Fundamental theorem for line integrals: Foundations

4.4.28 · D1 · Maths › Multivariable Calculus › Line Integrals ka Fundamental Theorem

Is page mein kuch bhi assume nahi kiya gaya. Parent note (Fundamental theorem for line integrals) mein jo bhi symbol aate hain — $F$ , $\nabla f$ , $d r$ , dot, integral $\int_{C}$ — sab yahan blank page se build kiye gaye hain. Upar se neeche padho; har block agli cheez earn karta hai.

0. Woh picture jis par hum baar baar aate hain

Ek geeli pahadiyon wali landscape imagine karo sunset ke waqt. Ground ke har point par tum pooch sakte ho: main kitni unchai par hoon? Woh ek number, height, hamaara pehla object hai.

Figure — Fundamental theorem for line integrals

1. Plane mein ek point: $(x, y)$

Simple words: ground par ek address.
Picture: landscape ke peach floor par ek single dot.
Topic ko iske zarorat kyun hai: $f (B) - f (A)$ mein endpoints $A$ aur $B$ bas aisi hi do addresses hain.

Hum $(x, y, z)$ baad mein milenge — same idea, bas "upar" ke liye ek teesra number.

2. Ek scalar function: $f (x, y)$ — height map

Simple words: ek machine: address daalo, height nikalo.
Picture: pahaadi ki surface — lekin ek flat map ki tarah paint ki gayi jahan colour = height.
Topic ko iske zarorat kyun hai: yahi $f$ potential function hai. FTLI ka poora payoff yeh hai ki answer $f (B) - f (A)$ hai, is machine ki do readings.

Us figure mein curved lines contour lines (ya level curves) hain: woh saari addresses jo same height par hain, bilkul real hiking map ki tarah. Jahan yeh bheed ho jaati hain, pahaadi steep hai.

3. Partial derivatives: $f_{x}$ aur $f_{y}$

Arrows se pehle, humein ek direction mein ek baar steepness measure karni hai.

Simple words: "Agar main $x$ ko thoda sa nudge karoon aur $y$ ko freeze karoon, toh height kitni tezi se change hoti hai?"
Yeh tool kyun aur ordinary derivative kyun nahi? Ek ordinary derivative $\frac{d g}{d x}$ ko ek variable wale function ki zaroorat hoti hai. Hamare $f$ ke do inputs hain. Partial derivative "sirf ek input mein change" poochne ka honest tarika hai — hum simply doosre input ko ek constant maante hain.

4. Gradient: $\nabla f$ — uphill arrow

Ab hum dono slopes ko ek single object mein bundle karte hain jo upar ki taraf point karta hai.

Simple words: har point par, ek arrow jiska direction seedha uphill hai aur jiska length wahan pahaadi ki steepness hai.
Picture: map par scattered chhote arrows, sab higher ground ki taraf jhuke hue, jahan contours bheed ho wahan sabse lambe.
Topic ko iske zaroorat kyun hai: FTLI sirf un fields ke baare mein hai jinki special form $F = \nabla f$ ho. Poora theorem yeh hai ki "jab field ek gradient ho, toh integral aasaan ho jaati hai."

5. Ek vector: bold $v$ , arrows, aur length

Gradient ek vector hai, toh chalte hain us word ko pin karte hain.

Simple words: "is taraf, itni door jao."
Picture: map par float karta ek single arrow.
Topic ko iske zaroorat kyun hai: field $F$ , gradient $\nabla f$ , position $r (t)$ aur velocity $r^{'} (t)$ sab vectors hain. Bold = arrow, plain = number.

6. Ek vector field: $F (x, y)$

Simple words: weather map par hawa — har jagah ki apni gust hai.
Picture: arrows ka ek poora maidaan.
Topic ko iske zaroorat kyun hai: $\int_{C} F \cdot d r$ measure karta hai ki ek field tumhe "path ke saath kitna push karta hai." Jab $F$ kabhi $\nabla f$ ke barabar ho jaata hai, hum ise conservative kehte hain (dekho Conservative vector fields) aur FTLI apply hoti hai.

7. Ek parametrised curve: $r (t)$

Humein woh trail describe karni hai jo hum chalte hain.

Simple words: ek hiker ki movie; $t$ ghadi hai.
Picture: map par ek curve jisme ek moving dot aur ek timestamp hai.
Topic ko iske zaroorat kyun hai: parent ka Step 1 line integral ko bilkul isi $r (t)$ ka use karke $t$ mein ek ordinary integral mein convert karta hai.

Parent note mein tiny displacement, $d r = r^{'} (t) d t$ , bas "velocity $\times$ tiny time = tiny step" hai.

8. Dot product: $a \cdot b$

$\nabla f$ aur $d r$ ke beech ka symbol ek dot hai, aur yeh ordinary multiplication nahi hai.

Simple words: bada aur positive jab arrows agree karein, zero jab perpendicular hon, negative jab oppose karein.
Yeh tool kyun aur multiplication kyun nahi? Tum do arrows ko numbers ki tarah multiply nahi kar sakte. Lekin tum zaroor pooch sakte ho "field ka kitna hissa mujhe mere step ke saath aage push karta hai?" Woh sawaal — field component along motion — precisely dot product hai. Isliye line integrand $F \cdot d r$ hai aur $F d r$ nahi.

9. Integral sign: $\int$ aur $\int_{C}$

Simple words: tiny contributions ka grand total.
Line integral $\int_{C} F \cdot d r$ : path $C$ ke saath har tiny step par, push-along-the-step number $F \cdot d r$ lo, aur unhe total karo. Dekho Line integrals of vector fields.
Loop $\oint_{C}$ : same, bas ek chhota circle tumhe warn karta hai ki path closed hai (jahan se shuru wahan khatam).

10. Fundamental Theorem of Calculus (1D ancestor)

Simple words: poora = (final value) − (starting value).
Topic ko iske zaroorat kyun hai: FTLI bilkul yahi statement hai, dress-up mein. Parent proof line integral ko $\int_{a}^{b} \frac{d g}{d t} d t$ tak reduce karta hai aur phir yahi quote karta hai. Dekho Fundamental Theorem of Calculus.

11. Symbols ko ek sentence mein rakhna

Ab parent ke headline formula ka har piece defined hai: $§9 path ke saath sum \int_{C} §4 uphill arrows \nabla f \cdot §7 tiny step d r = §2 do height readings f (B) - f (A) .$

Zor se padho: "Trail ke har tiny step ke saath uphill-push ko add karna, end par height minus start par height ke barabar hai."

Prerequisite map

Equipment checklist

Khud test karo — right side cover karo aur reveal karne se pehle jawab do.

Scalar function

f (x, y)

kya hai?

Ek machine jo ek point leta hai aur ek number return karta hai — yahan, pahaadi ki height.

Partial derivative

f_{x}

kya measure karta hai?

f

ki rate of change jab tum

x

-direction mein kadam badhate ho,

y

ko frozen rakh ke.

\nabla f

components mein kya hai, aur yeh kahan point karta hai?

\nabla f = (f_{x}, f_{y})

; yeh steepest ascent (uphill) ki direction mein point karta hai, length steepness ke barabar hoti hai.

Vector field

F (x, y)

kya hai?

Ek rule jo plane ke har point par ek arrow

(P, Q)

attach karta hai.

r (t)

kya hai aur

r^{'} (t)

kya hai?

r (t)

curve par time

t

par tumhari position hai;

r^{'} (t)

velocity hai, curve par tangent.

Dot product

(2, 3) \cdot (4, - 1)

compute karo.

2 \cdot 4 + 3 \cdot (- 1) = 8 - 3 = 5

Dot product do arrows ke baare mein tumhe kya batata hai?

Woh kitna same taraf point karte hain — positive agar aligned hon, zero agar perpendicular hon, negative agar oppose karein.

Ordinary Fundamental Theorem of Calculus state karo.

\int_{a}^{b} g^{'} (t) d t = g (b) - g (a)

: tiny changes ka sum total change ke barabar hai.

\int_{C} \nabla f \cdot d r

ek line mein

f (B) - f (A)

kyun hota hai?

Integrand

g (t) = f (r (t))

ki derivative hai, toh 1D FTC deta hai

g (b) - g (a) = f (B) - f (A)

Connections

Fundamental theorem for line integrals — parent topic jisko yeh foundations feed karte hain.
Gradient — uphill-arrow operator $\nabla$ jo §4 mein build kiya gaya.
Line integrals of vector fields — §9 mein path-ke-saath-sum.
Conservative vector fields — fields ki special form $F = \nabla f$ .
Potential functions — §2 ka height map $f$ .
Fundamental Theorem of Calculus — §10 ka 1D ancestor.
Green's theorem — jahan yahi $P, Q$ dobara appear hote hain.
Curl — §3 jaise partials se bana 3D test.