Change of variables — general Jacobian — Hinglish notes

4.4.21 · Maths › Multivariable Calculus

KYA ho raha hai

KYUN Jacobian aata hai — scratch se derivation

Hum 2D scaling factor ko koi formula yaad kiye bina derive karenge.

$uv$ -plane mein ek tiny rectangle lo jiska corner $(u_{0}, v_{0})$ par hai, width $d u$ aur height $d v$ hai. Iske corners $T$ ke under $x y$ -plane mein points pe map hote hain. $(u_{0}, v_{0})$ ke paas $T$ ki first-order Taylor (linear) approximation use karte hue:

Corner $r (u_{0}, v_{0})$ par hi rehta hai.
$u$ mein $d u$ move karne par: displacement $\approx r_{u} d u = (x_{u}, y_{u}) d u$ . Kyun? Partial derivative $u$ -direction mein velocity vector hota hai.
$v$ mein $d v$ move karne par: displacement $\approx r_{v} d v = (x_{v}, y_{v}) d v$ .

Toh image (first order tak) do edge vectors se bana parallelogram hai: $a = (x_{u}, y_{u}) d u, b = (x_{v}, y_{v}) d v .$

2D mein $a, b$ se bane parallelogram ki area cross product ki magnitude hoti hai ( $z$ -component): $Area = ∣ a_{x} b_{y} - a_{y} b_{x} ∣ = ∣ x_{u} y_{v} - y_{u} x_{v} ∣ d u d v .$

Yeh step kyun? 2D mein $(a_{x}, a_{y}, 0)$ aur $(b_{x}, b_{y}, 0)$ ke cross product ka sirf $z$ -component hota hai jo $a_{x} b_{y} - a_{y} b_{x}$ ke barabar hai; uski magnitude area hoti hai.

Isliye $d A = d x d y \approx \frac{\partial ( x , y )}{\partial ( u , v )} d u d v . ■$

Figure — Change of variables — general Jacobian

KAISE use karein — recipe

Naye variables $u, v$ identify karo (aksar $R$ ya $f$ ko simple banane ke liye).
$x, y$ ko $u, v$ ke terms mein express karo (taaki differentiate aasaan ho).
$\frac{\partial ( x , y )}{\partial ( u , v )}$ compute karo.
$f$ rewrite karo, $d A \to ∣ J ∣ d u d v$ swap karo, aur naya region $S$ dhundo.

Worked Examples

Common Mistakes

Active Recall

Recall Quick self-test (peekne se pehle socho)

$d x d y = r d r d θ$ mein $r$ kyun hai? → kyunki ek polar cell ek curved wedge hai jiska area radius ke saath badhta hai; Jacobian $= r$ .
$det J$ kya measure karta hai? → linearized map ka local area/volume scaling factor.
Absolute value kyun? → area non-negative hoti hai; sign sirf orientation flip encode karta hai.

Recall Feynman: ek 12-saal ke bachche ko explain karo

Socho ek stretchy rubber sheet par ek tiny square bana rahe ho. Jab tum sheet ko kheencho aur marorte ho, woh chota square ek tedha, bada (ya chhota) shape ban jaata hai. Agar tum sheet par paint add karna chahte ho, tumhe pata hona chahiye ki har square kitna bada hua — warna tumhe lagega ki paint zyada ya kam hai jitni actually hai. Jacobian woh number hai jo batata hai "yeh square 3 guna bada hua" (ya aadha hua). Tum isse isliye multiply karte ho taaki tumhara total sahi rahe.

2D change-of-variables formula kya hai?

\iint_{R} f d A = \iint_{S} f (x (u, v), y (u, v)) ∣ \partial (x, y) / \partial (u, v) ∣ d u d v

Jacobian determinant

\partial (x, y) / \partial (u, v)

define karo.

x_{u} y_{v} - x_{v} y_{u}

, yaani

det (x_{u} y_{u} x_{v} y_{v})

Integral mein Jacobian kyun aata hai?

Ek tiny

uv

-cell ek parallelogram pe map hota hai jiska area

∣ det J ∣

se scale hota hai; hum sahi purani-area add karte hain.

det J

ka geometric meaning kya hai?

Linearized map

T

ka local signed area/volume scaling factor.

Jacobian ki absolute value kyun lete hain?

Area non-negative honi chahiye; sign sirf orientation reversal encode karta hai.

Polar coordinates

x = r cos θ, y = r sin θ

ke liye Jacobian?

r

, isliye

d x d y = r d r d θ

\partial (x, y) / \partial (u, v)

aur

\partial (u, v) / \partial (x, y)

mein kya relation hai?

Ye reciprocals hain: product

= 1

(inverse maps par chain rule).

Mapped cell ke do edge vectors kya hain?

r_{u} d u = (x_{u}, y_{u}) d u

aur

r_{v} d v = (x_{v}, y_{v}) d v

T

ki validity ke liye kya requirement hai?

T

smooth, injective, continuous partials ke saath aur region par nonzero Jacobian hona chahiye.

Connections

Polar Coordinates — special case jo $r d r d θ$ deta hai
Spherical and Cylindrical Coordinates — 3D Jacobians ( $ρ^{2} sin ϕ$ , $r$ )
Determinant as Volume Scaling — idea ki linear-algebra root
Cross Product and Area — derivation mein use hua
Inverse Function Theorem — local invertibility guarantee karta hai jab $det J \neq = 0$
Double Integrals over General Regions
Chain Rule (Multivariable) — inverse-Jacobian reciprocal justify karta hai

Change of variables — general Jacobian