Foundations — Line integrals — scalar and vector, work done

4.4.27 · D1 · Maths › Multivariable Calculus › Line integrals — scalar and vector, work done

Isse pehle ki tum parent note padh sako, tumhe kuch symbols mein fluent hona hoga. Yeh page har ek ko zero se build karta hai, us order mein jisme ek dusre par depend karte hain. Agar neeche koi symbol scary lagta hai, toh yeh ek sign hai ki parent note ne use assume kiya tha — toh hum use yahan pehle earn karte hain.

1. Number line aur ordinary integral ${\int }_a^{b}f(x)dx$

Chalte hain us cheez se shuru karte hain jo tumhe pehle se pata hai.

• Seedhi baat: $f$ ki total accumulated amount us interval ke paar.
• Picture: ek curve ke neeche area, patale rectangles se bana hua (figure mein blue strips dekho).
• Topic ko yeh kyun chahiye: ek line integral bilkul yahi hai, sirf "interval" space mein ek curve mein bend ho jaata hai. Jo bhi line integral hum compute karenge, woh aakhiri step mein inhi mein se ek ban jaayega.

Do cheezein dhyaan se dekho: $f(x)$ height hai, aur $dx$ choti width hai. Line integrals ka poora khel yeh hai: jab raasta curved ho toh choti width ka kaam kya karta hai?

2. Plane mein points aur vectors — $(x,y)$ aur ${\mathbb{R}}^n$

• Seedhi baat: ek location, ya equivalently ek arrow jo us location ki taraf point karta hai.
• Picture: ek dot jisme ek arrow corner $(0,0)$ se us tak pahunch raha ho.
• Topic ko yeh kyun chahiye: curves is space mein rehte hain, aur forces har point par attached arrows hain.

3. Parameter $t$ aur parametrised curve $\mathbf{r}(t)$

Yeh woh trick hai jo sab kuch kaam karvati hai.

• Seedhi baat: $t$ ek ghadi jaisi hai; $\mathbf{r}(t)$ woh jagah hai jahan ek walker time $t$ par khada hai.
• Picture: ek moving dot jo ek bent path trace karta hai, jiske saath kuch time-stamps likhe hain.
• Topic ko yeh kyun chahiye: curve 1-dimensional hai — tumhe sirf ek number chahiye yeh kehne ke liye ki tum us par kahan ho. Woh ek number $t$ hai, aur wahi hai jo ek curve integral ko plain interval integral mein collapse karne deta hai (§1).

Poore toolkit ke liye Parametric curves dekho; yahan humein sirf "ek dial $t$ dot ko move karta hai" chahiye.

4. Velocity ${\mathbf{r}}^{\prime }(t)$ — walk ki direction aur rate

• Seedhi baat: dot abhi kis taraf, aur kitni tezi se, move kar raha hai.
• Picture: ek chota arrow curve ke saath sath chalta hua, usse tangent (orange arrows dekho).
• Yeh tool kyun, koi aur kyun nahin? Curve par choti step measure karne ke liye humein motion ki direction jaanni hogi. Derivative bilkul wahi machine hai jo "instantaneous direction aur rate" report karti hai, toh yeh kaam ke liye sahi (aur ek hi) tool hai.

Short time $\Delta t$ mein chota displacement hai $\Delta \mathbf{r}\approx {\mathbf{r}}^{\prime }(t)\Delta t,$ yaani (direction-and-rate) × (kitne der hum move kiye) = (actual chota jump). Yeh ek line dono line integrals ka seed hai.

5. Ek arrow ki length — $|\mathbf{v}|$ aur speed $|{\mathbf{r}}^{\prime }(t)|$

• Seedhi baat: arrow kitna lamba hai, direction ignore karke — hamesha $\ge 0$.
• Picture: ek box ka diagonal jiske sides $v_1$ aur $v_2$ hain.
• Topic ko yeh kyun chahiye: velocity par apply karne par, $|{\mathbf{r}}^{\prime }(t)|$ speed hai — aur speed exactly wahi hai jo time mein step ko length mein step mein convert karti hai: $\Delta s\approx |{\mathbf{r}}^{\prime }(t)|\Delta t.$

Kyunki square root kabhi negative nahi hota, $ds\ge 0$: iska koi direction nahi. Yeh yaad rakho — isi liye scalar integral ko parwah nahi ki tum kis taraf chal rahe ho. Arc length bhi dekho, jo aur kuch nahi bas ${\int }_{C}1ds$ hai.

6. Dot product $\mathbf{a}\cdot \mathbf{b}$ — "ek arrow ka kitna hissa doosre ke saath hai"

• Seedhi baat: measure karta hai ki do arrows direction mein kitna agree karte hain. Same taraf → positive; perpendicular → zero; opposite → negative.
• Picture: $\mathbf{a}$ ki shadow $\mathbf{b}$ ki line par daalo; dot product us shadow ki length track karta hai ($|\mathbf{b}|$ se multiply karke).
• Yeh tool kyun, koi aur kyun nahin? Work sirf force ke us hisse ko count karta hai jo motion ke saath point karta hai. Dot product exactly wahi machine hai jo "along-part" extract karta hai, toh yahan yeh unavoidable hai.

Poori details Dot product mein hain; yahan humein sirf "yeh along-part rakhta hai aur sideways part fek deta hai" chahiye.

7. Force field $\mathbf{F}$ aur uske components $(P,Q)$

• Seedhi baat: pushes ka ek map — har jagah, ek arrow batata hai kis taraf aur kitni force se.
• Picture: chote arrows ki ek grid (ek "wind map"), har jagah alag alag.
• Topic ko yeh kyun chahiye: yeh woh "hawa" hai jo walker ko feel hoti hai. Vector line integral poore safar mein uski forward-push add karta hai.

8. Unit tangent $\mathbf{T}$ — travel ki pure direction

• Seedhi baat: "main kis taraf face kar raha hoon", "kitni tezi se" ke bina.
• Picture: orange velocity arrow, length ek tak shrink ya stretch kiya gaya.
• Topic ko yeh kyun chahiye: yeh vector integral ko scalar ke roop mein likhne deta hai, $\mathbf{F}\cdot d\mathbf{r}=(\mathbf{F}\cdot \mathbf{T})ds$ — "force ka forward-part" times "tiny length". Yeh integral ke dono flavors ko bridge karta hai. (Undefined jab $|{\mathbf{r}}^{\prime }(t)|=0$, yaani jab walker momentarily ruk jaata hai — ek degenerate case jisse bachna chahiye.)

9. Poori vocabulary ek hi nazar mein

Upar har ek symbol, aur yeh kahan flow karta hai:

Ise upar se neeche padho: ordinary integral aur point ka idea milke parametrised curve deta hai; use differentiate karne par velocity milti hai; velocity ki length step size $ds$ deti hai; dot product aur force field milke work dete hain.

Equipment checklist

Right side cover karo aur khud test karo. Agar koi bhi jawab shaky lage, toh us section ko dobara padho.

Connections

• Parametric curves — $\mathbf{r}(t)$ aur $t$ ka source.
• Arc length — case ${\int }_{C}1ds$ jo poori tarah $ds=|{\mathbf{r}}^{\prime }|dt$ par bana hai.
• Dot product — §6 aur §8 ka engine.
• Line integrals — scalar and vector, work done — parent topic jise yeh page feed karta hai.