4.4.26 · D1Conservative vector fields — potential functions
Ek vector field space mein arrows ki ek picture hai; kabhi kabhi ye arrows secretly kisi chupi hui landscape par "downhill" point kar rahe hote hain. Jab wo landscape exist karta hai, tab us field mein chalné ka energy cost sirf aapki altitude ki change ke barabar hoti hai — isliye loops ka cost kuch nahi hota, aur us field ko conservative kehte hain.
Isse pehle ki tum parent topic ki ek bhi line padh sako, tumhe un symbols mein fluent hona padega jo wo use karta hai. Ye page har ek cheez ko bilkul kuch nahi se build karta hai, ek aisi order mein jahan har naya idea sirf pehle waale ideas par depend karta hai.
Picture ek grid hai, jaise graph paper. Paper par har dot ka ek address hai. Parent topic baar baar f ( x , y ) A = ( 0 , 0 ) B = ( 1 , 2 )
Topic ko ye kyun chahiye: yeh poora subject un quantities ke baare mein hai jo jagah jagah alag hoti hain . Tum "jagah jagah alag hoti hain" nahi keh sakte jab tak jagahon ko name dene ka koi tarika na ho.
Definition Scalar function (scalar field)Ek scalar function f f ( x , y )
Figure dekho. Neeche ka flat grid points ( x , y ) f ( x , y ) hill — ek landscape. Yahi wo object hai jise parent topic potential function kehta hai.
"Scalar" ka matlab sirf "ek plain number, koi direction nahi." Height ek scalar hai: 500 hota hai kisi taraf.
Topic ko ye kyun chahiye: "conservative" hone ka poora fayda yahi hai ki arrows ka ek pura field ek AKELE aise hill se summarise ho sakta hai. Sab kuch is surface se heights padhne mein reduce ho jata hai.
Ek vector ek arrow hai: iske paas ek length hai (kitna strong) aur ek direction (kaunsi taraf). Hum use components ki list ke roop mein store karte hain — har axis ke saath kitna pahunchta hai.
2D: F = ( P , Q ) P Q
3D: F = ( P , Q , R )
Bold letter F P Q R components hain — andar ke individual numbers.
P Q P Q fixed numbers nahi hain. Ye point se point par change hote hain, to actually P ( x , y ) Q ( x , y ) ( 1 , 0 ) ( 2 , 5 )
Ek vector field ek vector ko space ke har point par ek saath chipka deta hai: har address ( x , y ) F ( x , y ) = ( P ( x , y ) , Q ( x , y ))
Figure plane ko arrows se bhara hua dikhata hai — ek "wind map" ya "force field." Ise aise padho: kisi bhi dot par khado, aur wahan ka arrow tumhe us jagah hawa ki strength aur direction batata hai.
Common mistake Vector field vs single vectorEk single vector ek arrow hai. Ek vector field unka poora maidan hai, ek per point. Parent ka F
Topic ko ye kyun chahiye: "conservative" arrows ke poore maidan ki ek property hai, yeh poochhti hai ki kya unhe kisi chupi hui hill par downhill point karne ke liye arrange kiya ja sakta hai.
Definition Partial derivativeHill f ( x , y ) partial derivative ∂ x ∂ f f x steepness hai agar tum purely x , y f y y
∂ d d x d ek variable ke functions ke liye hai — chalné ka sirf ek tarika hai. Ek hill par bahut saari directions hain. Curly ∂ f x
Intuition Topic ko "har direction mein steepness" kyun chahiyeAgar arrows downhill point karte hain, to arrow ka rightward push hill ki rightward steepness se match hona chahiye, aur uski upward push upward steepness se match honi chahiye. Yahi exactly woh statement P = f x Q = f y
Gradient ∇ f f ∇ f = ( ∂ x ∂ f , ∂ y ∂ f ) = ( f x , f y ) . ∇ f
Intuition Gradient arrow kahan point karta haiHill par kisi bhi jagah, ∇ f steepest ascent ki direction mein point karta hai — seedha upar — aur uski length batati hai kitna steep hai. To − ∇ f Gradient and directional derivative dekho.
Topic ko ye kyun chahiye: parent ki central definition literally F = ∇ f
Do vectors a = ( a 1 , a 2 ) b = ( b 1 , b 2 ) dot product ek single number hai:
a ⋅ b = a 1 b 1 + a 2 b 2 .
Intuition Dot product kya measure karta haiYe measure karta hai ek arrow dusre ke saath kitna jata hai . Agar dono same direction mein point kar rahe hain, number large aur positive hai. Perpendicular hain, to zero hai. Opposite hain, to negative hai. Picture mein, amber arrow ka shadow (projection) cyan waale par jo hai, wahi dot product capture karta hai.
Intuition "Ye tool kyun?" kyunParent work ko F ⋅ d r force ki direction mein " hai. Force ka sirf woh part jo tumhari motion ke saath aligned hai, work karta hai — sideways force kuch nahi karta. Dot product precisely woh tool hai jo "aligned part" extract karta hai, isliye ye yahan sahi aur akela natural tool hai.
Definition Parametrised pathEk path r ( t ) t a b r ( t ) = ( x ( t ) , y ( t )) C velocity r ′ ( t )
Intuition Notation padhna
t r ( t ) t ho .r ′ ( t ) t move kar rahe ho.d r = r ′ ( t ) d t
Topic ko ye kyun chahiye: "ek path par travel karke kiya gaya work" ke baare mein baat karne ke liye pehle path ko aur us par motion ko describe karne ka koi tarika hona chahiye. Pieces F ⋅ d r Line integrals of vector fields dekho.
Definition Line integral aur loop integral
∫ C F ⋅ d r C F ⋅ d r ∮ C C closed loop hai (tum wahin khatam hote ho jahan se shuru kiya tha). Sign par chota circle loop ki picture hai.
Intuition Loop symbol kyun apna notation deserve karta haiParent ka headline result hai ki conservative fields ke liye ∮ = 0
Definition Clairaut's theorem (equality of mixed partials)Agar tum ek acchi hill ko pehle x y y x f x y = f y x . Clairaut's theorem (equality of mixed partials) dekho.
Intuition Topic ko Clairaut kyun chahiye2D test P y = Q x P = f x Q = f y P y = f x y Q x = f y x
Definition Curl (swirl detector)Ek field ka curl uska microscopic swirl measure karta hai. 2D mein sirf woh piece matter karta hai jo hai Q x − P y Curl of a vector field dekho aur, loops kyun vanish hone chahiye iski gehra reason ke liye, Green's theorem dekho.
Intuition Conservation of energy"Conservative" naam seedha Conservation of energy in physics se liya gaya hai: agar F f f conserved rehti hai. Yahi poori wajah hai ki word "conservative" hai.
Scalar function f the hill
Vector field arrows everywhere
Partial derivatives f_x f_y
Conservative field F equals grad f
Loop integral equals zero
Apne aap ko test karo — right side cover karo aur zor se jawab do.
Scalar function f ( x , y ) Ek hill: har grid point ke upar ye ek height number deta hai.
Vector field F = ( P , Q ) Har point par ek arrow lagaya gaya; P , Q
∂ x ∂ f ∂ Hill ki steepness purely x y ∂
∇ f Gradient ( f x , f y )
( 1 , 2 ) ⋅ ( 3 , − 1 ) 1 ⋅ 3 + 2 ⋅ ( − 1 ) = 1
r ( t ) r ′ ( t ) Path par time t
∮ Ye ek closed loop ke around line integral hai (start = end).
Clairaut's theorem symbols mein batao. f x y = f y x f
Work ke liye dot product sahi tool kyun hai? Motion ke saath aligned force ka sirf woh part work karta hai; dot product exactly woh aligned part extract karta hai.
