Foundations — Vector fields — definition, visualization

4.4.23 · D1 · Maths › Multivariable Calculus › Vector fields — definition, visualization

Yeh page kuch bhi assume nahi karta. Hum har ek symbol ko build karenge jo parent parent note mein use hota hai, ek ek brick karke, aur har brick pehle wali ke upar tikti hai.

1. Ek point — "kahan" ka jawab

Picture: figure s01 dekho. Kala dot $(3,2)$ par baitha hai — teen across, do upar.

Yeh topic ko kyun chahiye: ek vector field ka input ek point hota hai. Yeh poochne se pehle ki "yahan kaunsa arrow rehta hai?" humein yahan kehne ka ek solid tarika chahiye. Symbol $x$ horizontal coordinate ko mean karta hai; $y$ vertical ko. Bas itna hi.

2. $\mathbb{R}$ aur ${\mathbb{R}}^n$ — jis space mein hum rehte hain

Picture: $\mathbb{R}$ ek line hai, ${\mathbb{R}}^2$ kaagaz ki ek sheet hai, ${\mathbb{R}}^3$ woh room hai jisme tum baithe ho.

Yeh topic ko kyun chahiye: parent likhta hai $\mathbf{F}:D\subseteq {\mathbb{R}}^n\to {\mathbb{R}}^n$. Ab tum scenery padh sakte ho: field $n$-dimensional space mein rehta hai. Symbol $\subseteq$ (padho "is a subset of / is contained in") bas yeh kehta hai ki hamara region $D$ us space ka koi patch hai — shayad poora, shayad ek disk.

3. Ek vector — "kis direction mein aur kitni force se" ka jawab

Picture: figure s02 mein wahi pair $(3,2)$ do baar dikhaya hai — ek baar point ke roop mein (dot), ek baar arrow ke roop mein (origin par rooted). Same numbers, alag meaning.

Yeh topic ko kyun chahiye: ek vector field ka output ek vector hota hai — har point par arrow.

4. Components $P,Q,R$ — arrow ke ingredients

Picture: ek arrow pehle $P$ right step lekar, phir $Q$ upar step lekar banta hai — arrow woh diagonal shortcut hai (figure s03).

Yeh topic ko kyun chahiye: parent likhta hai $\mathbf{F}(x,y)=(P(x,y),Q(x,y))$. Iska matlab hai: ek point daalo, aur do choti machines $P$ aur $Q$ har ek ek number ugalti hain; unhe arrow mein jodo. Radial field $\mathbf{F}=(x,y)$ mein, machine $P$ bas $x$ return karta hai aur $Q$ return karta hai $y$.

5. Unit vectors $\mathbf{i},\mathbf{j},\mathbf{k}$

Picture: figure s03 — teen chhote kale arrows, har axis direction ke liye ek, har ek length $1$.

Yeh topic ko kyun chahiye: $P\mathbf{i}+Q\mathbf{j}$ bas $(P,Q)$ likhne ka ek aur tarika hai. Padho "$P$ steps right-direction mein plus $Q$ steps up-direction mein." Dono notations identical arrow mean karte hain — parent dono ko interchangeably use karta hai.

6. Magnitude $|\mathbf{F}|$ aur square root

Picture: figure s04 — right triangle jiske legs $P$ aur $Q$ hain (kale) aur arrow red hypotenuse ke roop mein.

Yeh topic ko kyun chahiye: magnitude teen cheezein mein se ek hai jo har drawn arrow encode karta hai (length ya color). x​ kabhi negative nahi ugalti — ek length negative nahi ho sakti, jo bilkul sahi hai.

7. Direction $\frac{\mathbf{F}}{|\mathbf{F}|}$ — normalizing

Yeh topic ko kyun chahiye: fields draw karte waqt hum aksar saare arrows ko same length dete hain taaki woh overlap na karein; $\frac{\mathbf{F}}{|\mathbf{F}|}$ se hum woh uniform length paate hain direction preserve karte hue. Magnitude tab color ke roop mein saath chali aati hai.

8. Dot product $\mathbf{a}\cdot \mathbf{b}$ — perpendicularity detector

Yeh topic ko kyun chahiye: parent prove karta hai ki rotation field $(-y,x)$ pure swirl hai yeh dikhaakar ki $\mathbf{F}\cdot (x,y)=0$ — arrow origin se line ke perpendicular hai, isliye circle ke tangent hai. Woh poora argument tab tak unreadable hai jab tak $\cdot$ define na ho.

9. Partial derivative $\frac{\partial f}{\partial x}$ aur gradient $\nabla f$

Yeh topic ko kyun chahiye: parent ka Example 4 field $\nabla f=(2x,2y)$ banata hai $f=x^2+y^2$ se. Kyunki $\nabla f$ khud numbers ka ek pair hai jo point par depend karta hai, yeh ek vector field hai. Puri detail Gradient and Directional Derivatives mein hai — yahan tumhe bas symbol pehchanna hai.

10. Field khud: $\mathbf{F}:D\to {\mathbb{R}}^n$

Ab parent ki headline definition ka har piece earn ho gaya hai:

Prerequisite map

Equipment checklist

Khud test karo — right side cover karo, jawab do, phir reveal karo.

Connections

• Parent: Vector fields — yahan ki har cheez us definition ko feed karti hai.
• Gradient and Directional Derivatives — jahan $\nabla f$ aur partial derivatives properly build hote hain.
• Divergence and Curl — component functions $P,Q,R$ ke agle uses.
• Parametric Curves and Velocity — vectors as velocity along a path.