Foundations — Directional derivative — definition, formula

4.4.8 · D1 · Maths › Multivariable Calculus › Directional derivative — definition, formula

Yeh page woh alphabet build karta hai jo parent Directional derivative note secretly assume karta hai. Upar se neeche padho: har symbol earn hota hai pehle, tab agle mein use hota hai.

1. Ek point $\mathbf{a}$ — jahan tum khade ho

Pehle yeh poochhne se pehle ki "height kitni fast change hoti hai," humein kehna hoga ki kahan khade hain hum. Flat land mein, ek location ke liye do numbers chahiye: kitna east ($x$) aur kitna north ($y$).

Picture hai ek grid par ek single dot. Abhi kuch move nahi ho raha — yeh sirf "you are here" hai.

2. Ek function $f$ — zameen ki height

Ab har location ko ek height attach karo. Yahi kaam karta hai do variables ka function.

Picture hai ek landscape: flat grid ke upar floating ek bumpy surface. Har grid point surface ki taraf upar (ya neeche) jaata hai amount $f(x,y)$ se.

Related vault reading: Level curves and level sets is surface ko constant heights par slice karta hai.

3. Ek direction $\mathbf{u}$ aur uski length $\Vert \mathbf{u}\Vert$ — jis taraf tum face kar rahe ho

Ek vector ek arrow hai: iska ek direction hota hai aur ek length. Hum likhte hain $\mathbf{u}=(u_1,u_2)$ — wahi pair-of-numbers notation jaise ek point, lekin ab iska matlab hai "jahan bhi ho, wahan se $u_1$ east aur $u_2$ north jao."

4. Partial derivatives $\frac{\partial f}{\partial x}$ — due east face karte waqt steepness

Koi bhi direction allow karne se pehle, do easy ones master karo: due east aur due north.

Picture: landscape ko ek vertical wall se slice karo jo east–west run karta hai. Cut edge ek ordinary 1-D curve hai, aur $f_x$ uska slope hai.

5. Gradient $\nabla f$ — arrow jo sabse zyada uphill point karta hai

Do compass slopes ko ek arrow mein bundle karo.

6. Dot product $\nabla f\cdot \mathbf{u}$ — tumhari facing kitna uphill ki taraf lean karti hai

Do arrows, ek number.

7. Directional derivative $D_{\mathbf{u}}f$ — woh ek number jiske liye saare symbols build hue the

Ab hum finally us cheez ko naam de sakte hain.

Us number ko precise banane ke liye humein ek aur tool chahiye — the limit.

Yahan helper $g$ hai jo haare pieces se tied hai. $\mathbf{a}$ par khade ho aur unit direction $\mathbf{u}$ ke along distance $h$ step lo toh tum point $\mathbf{a}+h\mathbf{u}$ par pahuncho. Maano $g(h)=f(\mathbf{a}+h\mathbf{u})$ height hai jaise tum direction $\mathbf{u}$ mein straight line walk karte ho. Toh $g(0)=f(\mathbf{a})$ (tum hile nahi), aur difference quotient exactly hai "height mein rise $÷$ chali gayi distance." Is $g$ ko limit mein feed karna wo full definition deta hai jiske liye is page ka har symbol build ho raha tha:

Aur yeh limit collapse ho jaata hai (jab $f$ differentiable ho) §5–§6 ke gradient aur dot product use karke ek clean shortcut mein: $D_{\mathbf{u}}f=\nabla f\cdot \mathbf{u}=\Vert \nabla f\Vert \cos \theta ,$ jo sabse bada hota hai ($\theta =0$, seedha uphill face karo), zero ($\theta =90^{\circ }$, level curve ke along chalo), aur sabse negative ($\theta =180^{\circ }$, seedha downhill).

Prerequisite map

Equipment checklist

Khud test karo — sirf tab reveal karo jab zor se answer de diya ho.