Change of variables — general Jacobian: Visual walkthrough

4.4.21 · D2 · Maths › Multivariable Calculus › Change of variables — general Jacobian

Hum do flat pages side by side use karte hain:

input page, jahaan hamare coordinates $u$ (horizontal) aur $v$ (vertical) kehlate hain;
output page, jahaan coordinates $x$ (horizontal) aur $y$ (vertical) kehlate hain.

Ek rule $T$ input page ke har point $(u, v)$ ko leta hai aur use output page par kahin $(x, y)$ par drop karta hai. $T$ ko ek machine ki tarah padho: $(u, v)$ feed karo, $(x, y)$ bahar aata hai.

Step 1 — Do pages aur unke beech ek machine

KYA. Hum input page (left) aur output page (right) draw karte hain. Hum input page par ek point $(u_{0}, v_{0})$ mark karte hain aur use machine $T$ ke through follow karte hain output page par uski landing spot tak.

KYUN. Kuch move ya stretch hone se pehle, humein agree karna hoga ki "machine" ka matlab kya hai. Baad ki saari cheezein bas yahi hain: $(u_{0}, v_{0})$ ke paas ke points $(u_{0}, v_{0})$ ke comparison mein kaise move karte hain? Toh pehle hum woh home base pin down karte hain.

PICTURE. Left par magenta dot aur right par orange dot dekho — same point, $T$ se pehle aur baad mein.

Figure — Change of variables — general Jacobian

Step 2 — Woh tiny cell draw karo jis ki hume parwah hai

KYA. Input page par hum ek tiny rectangle draw karte hain: iska bottom-left corner $(u_{0}, v_{0})$ par baitha hai, iski width $d u$ hai ( $u$ -direction mein ek tiny step) aur height $d v$ hai ( $v$ -direction mein ek tiny step).

KYUN. Ek integral tiny cells par ek sum hai. Agar hum samjhein ki ek tiny cell ka area kaise change hota hai, toh hum poora integral samajh jaate hain — kyunki map itne chhote scale par har jagah same dikhta hai. Hum rectangle choose karte hain kyunki rectangles plane ko perfectly tile karte hain, toh unhe add karne se region recover ho jaata hai.

PICTURE. Violet rectangle ke four labelled corners hain. Iski input area bilkul simple hai: $d u \times d v$ .

Step 3 — Corners kahaan land karte hain? (velocity vectors)

KYA. Hum poochhte hain: jab main input ko $u$ -direction mein $d u$ se nudge karta hoon, toh output kitna door aur kis direction mein move karta hai? Answer hai partial derivative vector $r_{u}$ , times $d u$ .

YEH TOOL KYUN — partial derivative. Humein ek tarika chahiye yeh kehne ka ki "output kitni tezi se slide karta hai jab main sirf $u$ wiggle karta hoon." Yahi ek partial derivative measure karta hai: ek function ki rate of change jab ek input move kare aur baaki freeze hon. Hum $\frac{\partial x}{\partial u}$ (likha $x_{u}$ ) use karte hain ordinary derivative ki jagah precisely kyunki do inputs hain aur hum $v$ ko freeze kar rahe hain.

PICTURE. Orange landing dot se, do arrows nikaltein hain: magenta arrow $d u$ -step ka image hai, violet arrow $d v$ -step ka image hai.

Step 4 — Cell ek parallelogram ban jaata hai

KYA. Do mapped edge-arrows $a$ aur $b$ output page par ek shape span karte hain. Kyunki woh dono same corner se start hote hain aur do straight edges hain, hamare tiny rectangle ka image (first order tak) ek parallelogram hai — fourth corner bas $a + b$ door hai.

KYUN. Original ek rectangle tha jiske do edges the " $d u$ across" aur " $d v$ up." Ek linear map straight edges ko straight edges mein bhejta hai aur parallel edges ko parallel rakhta hai — toh ek rectangle sirf parallelogram ban sakta hai (possibly rotated, stretched, sheared). Yeh generally rectangle nahi hoga: do arrows zaruri perpendicular nahi honge. Woh tilt hi exactly woh reason hai ki naive "new width × new height" galat hota.

PICTURE. Violet input rectangle aur filled parallelogram jisme woh map hota hai, fourth corner $a$ aur $b$ ko tip-to-tail add karke reach kiya jaata hai.

Step 5 — Ek parallelogram ka area (cross product kyun)

KYA. Hum $a = (a_{x}, a_{y})$ aur $b = (b_{x}, b_{y})$ se span hone wale parallelogram ka area compute karte hain. Answer hai $∣ a_{x} b_{y} - a_{y} b_{x} ∣$ .

YEH TOOL KYUN — cross product. Ek slanted parallelogram ka area $(base) \times (true height)$ hai, aur true height mein edges ke beech ke angle ka sine involve hota hai. Cross product exactly wahi package karta hai: $∣ a \times b ∣ = ∣ a ∣ ∣ b ∣ sin ϑ$ area hai, aur yeh automatically zero ho jaata hai jab edges parallel hon (zero area). 2D mein, arrows ko 3D mein $(a_{x}, a_{y}, 0)$ aur $(b_{x}, b_{y}, 0)$ ke roop mein embed karo, cross product seedha page se bahar point karta hai sirf ek $z$ -component ke saath: $a \times b = (0, 0, a_{x} b_{y} - a_{y} b_{x}) .$

PICTURE. Parallelogram with base $∣ a ∣$ (magenta), $b$ ki tip se perpendicular height (dashed), aur edges ke beech angle $ϑ$ marked.

Step 6 — Arrows substitute karo: Jacobian paida hota hai

KYA. $a = (x_{u}, y_{u}) d u$ aur $b = (x_{v}, y_{v}) d v$ ko parallelogram-area formula mein plug karo.

KYUN. Hum finally geometry (mapped cell ka area) ko calculus ( $T$ ke partial derivatives) se connect karte hain. Yeh substitution hi poora climax hai.

PICTURE. Same parallelogram, ab har arrow apne component formula se labelled hai, aur resulting area expression iske upar likhi hai.

Step 7 — Sign, aur degenerate (zero-area) case

KYA. Hum examine karte hain ki $x_{u} y_{v} - x_{v} y_{u}$ ka sign $∣ \cdot ∣$ se throw away karne se pehle kya matlab rakhta hai, aur kya hota hai jab woh zero ho.

KYUN. Contract yeh hai: har case cover karo. Positive Jacobian, negative Jacobian, aur zero Jacobian — yeh teen genuinely alag geometric situations hain, aur reader ko har ek pehchanna chahiye.

PICTURE. Teen chhote cells side by side: orientation preserved (positive), mirror ki tarah flip (negative), ek line segment mein collapse (zero).

Definition Sign padhna

Positive $det J$ : map orientation preserve karta hai — input cell ke around counter-clockwise jao, image ke around bhi counter-clockwise hi jaoge. (Polar ke baare mein socho: $det J = r > 0$ .)
Negative $det J$ : map plane ko flip karta hai, mirror ki tarah. Area ab bhi real aur positive hai; minus sign sirf flip record karta hai. Isliye hum $∣ J ∣$ se integrate karte hain — parent ki "absolute value bhool jaana" wali mistake dekho.
Zero $det J$ : do arrows $a, b$ parallel ho gaye (ya ek collapse ho gaya). Parallelogram ek zero area wale line segment mein squash ho gaya. Yahaan $T$ locally invertible nahi hai — Inverse Function Theorem ke hisaab se $T$ ke reversible hone ke liye $det J \neq = 0$ chahiye us point ke paas. Change-of-variables theorem aise points ko exclude karta hai (unhe measure-zero set banana hoga, koi area contribute nahi karte).

Ek-picture summary

Upar ki saari cheezein ek single diagram mein compress ki gayi hain: input rectangle $d u d v$ → velocity arrows $r_{u} d u, r_{v} d v$ ke zariye linear approximation → parallelogram → iski area hai $∣ det J ∣ d u d v$ .

Recall Feynman retelling — ek dost ko bolke sunao

Ek stretchy sheet par drawn graph paper ki picture karo. Ek tiny square chuno: woh $d u$ wide aur $d v$ tall hai, toh iski area $d u d v$ hai — aasaan. Ab sheet ko khiincho aur twist karo. Square ke bottom-left corner ko home base track karo. Jab tum home base ko $d u$ se rightward nudge karo, uske paas wala corner kisi nayi jagah slide ho jaata hai — us arrow ko $a$ kaho. Upar $d v$ se nudge karo, aur doosra neighbour arrow $b$ ke saath slide karta hai. Woh do arrows nayi shape ke sides hain, aur kyunki straight edges straight rehte hain, shape ek slanted parallelogram hai. Iski area width-times-height nahi hai (sides tilted hain), yeh $∣ a_{x} b_{y} - a_{y} b_{x} ∣$ hai — cross-product rule, jo quietly slant ko subtract karta hai aur zero deta hai jab sides line up ho jaayein. Feed karo ki $a$ aur $b$ actually kya hain, aur woh expression $∣ x_{u} y_{v} - x_{v} y_{u} ∣ d u d v$ hai: tiny square ki area Jacobian se multiply ho gayi. Negative Jacobian bas matlab hai sheet mirror-flip ho gayi, toh hum sirf iski size use karte hain. Aur agar Jacobian zero hai, toh square ek line mein crush ho gaya — map wahan reversible nahi hai. Har square ke liye is stretch factor se multiply karo, unhe add karo, aur tumhara total (paint ka, mass ka, probability ka — jo bhi $f$ ho) honest rehta hai.

Recall Quick self-test

Mapped cell ek parallelogram kyun hai rectangle nahi? ::: Do edge-arrows $r_{u} d u$ aur $r_{v} d v$ perpendicular nahi honge zaruri; ek linear map edges ko straight aur parallel rakhta hai lekin unhe tilt karta hai. Subtraction $x_{u} y_{v} - x_{v} y_{u}$ kahaan se aata hai? ::: Cross-product area formula $a_{x} b_{y} - a_{y} b_{x}$ se; minus sign slant cancel karta hai aur parallel edges ke liye $0$ deta hai. $∣ det J ∣$ kyun, $det J$ kyun nahi? ::: Area non-negative hota hai; negative determinant sirf orientation flip (mirror image) record karta hai. $det J = 0$ ka geometrically kya matlab hai? ::: Mapped cell ek segment mein collapse ho gaya (edges parallel); $T$ wahan locally invertible nahi hai.

Parent: Change of variables — general Jacobian