Gradient vector ∇f — definition, properties

4.4.9 · Maths › Multivariable Calculus

1. Scratch se banana (HOW)

1.1 Partial derivatives — ek axis ke saath slope

$y$ ko fix rakho, $x$ ko thoda hilao: $f_x=\frac{\partial f}{\partial x}={\lim }_{h\to 0}\frac{f(x+h,y)-f(x,y)}{h}$ Similarly $f_y=\partial f/\partial y$. Har ek 1D slice ka slope hai.

1.2 Directional derivative (WHAT hum actually chahte hain)

$f$ ke change ki rate jab hum ek unit direction $\hat{u}=(u_1,u_2)$ mein step lete hain: $D_{\hat{u}}f={\lim }_{h\to 0}\frac{f(\mathbf{p}+h\hat{u})-f(\mathbf{p})}{h}$

Derive karo. Maano $g(h)=f(p_1+h{u}_1,p_2+h{u}_2)$. Chain rule se: $D_{\hat{u}}f=g^{\prime }(0)=f_x\cdot \frac{d(p_1+h{u}_1)}{dh}+f_y\cdot \frac{d(p_2+h{u}_2)}{dh}=f_x{u}_1+f_y{u}_2$

Yeh step kyun? Chain rule "$\hat{u}$ direction mein move karna" ko tod deta hai — "kitni tezi se $x$ change hota hai" times "kitni tezi se $f$, $x$ ke saath change hota hai", plus $y$ ke liye bhi yehi. Yeh ek dot product hai: $D_{\hat{u}}f=(f_x,f_y)\cdot (u_1,u_2)$

Woh pehla vector $(f_x,f_y)$ zaroori ban jaata hai. Hum isko naam dete hain:

2. Teeno badi properties (WHY ke saath)

2.1 Steepest ascent ki direction

Kyunki $D_{\hat{u}}f=|\nabla f|\cos \theta$ jahan $\theta$, $\hat{u}$ aur $\nabla f$ ke beech ka angle hai:

• $\cos \theta$ maximise hota hai $\theta =0$ par ⟹ $\nabla f$ ke saath chalo, rate milta hai $|\nabla f|$.
• Minimise hota hai $\theta =\pi$ par ⟹ steepest descent hai $-\nabla f$, rate $-|\nabla f|$.
• Zero hota hai $\theta =\pi /2$ par ⟹ $\nabla f$ ke perpendicular move karna $f$ ko constant rakhta hai.

2.2 Gradient ⟂ level sets

Level curve $f=c$ par, $f$ change nahi hota, to $D_{\hat{u}}f=0$ jab $\hat{u}$ curve ke tangent ho. Lekin $0=\nabla f\cdot \hat{u}$ ka matlab hai $\nabla f\perp \hat{u}$. Isliye $\nabla f$ level set ke normal hota hai.

2.3 Linearity & tangent plane

$\nabla (af+bg)=a\nabla f+b\nabla g$. $z=f(x,y)$ ka tangent plane $\mathbf{p}$ par: $z=f(\mathbf{p})+\nabla f(\mathbf{p})\cdot (\mathbf{x}-\mathbf{p})$ kyunki $\mathbf{p}$ ke paas, $f$ ka best linear approximation exactly uske partial slopes use karta hai.

3. Worked examples

Connections

• Partial derivatives — $\nabla f$ ke components.
• Directional derivative — $\nabla f\cdot \hat{u}$ se define hota hai.
• Chain rule (multivariable) — dot-product formula derive karne mein use hota hai.
• Tangent plane and linearisation — gradient linear part deta hai.
• Level curves and surfaces — gradient unka normal hai.
• Gradient descent — $-\nabla f$ ke saath move karta hai.
• Lagrange multipliers — $\nabla f=\lambda \nabla g$ normal directions use karta hai.

Flashcards

Concept Map