Line integrals — scalar and vector, work done

4.4.27 · Maths › Multivariable Calculus

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1. Setup: curve ko parametrise karna

Parametrise kyun karein? Ek curve intrinsically 1-dimensional hoti hai — aapko yeh batane ke liye sirf ek number ($t$) chahiye ki aap uस par kahan hain. Ek baar jab humare paas $t$ hai, $C$ par har integral ek ordinary single integral mein $t$ mein collapse ho jaata hai, jo hum pehle se jaante hain kaise karna hai.

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2. Scalar line integral — ${\int }_{C}fds$ ko scratch se derive karna

Hum $ds$ ko kuch computable mein kaise convert karein. $t$ se $t+\Delta t$ tak jaayein. Displacement hai $\Delta \mathbf{r}\approx {\mathbf{r}}^{\prime }(t)\Delta t.$ Us tiny step ki length hai $\Delta s=|\Delta \mathbf{r}|\approx |{\mathbf{r}}^{\prime }(t)|\Delta t.$

Key fact (WHY it's robust): result parametrisation ki direction ya speed par depend nahin karta, kyunki $ds$ ek unsigned length hai. Curve ko reverse karein aur aapko wohi answer milega. $f=1$ set karne se arc length $L={\int }_{C}ds$ milta hai.

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3. Vector line integral aur work — ${\int }_C\mathbf{F}\cdot d\mathbf{r}$ derive karna

Ek tiny step par, kiya gaya work $\approx \mathbf{F}\cdot \Delta \mathbf{r}\approx \mathbf{F}\cdot {\mathbf{r}}^{\prime }(t)\Delta t.$ Sum karke aur limit lete hue:

Usi cheez ko padhne ke do tarike — scalar aur vector ko connect karna: ${\int }_C\mathbf{F}\cdot d\mathbf{r}={\int }_C(\mathbf{F}\cdot \mathbf{T})ds,\mathbf{T}=\frac{{\mathbf{r}}^{\prime }(t)}{|{\mathbf{r}}^{\prime }(t)|}.$ Kyun? $d\mathbf{r}={\mathbf{r}}^{\prime }dt=\mathbf{T}|{\mathbf{r}}^{\prime }|dt=\mathbf{T}ds$. Toh ek vector line integral $\mathbf{F}$ ke tangential component ka scalar integral hai.

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4. Common mistakes (Steel-manned)

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5. Active recall

Recall Quick self-test (answers cover karein)

• Kaunsa factor $dt$ ko $ds$ mein convert karta hai? → $|{\mathbf{r}}^{\prime }(t)|$.
• Kaunsa line integral curve reverse karne par sign change karta hai? → Vector wala.
• ${\int }_C\mathbf{F}\cdot d\mathbf{r}$ physically kya represent karta hai? → $\mathbf{F}$ dwara kiya gaya work.
• ${\int }_C\mathbf{F}\cdot d\mathbf{r}$ ko ek scalar integral ke roop mein express karein. → ${\int }_C(\mathbf{F}\cdot \mathbf{T})ds$.

Recall Feynman: ek 12-saal ke bache ko explain karein

Ek cheenti ki kalpana karein jo ek bent wire par chal rahi hai. Scalar line integral: har jagah wire mein ek "heaviness" hoti hai. Heaviness ko us tiny bit of wire se multiply karke add karein jo cheenti cover karti hai — aapko total weight milta hai. Vector line integral: ab ek hawa chal rahi hai. Hume sirf woh hissa chahiye jo hawa cheenti ko aage dhakelta hai. "Forward-push × tiny step" ko poore safar mein add karein — itni help (ya takkar) hawa ne cheenti ko di. Wire ko ulta chalein aur hawa ki help hindrance ban jaati hai — toh yeh direction ki parwaah karta hai, weight wala nahin karta.

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6. Connections

• Arc length — special case ${\int }_{C}1ds$.
• Gradient and conservative fields — jab $\mathbf{F}=\nabla \varphi$, work path-independent hota hai.
• Fundamental Theorem for Line Integrals — sirf endpoints.
• Green's Theorem — ek closed-curve vector line integral ko double integral mein convert karta hai.
• Dot product — "path ke saath component" ka engine.
• Parametric curves — $C$ ki prerequisite description.

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Concept Map

Deep Dive

• D1 · Foundations — har symbol zero se
• D2 · Visual walkthrough — derivation pictures mein
• D3 · Worked examples — har scenario
• D4 · Exercises — graded, full solutions
• D5 · Question bank — concept traps