Exact equations — exactness condition, finding potential function: Foundations

4.6.6 · D1 · Maths › Ordinary Differential Equations › Exact equations — exactness condition, finding potential fun

Tumhaar parent note ki kisi bhi line par trust karne se pehle, tumhe genuinely dekhna hoga ki har symbol ka matlab kya hai. Yeh page unhe ek ek karke build karta hai, ek aisi reader ke liye jo kabhi partial derivative se nahi mili. Yahan kuch bhi assume nahi kiya gaya hai; har tool use hone se pehle samjhaya gaya hai.

1. Starting object: do variables wali function $F (x, y)$

Is chapter mein sab kuch ek aisi function ke aas paas ghoomta hai jo do numbers leta hai aur ek return karta hai.

Plain words → picture → kyun.

Plain words: do dials $x, y$ aur ek output number wali machine.
Picture: ek landscape. Har floor point $(x, y)$ ke upar ek hill surface hai height $F$ par.
Topic ko kyun chahiye: exact equation ka poora solution sentence hai " $F (x, y) = c$ ". Yeh jaane bina ki do-input function kya hoti hai, woh sentence meaningless hai. Is par build hone wali machinery ke liye Total Differential & Partial Derivatives dekho.

2. Level curves: contour lines $F (x, y) = c$

Ek hill 3D hoti hai aur draw karna mushkil hota hai. Mapmakers ise flatten karte hain un sab floor points ka set draw karke jo ek hi height par hain.

Plain words → picture → kyun.

Plain words: "ek hi altitude par sab jagah."
Picture: flat floor par draw kiye gaye loops (figure mein coloured rings dekho). Ek ring par chalo aur tumhari height frozen rehti hai.
Topic ko kyun chahiye: ek exact ODE ka answer inhi rings mein se ek hota hai. Equation solve karna = yeh identify karna ki tumhe kaunsi ring par rehne ko kaha gaya tha. Deep dive: Level Curves & Contour Lines.

Recall Agar tum ek level curve par chalte ho, toh

$F$ kaise change hota hai? Bilkul bhi nahi — by definition curve par har point ki same value $c$ hai, toh height mein change exactly zero hai. Woh "zero change" hi equation $d F = 0$ ka beej hai.

3. Partial derivatives: $\frac{\partial F}{\partial x}$ aur $\frac{\partial F}{\partial y}$

Hill describe karne ke liye humen uski slopes chahiye. Lekin ek hill par har point par do independent slopes hoti hain: east chalte waqt steepness, aur north chalte waqt steepness.

Plain words → picture → kyun.

Plain words: east ki taraf slope ( $F_{x}$ ) aur north ki taraf slope ( $F_{y}$ ).
Picture: ek point par hill se nikale gaye do perpendicular ramps (red ramp $x$ -slope hai, mint ramp $y$ -slope).
Topic ko kyun chahiye: parent note ki central boxed equations hain $F_{x} = M$ aur $F_{y} = N$ . Tumhari ODE mein coefficients $M$ aur $N$ secretly yahi do slopes hain. Full treatment: Total Differential & Partial Derivatives.

4. Differentials $d x$ , $d y$ , aur total differential $d F$

Plain words → picture → kyun.

Plain words: thoda east climb karna aur thoda north climb karna mil kar tumhara total climb deta hai.
Picture: figure s03 ke wahi do ramps, ab use kiye ja rahe hain estimate karne ke liye ki ek diagonal micro-step mein kitni height gain hoti hai.
Topic ko kyun chahiye: parent ki "exact" ki definition exactly yahi hai ki " $M d x + N d y$ ek total differential $d F$ hai." Toh $M$ pair karta hai $F_{x}$ ke saath aur $N$ pair karta hai $F_{y}$ ke saath. Yahan se $M = F_{x}, N = F_{y}$ janm leta hai.

Recall Level curve par rehna

$d F = 0$ kyun force karta hai? Ek contour par height kabhi nahi badlti, toh curve ke saath kisi bhi step par iska change $d F$ $0$ hota hai. Isliye $M d x + N d y = 0$ hi yeh sentence hai " $d F = 0$ ", yaani "contour mat chhodo."

5. Coefficients $M (x, y)$ aur $N (x, y)$

6. Mixed second partials & Clairaut: $F_{x y} = F_{y x}$

Ek slope lo, phir us slope ki slope doosri direction mein lo. Woh hai mixed second partial.

Plain words → picture → kyun.

Plain words: "east-slope north jaane par kitna badalti hai" equals "north-slope east jaane par kitna badalti hai."
Picture: hill ka ek tiny square patch; tum iske around clockwise ya anticlockwise chadh sakte ho — corner heights dono mixed slopes ko agree karne par force karti hain.
Topic ko kyun chahiye: kyunki $M = F_{x}$ aur $N = F_{y}$ , hume milta hai $M_{y} = F_{x y}$ aur $N_{x} = F_{y x}$ . Clairaut kehta hai yeh equal hain, jo exactness test $M_{y} = N_{x}$ deta hai. Yahi poora justification hai. Dekho Clairaut's Theorem (Equality of Mixed Partials).

F (x, y)

kya return karti hai aur use picture kar sakta/sakti hoon

Floor point

(x, y)

ke upar ek single height number — ek hill surface.

Main ek sentence mein level curve describe kar sakta/sakti hoon

Ek hi height share karne wale sab floor points,

F (x, y) = c

— ek contour ring.

Main jaanta/jaanti hoon ki

\partial

symbol

d

ki jagah kyun use hota hai

Kyunki do inputs hain;

\partial

ka matlab hai "ek direction mein slope jabki doosra input fixed rakha gaya ho."

Main

F = x^{2} y

ke liye

F_{x}

aur

F_{y}

compute kar sakta/sakti hoon

F_{x} = 2 x y

aur

F_{y} = x^{2}

Main total differential

d F

likh sakta/sakti hoon

d F = F_{x} d x + F_{y} d y

Main jaanta/jaanti hoon

M

aur

N

kya hain

M d x + N d y = 0

mein

d x

aur

d y

ke coefficients.

Main Clairaut's theorem state kar sakta/sakti hoon

Continuous second partials ke liye,

F_{x y} = F_{y x}

Main explain kar sakta/sakti hoon ki Clairaut exactness test kaise deta hai

M = F_{x}, N = F_{y} \Rightarrow M_{y} = F_{x y} = F_{y x} = N_{x}

Main jaanta/jaanti hoon

g (y)

(sirf

+ C

nahi) kyun aata hai

x

mein integrate karte waqt

y

constant mana jata hai, toh missing piece

y

par depend kar sakta hai.

Main jaanta/jaanti hoon "simply connected" kyun matter karta hai

No holes slopes ko ek single-valued potential

F

mein jodne deta hai.

Recall Jaane se pehle ek-line summary

$M$ aur $N$ ek hidden hill $F$ ki do slopes hain; agar unki cross-slopes match karti hain ( $M_{y} = N_{x}$ ) toh hill actually exist karti hai, aur ODE sirf yeh kehti hai "contour $F = c$ par raho."

Foundations — Exact equations — exactness condition, finding potential function

1. Starting object: do variables wali function $F (x, y)$

2. Level curves: contour lines $F (x, y) = c$

3. Partial derivatives: $\frac{\partial F}{\partial x}$ aur $\frac{\partial F}{\partial y}$

4. Differentials $d x$ , $d y$ , aur total differential $d F$

5. Coefficients $M (x, y)$ aur $N (x, y)$

6. Mixed second partials & Clairaut: $F_{x y} = F_{y x}$

7. "Simply-connected region" — no-holes condition

8. Integration ka constant jo secretly ek function hai: $g (y)$

9. Symbol $=!$ aur " $c$ "

Prerequisite map

Equipment checklist