Foundations — Gradient vector ∇f — definition, properties

4.4.9 · D1 · Maths › Multivariable Calculus › Gradient vector ∇f — definition, properties

Pehle us ek arrow par trust karne se pehle, tumhe page pe har ek mark padhna aana chahiye. Neeche har ek symbol hai jo parent note use karta hai, bilkul zero se build kiya gaya hai, har ek apna next earn karta hai.

0. Stage: ek function $f(x,y)$

Ek flat table ($x$–$y$ plane) socho. Table ke har point ke upar ek height hai. Saare points pe jao aur woh heights ek hill banati hain — ek curved surface jo table ke upar float karti hai.

• $x$ aur $y$ ::: flat map pe tumhari position ke do coordinates (east–west aur north–south).
• $z=f(x,y)$ ::: hill ki height seedha wahan ke upar jahan tum khade ho.

Yeh kyun chahiye: poora topic is hill ki slopes ke baare mein hai, aur slope ki baat tab tak nahi kar sakte jab tak koi surface ho hi na.

1. Point $\mathbf{p}$ aur bold letters

Toh jab parent $f(\mathbf{p})$ likhta hai toh bas matlab hai "jagah $\mathbf{p}$ pe height." Bold letters jaise $\mathbf{p}$, $\mathbf{x}$, $\mathbf{v}$ — har ek ke andar secretly do numbers hote hain.

• $\mathbf{p}$ ::: tum kahan khade ho (ek point).
• $\mathbf{x}$ ::: koi nearby general point jahan tum move kar sakte ho.
• $\mathbf{x}-\mathbf{p}$ ::: $\mathbf{p}$ se $\mathbf{x}$ tak ka arrow — coordinate by coordinate subtract karo.

Yeh kyun chahiye: har jagah $(p_1,p_2)$ likhna noisy hai. Ek bold letter formulas ko readable rakhta hai.

2. Limit $\underset{h\to 0}{\lim }$

Fraction $\frac{f(x+h)-f(x)}{h}$ ko ek stick socho jo hill ko do points pe touch kar rahi hai jo $h$ door hain. Jaise $h$ shrink hota hai, stick pivot karti hai aur ek point pe sahi tangent slope pe settle ho jaati hai.

• $h$ ::: ek tiny step size, zero ki taraf ja raha hai.
• $\to$ ::: "approaches."

Yeh kyun chahiye: slope hai "rise over run," aur sahi slope ke liye infinitesimally small run chahiye. Limit hi ek honest tarika hai yeh kehne ka "chhota run."

3. Partial derivatives $f_x,f_y$

Ab ek coordinate ko freeze karo taaki do-variable hill ek simple one-variable curve ban jaaye.

Curly $\partial$ (kaho "partial dee") ek warning flag hai: "doosre variables abhi bhi hain." Ise seedhe $d$ se compare karo ordinary one-variable calculus ke, jahan hold karne ke liye kuch aur hota hi nahi.

• $\partial$ ::: "derivative, lekin baaki variables constant held hain."
• $f_x$ ::: height kitni tezi se chadhti hai agar tum sirf due east chalo.
• $f_y$ ::: height kitni tezi se chadhti hai agar tum sirf due north chalo.

Topic ko yeh kyun chahiye: $f_x$ aur $f_y$ literally gradient ke do components hain. Inhe build karo aur gradient aadha ban jaata hai. Dekho Partial derivatives.

4. Unit direction $\hat{u}$

Tum kisi bhi compass direction mein chal sakte ho, sirf east ya north mein nahi. Hum ek direction ko ek arrow se naam dete hain.

Length 1 kyun honi chahiye: "climb rate per step" tabhi sense karta hai jab har step same size ka ho. Length-1 arrow guarantee karta hai ek unit of walking, isliye rate honest hai. Length-2 arrow daalo aur har rate double ho jaata hai — ek classic mistake jiske baare mein parent warn karta hai.

• $|\mathbf{v}|$ (bars) ::: arrow $\mathbf{v}$ ki length, $\sqrt{v_1^2+v_2^2}$ se.
• Normalise ::: ek arrow ko length 1 tak shrink karo apni hi length se divide karke: $\hat{u}=\mathbf{v}/|\mathbf{v}|$.

5. Dot product $\mathbf{a}\cdot \mathbf{b}$

Yeh woh machine hai jo "gradient + direction" ko "climb rate" mein badal deti hai.

Geometrically, dot product measure karta hai ek arrow doosre ke saath kitna point karta hai — $\mathbf{a}$ par $\mathbf{b}$ ki shadow ki length, $|\mathbf{a}|$ se scale ki hui.

• Same direction ($\theta =0$): $\cos \theta =1$, dot sabse bada hai.
• Perpendicular ($\theta =90^{\circ }$): $\cos \theta =0$, dot zero hai.
• Opposite ($\theta =180^{\circ }$): $\cos \theta =-1$, dot sabse negative hai.

Topic is par kyun jeeta hai: directional derivative ek dot product hai. Toh "sabse steep direction kaun si hai?" ban jaata hai "kaun si direction $\cos \theta =1$ banati hai?" — aur jawab forced hai: gradient ke saath same direction mein point karo. Yeh ek fact steepest ascent, descent, aur level-curve-ke-perpendicular property ko power karta hai. Dekho Directional derivative.

6. Angle $\theta$ aur $\cos \theta$

$\theta$ (theta) bas woh Greek letter hai jo hum do arrows ke beech ke angle ke liye use karte hain. Yahan yeh tumhari walking direction $\hat{u}$ aur gradient ke beech ka angle hai. $\cos \theta$ woh number hai, $-1$ aur $+1$ ke beech, jo kehta hai woh kitne aligned hain (Section 5). Bas itna hi hai parent ka $|\nabla f|\cos \theta$: gradient ki length, is baat se dim ki gayi ki tum kitne off-direction ho.

7. Gradient symbol $\nabla f$

Ab har part exist karta hai, toh hum star assemble kar sakte hain.

• $\nabla f$ ::: ek point pe saare partial slopes ka arrow.
• $|\nabla f|$ ::: iski length = sabse steep possible climb rate.
• $\nabla f\cdot \hat{u}$ ::: tumhara actual climb rate jab direction $\hat{u}$ mein chalte ho.

Ek symbol kyun: $f_x,f_y$ ko $\nabla f$ mein pack karna puri directional-derivative story ko ek clean line mein shrink kar deta hai, $D_{\hat{u}}f=\nabla f\cdot \hat{u}$.

8. Chain rule (yeh glue hai)

Parent dot-product formula ko chain rule use karke derive karta hai. Ek variable mein yeh kehta hai: agar $z$ depend karta hai $u$ par aur $u$ depend karta hai $h$ par, toh $\frac{dz}{dh}=\frac{dz}{du}\frac{du}{dh}$ — rates ek chain ke along multiply hote hain. Kai variables mein, direction $\hat{u}$ mein chalna $x$ aur $y$ dono ko ek saath badalta hai, toh rates add up ho jaate hain: $\frac{d}{dh}f(p_1+h{u}_1,p_2+h{u}_2)=f_x{u}_1+f_y{u}_2.$ Yeh kyun chahiye: yeh woh bridge hai jo dikhata hai ki climb rate is a dot product. Dekho Chain rule (multivariable).

9. Level curve $f=c$

Contour ke along height kabhi nahi badlti, toh climb rate zero hai, toh along-contour direction $\hat{u}$ ke liye $\nabla f\cdot \hat{u}=0$ — matlab $\nabla f$ contour ke perpendicular baithta hai. Yahi parent ki teesri badi property hai. Dekho Level curves and surfaces.

Prerequisite map

Equipment checklist

Khud test karo — tum parent note ke liye ready ho agar bina dekhey har ek ka jawab de sako.

Connections

• Yeh note Hinglish mein padho →
• Partial derivatives — yahan build kiye gaye components $f_x,f_y$.
• Directional derivative — yahan assemble kiya gaya dot product $\nabla f\cdot \hat{u}$.
• Chain rule (multivariable) — dot-product formula ke peeche ka glue.
• Level curves and surfaces — jahan perpendicularity ka picture rehta hai.
• Tangent plane and linearisation — gradient build hone ke baad ka agla step.
• Gradient descent — $-\nabla f$ use karta hai.
• Lagrange multipliers — gradient normals use karta hai.