4.4.21 · D1Change of variables — general Jacobian
Jab tum ek plane ke points ko naye coordinates se re-label karte ho, to area ke chhote patches stretch ya squeeze ho jaate hain ek aisi number se jo depend karti hai ki tum kahaan ho. Integral ko honest rakhne ke liye, tumhe us local stretch factor se multiply karna padta hai — aur woh factor hai Jacobian determinant ki absolute value .
Yeh page har woh symbol build karta hai jis par parent note Change of Variables rely karta hai, bilkul scratch se. Agar wahan koi word ya squiggle appear ho aur confuse kare, toh woh neeche explain kiya gaya hai, us order mein jo har idea ko pichle idea ke upar rest karne deta hai.
Yahan sab kuch do flat planes mein side by side hota hai.
uv choose karte hain acche coordinates.x y
Definition Ek coordinate pair( u , v )
Hum constantly ek point ko left sheet se right sheet par move karenge. Us move ko map kehte hain.
Definition Map / transformationT rule jo left sheet ke har point ( u , v ) ( x , y ) T ( u , v ) = ( x , y )
Arrow ↦ yahan bheja jaata hai ". Toh ( u , v ) ↦ ( x , y )
T : ( u , v ) ↦ ( x , y ) woh rule hai jo ".
Concretely ek map actually do ordinary functions ek saath bundled hote hain:
x = x ( u , v ) , y = y ( u , v ) . x ( u , v ) x u v
Intuition Map kyun chahiyeHum coordinates ko andar ek integral ke swap karna chahte hain. Map ek exact dictionary hai jo har naye point ko translate karta hai ki woh purani duniya mein kahan land karta hai. Iske bina hum integrand ya region ko rewrite nahi kar sakte.
Do words jo parent T
Map mein koi rips, jumps, ya sharp corners nahi hain — tum x ( u , v ) y ( u , v )
Definition Injective (one-to-one)Koi bhi do alag naye points ek hi purane point par land nahi karte. Imagine karo: map sheet ko kabhi apne upar wapas fold nahi karta. Yeh important hai kyunki agar do cells ek hi spot par pile ho jaate, toh hum us spot ki area do baar count kar lete.
Parent likhta hai r ( u 0 , v 0 ) r u r
Definition Position vectorr r ( u , v ) = ( x ( u , v ) , y ( u , v )) arrow ke roop mein dekha jaata hai origin se us point tak. Bold letters matlab "yeh ek vector (ek arrow) hai, single number nahi."
Ek point ko arrow mein badalne ki zahmat kyun? Kyunki arrows ko subtract kiya ja sakta hai displacement arrows paane ke liye, aur displacements exactly wahi hain jo hume ek mapped patch ka size measure karne ke liye chahiye.
Yeh woh tool hai jo measure karta hai ki output kaise move karta hai jab tum ek input ko nudge karte ho .
Partial derivative kyun, ordinary kyun nahi?
Humara output x do inputs u v the input change hota hai output kaise change hota hai" — lekin hamare paas do inputs hain. Toh hum ek sharpened sawaal poochhte hain: "agar main v u x partial derivative hai.
Definition Partial derivative notation∂ u ∂ x = x u = lim h → 0 h x ( u + h , v ) − x ( u , v ) .
Curly ∂ d doosre variables constant rakhe gaye hain ".
Subscript form x u
Plain words: u x v
Definition Velocity vectorsr u r v r u = ( x u , y u ) , r v = ( x v , y v ) . Picture: r u u u r v v
Intuition Yeh do arrows hi puri game kyun hainuv u v r u d u r v d v
Link Chain Rule (Multivariable) woh jagah hai jahan yeh partials combine hote hain jab maps compose hote hain; parent mein reciprocal trick usi par lean karti hai.
d u , d v d u u infinitely small width"; d v v
d A d A = d x d y x y tiny area hai. Topic ka poora point yeh hai ki d A d u d v nahi hoti — map use rescale karta hai.
"Infinitely small" kyun? Kyunki tabhi jab ek patch chota hota hai, smooth map us par straight (linear) lagta hai. Woh straightness hi hai jo ek triangle-aur-parallelogram argument ko kaam karne deti hai.
Mapped patch ek parallelogram (ek tilted rectangle) hai. Hume iske area ki zaroorat hai.
Definition Do arrows se spanned parallelogramEdge-arrows a b
Intuition Yeh exact combination area kyun measure karta haiQuantity a x b y − a y b x cross product hai — dekho Cross Product and Area . Yeh ∣ a ∣ ∣ b ∣ sin γ γ × = ∣ a ∣ × ( ∣ b ∣ sin γ )
Definition Answer bars se pehle negative kyun aa sakta haiAgar do edges swap ho jaayein (ya map sheet ko mirror ki tarah flip kar de), toh a x b y − a y b x negative nikalta hai. Area kabhi negative nahi hoti, isliye hum ise ∣ ⋯ ∣
Parent un chaar partials ko ek matrix mein bundle karta hai aur uska determinant leta hai. Dono symbols naye hain; yahan hain.
2 × 2 ( p r q s ) columns ke roop mein record karta hai: column one r u r v
2 × 2 det ( p r q s ) = p s − q r . det p s − q r
kyun stretch factor hai
Determinant ko unit square feed karo (edges ( 1 , 0 ) ( 0 , 1 ) det Determinant as Volume Scaling ka poora message hai. Negative determinant = map ne sheet ko flip kar diya (mirror flip); iska size phir bhi stretch hai.
Ab har piece topic ke star mein assemble hoti hai.
Definition Jacobian determinant∂ ( u , v ) ∂ ( x , y ) = det ( x u y u x v y v ) = x u y v − x v y u .
Stacked notation ∂ ( u , v ) ∂ ( x , y ) ek symbol hai jiska matlab hai "( x , y ) ( u , v ) Top = output variables, bottom = input variables.
Plain words: map T
Common mistake Fraction ko normal division ki tarah padhna∂ ( u , v ) ∂ ( x , y ) numerator over denominator nahi hai jo tum cancel kar sako. Yeh poora determinant hai. Ek hi fraction-like fact allowed hai reciprocal rule inverse map ke saath — jo Inverse Function Theorem se aata hai.
Absolute value ∂ ( u , v ) ∂ ( x , y )
Definition Double integral∬ R f d A R f d A ∫ R kaunsa region tum sum kar rahe ho. Dekho Double Integrals over General Regions .
R S
S naye (uv R purane (x y R = T ( S ) ∬ R ( purana ) = ∬ S ( naya ) × ∣ J ∣ dono integrand aur region translate karne honge.
partial derivatives x_u y_v
velocity edge arrows r_u and r_v
parallelogram area via cross product
determinant as area scaling
Jacobian dx dy over du dv
double integral over a region
change of variables theorem
Apne aap ko test karo — answer reveal karne se pehle awaaz mein bolo.
( u , v ) ↦ ( x , y ) ↦ "yahan bheja jaata hai" — left ka input point right ka output point produce karta hai.
"Injective" map ke baare mein kya guarantee karta hai? Koi bhi do alag points same output par land nahi karte, isliye sheet kabhi apne upar fold nahi hoti aur koi area double-count nahi hoti.
x u Woh rate jis par x u v
∂ d Yeh flag karne ke liye ki doosre variables constant rakhe ja rahe hain (ek se zyada input hai).
r u r v Arrows jo dikhate hain ki output kitni tezi se aur kis direction mein move karta hai u v
Ek tiny uv Ek parallelogram (first order tak), r u d u r v d v
a x b y − a y b x Arrows a b
Us quantity ki absolute value kyun lete hain? Uska sign sirf orientation record karta hai (clockwise vs counter-clockwise / mirror flip); area khud non-negative honi chahiye.
2 × 2 Woh factor jis se linear map area scale karta hai.
∂ ( u , v ) ∂ ( x , y ) Output (purane) variables x , y u , v
d A d u d v Tiny old-plane area d x d y d A = ∣ J ∣ d u d v
Variables change karte waqt, integrand ke alawa aur kya translate karna padta hai? Region — uv S
