depend karti hai:

x^{2} + y^{2}

ke liye radius

c

hai, isliye equal

c

-steps wale curves baahri taraf bheed-bhaadte hain.

The level set of

f (x, y, z) = c

is generally a curve.

Jhooth. Teen unknowns mein ek equation ek degree of freedom remove karti hai, ek 2D surface bachti hai, curve nahi.

Crowded level curves mean the terrain is nearly flat there.

Jhooth. Crowding ka matlab hai ki value chhoti horizontal distance par fast change ho rahi hai — yeh steep hai, flat ka ulta.

The graph of

z = f (x, y)

and the level curve

f (x, y) = c

live in the same space.

Jhooth. Graph

R^{3}

mein ek surface hai; level curve 2D domain

R^{2}

mein ek set hai.

Every value of

c

produces a non-empty level set.

Jhooth.

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

ke liye koi bhi

c < 0

empty set deta hai, kyunki squares ka sum kabhi negative nahi ho sakta.

Two different level curves of the same function can cross at a point.

Generally Jhooth. Cross karne ka matlab hoga ki ek input point

(x, y)

ke do alag outputs

c_{1} \neq = c_{2}

hain, jo single-valued function ke liye impossible hai.

A function of two variables can have a level "curve" that is a single point.

Sach.

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

ke liye

c = 0

par level set sirf origin hai — ek degenerate curve jahan

\nabla f = 0

hai.

Error dhundo

f (x, y) = 9 - x^{2} - y^{2}

ka domain poora

R^{2}

hai kyunki squaring hamesha defined hoti hai."

Error: square root ke liye andar non-negative hona zaroori hai, isliye

9 - x^{2} - y^{2} \geq 0

chahiye, jo sirf disk

x^{2} + y^{2} \leq 9

deta hai.

f = x^{2} - y^{2}

ke liye,

c = 0

set karne par koi level set nahi milti kyunki hyperbola disappear ho jaata hai."

Error:

c = 0

deta hai

x^{2} = y^{2}

, yani do crossing lines

y = \pm x

— yeh degenerate case hai, empty set nahi.

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

ke level surfaces har

c

ke liye spheres hain."

Error: sirf

c > 0

ke liye.

c = 0

par yeh origin par single point hai, aur

c < 0

par empty hai.

f (x, y, z)

ko graph karne ke liye mujhe sirf 3D mein points plot karne hain."

Error:

f (x, y, z)

ke graph ko 4 axes chahiye (3 inputs + 1 output) aur draw nahi kiya ja sakta; hum uski jagah ordinary 3D mein level surfaces use karte hain.

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

ke level curves baahri taraf dur hoti jaati hain kyunki

c

barhta hai."

Error: radius

= c

c

se dhheere barhta hai, isliye equal

c

-jumps circles ko baahri taraf paas-paas karte hain.

"Ek trace aur ek level curve same object hain."

Error: trace actual slice hai jo 3D mein height

z = c

par baith ke exist karti hai; level curve uski projection hai jo

x y

-plane mein neeche jaati hai.

R^{n}

par kisi bhi

f

ka level set hamesha dimension

n - 1

ki smooth surface hoti hai."

Error: iske liye set par

\nabla f \neq = 0

hona zaroori hai (implicit function theorem). Ek critical value par jahan

\nabla f = 0

ho, level set degenerate ho sakti hai — jaise ek point ya crossing lines — aur smooth

(n - 1)

-manifold nahi reh sakti.

Why questions

Hum directly

f (x, y)

graph kar sakte hain lekin

f (x, y, z)

nahi, kyun?

Graph ko har input ke liye ek axis aur output ke liye ek axis chahiye;

n = 2

mein 3 axes chahiye (draw ho sakta hai) jabki

n = 3

mein 4 axes chahiye (draw nahi ho sakta).

Cartographers ke contour maps aur level curves ek hi idea se kaise kaam karte hain?

Dono constant heights par height surface ko slice karte hain aur un slices ko flat ground par project karte hain, har ek ko uski height se label karte hain.

R^{n}

par

f

ka level set generally dimension

n - 1

ka kyun hota hai, aur yeh kab fail ho sakta hai?

Ek equation

f = c

ek degree of freedom remove karti hai, isliye jahan

\nabla f \neq = 0

hota hai implicit function theorem ek smooth

(n - 1)

-manifold deta hai; critical values par (

\nabla f = 0

) yeh points, crossings, ya aur bura kuch bhi ban sakta hai.

Ek level curve ko kabhi do alag values

c

se label kyun nahi kiya ja sakta?

Kyunki

f

single-valued hai: har point

(x, y)

exactly ek output par map hota hai, isliye woh exactly ek level curve mein belong karta hai.

Level curves ki spacing shape ke bajaay steepness ko encode kyun karti hai?

Steepness hai (height mein change)/(horizontal distance); equal height steps fix karne par, paas ke curves ka matlab hai har step mein kam horizontal distance, isliye steeper slope.

x^{2} + y^{2}

paraboloid ki signature concentric circles kyun hain?

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

fix karna origin se constant distance

c

force karta hai, aur radii

c

ke saath badhte hain, ek bowl ke nested rings produce karte hain.

Edge cases

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

ka level set

c = 0

par kya hai?

Ek single point, origin — circles ki family ka sabse chhota, degenerate member, aur woh critical value jahan

\nabla f = 0

hai.

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

ka level set

c \to 0^{+}

par kya hota hai?

Spheres origin ki taraf shrink hoti hain,

c = 0

par single point mein collapse ho jaati hain (radius

c \to 0

Ek constant function

f (x, y) = 5

ka level set

c = 5

aur

c = 3

par kya hai?

c = 5

par yeh poora domain hai (har point 5 output karta hai);

c = 3

par yeh empty hai, kyunki koi point 3 output nahi karta.

Saddle

f = x^{2} - y^{2}

ke liye,

c > 0

c = 0

, aur

c < 0

mein level sets kaise alag hain?

c > 0

: hyperbolas left/right khulte hain (

x

-axis ko cross karte hain);

c = 0

: do degenerate crossing lines

y = \pm x

;

c < 0

: hyperbolas up/down khulte hain (

y

-axis ko cross karte hain) —

9 0^{\circ}

rotated family.

c > 0

aur

c < 0

wale saddle hyperbolas perpendicular directions mein kyun point karte hain?

c > 0

ke liye,

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

x^{2} \geq c

chahiye isliye real points sirf

y

-axis se door exist karte hain (sideways khulta hai);

c < 0

ke liye,

y^{2} - x^{2} = ∣ c ∣

wahi condition

y

par force karta hai (vertically khulta hai).

f = 9 - x^{2} - y^{2}

ki highest-value level curve kya hai, aur uski shape kya hai?

c = 3

par (iska maximum),

9 - x^{2} - y^{2} = 9

force karta hai

x^{2} + y^{2} = 0

, isliye level set sirf origin point hai.

Kya ek level set ek curve ho sakti hai jo unbounded ho (infinity tak jaaye)?

Haan. Saddle

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

ke liye

c > 0

ke saath, hyperbola branches infinitely far tak extend hoti hain — level sets closed loops hona zaroori nahi.

Connections

Gradient vector — har level set ke perpendicular; $\nabla f = 0$ degenerate critical values ko mark karta hai
Partial derivatives — woh slopes build karte hain jo crowding reflect karti hai
Directional derivatives — zero hoti hain jab tum ek level curve along chalo
Quadric surfaces — woh paraboloids aur saddles jinke level sets yeh traps probe karte hain
Limits and continuity of multivariable functions — level sets ke across ek point approach karna