Vector fields — definition, visualization — Hinglish notes

4.4.23 · Maths › Multivariable Calculus

Vector field kya hota hai?

YEH actually kya hai: input ek point hai, output ek arrow hai jo us point par rooted hai. Components $P, Q, R$ tumhe arrow ke coordinates batate hain.

Same dimension KYU? Kyunki arrow ko us point ke saath same space mein "rehna" padta hai (plane ke point par plane ka arrow). Mathematically $R^{2} \to R^{3}$ function bhi valid hai, lekin "arrow attached at the point" wali geometric picture tabhi cleanly kaam karti hai jab input aur output dimension match karein.

Use kaise karein / draw karein (visualization)

Har arrow teen cheezein encode karta hai:

Base point — space mein kahan hai.
Direction — $\frac{F}{∣ F ∣}$ .
Magnitude — $∣ F ∣ = P^{2} + Q^{2}$ (arrow ki length ya color).

Figure — Vector fields — definition, visualization

Worked examples

Khaas fields jo jaanne chahiye

Flashcards

Vector field kya hota hai (formal)?

Ek function

F : D \subseteq R^{n} \to R^{n}

D

ke har point ko ek vector assign karta hai.

2D mein

F = P i + Q j

mein

P

aur

Q

kya hain?

Scalar component functions

P (x, y)

aur

Q (x, y)

Output dimension input dimension se match kyun karta hai?

Taaki arrow apne base point ke saath same space mein rahe (geometric "arrow attached at point" picture).

Ek drawn arrow teen cheezein kya encode karta hai?

Base point (location), direction

F /∣ F ∣

, aur magnitude

∣ F ∣

(length/color).

F (x, y) = (x, y)

describe karo.

Radial outward field; arrows origin se door point karte hain, length

= r

(ek source).

F (x, y) = (- y, x)

describe karo.

Rotation field; arrows circles ke tangent hote hain, counter-clockwise swirl.

(- y, x)

purely rotational hai yeh kaise dikhate hain?

F \cdot (x, y) = - y x + x y = 0

, toh yeh radius ke perpendicular hai ⇒ circles ka tangent.

Gradient (conservative) field kya hota hai?

Ek field jo kisi scalar function

f

ke liye

\nabla f

ke barabar ho.

\nabla (x^{2} + y^{2})

compute karo.

(2 x, 2 y)

Inverse-square field ki form kya hai?

F = c r /∣ r ∣^{3}

, magnitude

∣ c ∣/ r^{2}

Recall Feynman: 12-saal ke bachche ko explain karo

Ek bada sa invisible map imagine karo. Map par har ek jagah par ek chhota sa arrow drawn hai. Arrow batata hai ki agar tum wahan khade ho toh tumhe kis taraf push kiya jaayega aur kitni takat se. Ek fan ke paas khade ho toh arrow tumhe door push karta hai; ek whirlpool mein khade ho toh arrow tumhe circle mein ghuma deta hai. Ek vector field bas yeh hai — poora map inn "push arrows" se bhara hua — har jagah ke liye ek arrow. Isse draw karne ke liye tum bahut saari jagahon par jaate ho, poochte ho "kis taraf aur kitna tez?", aur woh chhota arrow sketch kar dete ho.

Connections

Gradient and Directional Derivatives — gradient fields scalar functions se aate hain.
Divergence and Curl — ek field ke spreading vs. swirling ko measure karte hain.
Line Integrals — ek field ko ek path ke along integrate karo (kaam kiya gaya).
Conservative Fields and Potential Functions — kab $F = \nabla f$ hota hai?
Flux and the Divergence Theorem — surfaces ke through ek field ka flow.
Parametric Curves and Velocity — velocity curve ke along ek vector field hai.

Vector fields — definition, visualization