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

4.4.1 · D1 · Maths › Multivariable Calculus › Functions of several variables — graphs, level curves, level

Yeh page yeh maanta hai ki tumne kuch bhi nahi dekha. Hum parent note par har ek squiggle ka naam lenge aur use ek picture mein badlenge, isse pehle ki woh dobara use ho.

0. Is topic ki alphabet

Neeche har woh symbol hai jo parent page use karta hai. Hum unhe ek aisi order mein banate hain jahan har ek sirf usse pehle waale par depend karta hai.

1. Ek number line aur symbol $\mathbb{R}$

Picture: ek seedhi horizontal line jo dono taraf forever stretch karti hai, beech mein $0$ ke saath. Us line par har ek point ek real number hai.

Topic ko yeh kyun chahiye: poora subject numbers ke andar jaane aur ek number ke bahar aane ke baare mein hai. $\mathbb{R}$ woh bag hai jis se woh numbers liye jaate hain. Jab tum baad mein "$f:{\mathbb{R}}^n\to \mathbb{R}$" dekhte ho, toh daayein taraf ka plain R bilkul yahi ruler hai — output hamesha uspar kahiin land karta hai.

2. Ordered pairs, triples, aur symbols ${\mathbb{R}}^2$, ${\mathbb{R}}^3$, ${\mathbb{R}}^n$

• ${\mathbb{R}}^2$ = "R to the power 2" = saare ordered pairs ka set = poora flat plane.
• ${\mathbb{R}}^3$ = saare triples ka set = poora space.
• ${\mathbb{R}}^n$ = $n$ numbers ki saari lists. Chhota raised $n$ ("superscript $n$") count karta hai ki tum ek saath kitne numbers feed karte ho.

Picture: $(x,y)$ dhundhne ke liye, horizontal ruler ke saath $x$ steps East chalo, phir ek doosre ruler ke saath $y$ steps North. Jahan tum ruko woh point hai. $(x,y,z)$ ke liye, phir $z$ steps seedhe upar jao.

Topic ko yeh kyun chahiye: several variables ka function ek input point ko feed kiya jaata hai. ${\mathbb{R}}^2$ mein woh input ek pair $(x,y)$ hai; ${\mathbb{R}}^3$ mein ek triple $(x,y,z)$. Yeh literally "several variables" mein "several" hai.

3. Arrow $\to$, letter $f$, aur "function" phrase

Picture: ek box jis par $f$ lika ho. Ek point $(x,y)$ left slot mein jaata hai; ek single number $z$ neeche se nikalti hai. Exactly ek number bahar — kabhi do nahi, kabhi zero nahi — yahi woh rule hai jo $f$ ko ek function banata hai.

Topic ko yeh kyun chahiye: sab kuch — graphs, level curves, level surfaces — is ek machine ko dekhne ke tarike hain. Machine ke bina kuch bhi draw karne layak nahi hai.

4. Symbols $\subseteq$ aur $D$ (domain), aur "range"

Picture: input plane par ek region shade karo. Shading ke andar ke points legal inputs hain; bahar ke points forbidden hain (woh division by zero, ya negative ka square root, etc. force karenge).

Topic ko yeh kyun chahiye: tum ek graph ya level set sirf domain ke upar draw kar sakte ho. "Picture exist kahan karti hai?" poochho toh jawaab $D$ hai.

5. Squares, $x^2+y^2$, aur distance

Picture: point $(x,y)$, origin $(0,0)$, aur unke beech ka seedha segment ek right triangle banate hain jiska horizontal leg $x$ hai aur vertical leg $y$ hai. Pythagorean theorem kehta hai ki segment ki length $r$ satisfy karti hai $r^2=x^2+y^2$. Toh $x^2+y^2$ woh length squared hai, aur $\sqrt{x^2+y^2}=r$ length khud hai.

Topic ko yeh kyun chahiye: parent ke headline examples ($x^2+y^2$ circles deta hai, $x^2+y^2+z^2$ spheres deta hai) pure distance formulas hain. Jab tum "$x^2+y^2=c$" ko "squared-distance equals a constant" ke roop mein dekhte ho, toh yeh obvious ho jaata hai ki yeh radius $\sqrt{c}$ ka circle hai — har direction mein same distance.

6. Output letters $z$, $w$ aur vertical axis

Picture: $z=f(x,y)$ ke liye, input plane se seedha upar ek teesra ruler lagao. Output number $z$ woh height hai jo tum input spot ke upar us vertical ruler par chadhte ho. Har input point ke liye yeh karo aur heights ek surface trace karti hain — graph.

7. Constant $c$ aur set-builder $\{\cdot :\cdot \}$

Picture: surface ko height $z=c$ par ek perfectly horizontal sheet se slice karo. Sheet surface ke saath ek curve mein milti hai; us curve ko seedha flat input plane par neeche drop karo. Shadow level curve hai — woh saari jagahein jo output $c$ share karti hain.

Topic ko yeh kyun chahiye: level curves aur level surfaces dono is ek hi template se bante hain — $c$ freeze karo, woh saare inputs collect karo jo $c$ dete hain. Ek word badlo (curve/surface) depending on ki tumhare paas 2 ya 3 inputs the.

8. Dimension bookkeeping ("$n$ minus one")

Topic ko yeh kyun chahiye: yahi wajah hai ki $f(x,y)=c$ curves deta hai lekin $f(x,y,z)=c$ surfaces deta hai — parent ka "Mistake 3." Arithmetic $n-1$ apni pocket mein rakho.

Foundations topic ko kaise feed karte hain

Equipment checklist

Khud ko test karo — daaya side cover karo aur zor se jawab do.

Connections

• Parent topic — poori picture
• Limits and continuity of multivariable functions — "${\mathbb{R}}^n$ mein ek point approach karne" ka kya matlab hai, in coordinates par based
• Partial derivatives — machine aur uska graph samajh aane ke baad agla tool
• Gradient vector — yahan define kiye gaye level sets ke perpendicular rehta hai
• Quadric surfaces — woh paraboloids aur spheres jinke distance formulas hum ne unpack kiye