Gradient vector ∇f — definition, properties

1. Building it from scratch (HOW)

1.1 Partial derivatives — slope along one axis

Hold $y$ fixed, wiggle $x$ : $f_x = \frac{\partial f}{\partial x} = \lim_{h\to 0}\frac{f(x+h,y)-f(x,y)}{h}$ Similarly $f_y = \partial f/\partial y$ . Each is the slope of a 1D slice.

1.2 The directional derivative (WHAT we really want)

The rate of change of $f$ as we step in a unit direction $\hat u = (u_1,u_2)$ : $D_{\hat u}f = \lim_{h\to 0}\frac{f(\mathbf p + h\hat u)-f(\mathbf p)}{h}$

Derive it. Let $g(h)=f(p_1+hu_1,\,p_2+hu_2)$ . By the chain rule: $D_{\hat u}f = g'(0) = f_x\cdot\frac{d(p_1+hu_1)}{dh}+f_y\cdot\frac{d(p_2+hu_2)}{dh} = f_x u_1 + f_y u_2$

Why this step? The chain rule splits "moving in direction $\hat u$ " into "how fast $x$ changes" times "how fast $f$ changes with $x$ ", plus the same for $y$ . This is a dot product: $D_{\hat u}f = (f_x,f_y)\cdot(u_1,u_2)$

That first vector $(f_x,f_y)$ is forced into existence. We name it:

2. The three big properties (with WHY)

2.1 Direction of steepest ascent

Since $D_{\hat u}f = |\nabla f|\cos\theta$ where $\theta$ is the angle between $\hat u$ and $\nabla f$ :

$\cos\theta$ is maximised at $\theta=0$ ⟹ walk along $\nabla f$ , getting rate $|\nabla f|$ .
Minimised at $\theta=\pi$ ⟹ steepest descent is $-\nabla f$ , rate $-|\nabla f|$ .
Zero at $\theta=\pi/2$ ⟹ moving perpendicular to $\nabla f$ keeps $f$ constant.

2.2 Gradient ⟂ level sets

On a level curve $f=c$ , $f$ doesn't change, so $D_{\hat u}f=0$ for $\hat u$ tangent to the curve. But $0=\nabla f\cdot\hat u$ means $\nabla f\perp\hat u$ . Hence $\nabla f$ is normal to the level set.

2.3 Linearity & tangent plane

$\nabla(af+bg)=a\nabla f+b\nabla g$ . The tangent plane to $z=f(x,y)$ at $\mathbf p$ : $z = f(\mathbf p)+\nabla f(\mathbf p)\cdot(\mathbf x-\mathbf p)$ because near $\mathbf p$ , $f$ 's best linear approximation uses exactly its partial slopes.

Figure — Gradient vector ∇f — definition, properties

3. Worked examples

Common mistake Steel-manned common errors

(a) "Plug a non-unit vector into $D_{\hat u}f$ ." Why it feels right: you have a direction, you dot it. The fix: $\nabla f\cdot\mathbf v$ with $|\mathbf v|\ne1$ gives the rate scaled by $|\mathbf v|$ — always normalise to $\hat u$ . (b) " $\nabla f$ is the tangent to the curve $f=c$ ." Why it feels right: gradients feel like "the direction of the curve." The fix: it's normal, because moving along the curve gives zero change. (c) "Gradient is a scalar / a number." Why: slope sounds scalar. Fix: in $n$ D you need a vector to encode all directional slopes. (d) "Max rate of change is $\nabla f$ ." Fix: the max rate is $|\nabla f|$ (a number); the direction is $\nabla f$ .

Recall Feynman: explain to a 12-year-old

Imagine standing on a bumpy hill in fog. You can't see the whole hill, but you can feel the ground tilt under your feet. The gradient is like an arrow lying on the ground that points straight uphill, and the longer the arrow, the steeper the climb. If you walk sideways across the arrow, you stay at the same height. If you want to walk a little uphill in some other direction, just shine the arrow's "shadow" onto your direction (that's the dot product) — that shadow length tells you how fast you'll climb.

Connections

Partial derivatives — the components of $\nabla f$ .
Directional derivative — defined via $\nabla f\cdot\hat u$ .
Chain rule (multivariable) — used to derive the dot-product formula.
Tangent plane and linearisation — gradient gives the linear part.
Level curves and surfaces — gradient is their normal.
Gradient descent — moves along $-\nabla f$ .
Lagrange multipliers — $\nabla f=\lambda\nabla g$ uses normal directions.

Flashcards

Define the gradient

\nabla f

The vector of partial derivatives

(f_x,f_y,\dots)

; a vector field giving all directional slopes.

How is the directional derivative related to

\nabla f

D_{\hat u}f=\nabla f\cdot\hat u

for unit

\hat u

In which direction does

f

increase fastest?

Along

\nabla f

; the rate is

|\nabla f|

What is the relationship between

\nabla f

and level curves

f=c

\nabla f

is perpendicular (normal) to them.

Why must

\hat u

be a unit vector in

D_{\hat u}f

Otherwise the rate is scaled by

|\mathbf v|

, not a true per-unit-length rate.

Maximum rate of change of

f

at a point?

|\nabla f|

(a scalar).

Derive

D_{\hat u}f=f_xu_1+f_yu_2

Set

g(h)=f(\mathbf p+h\hat u)

, apply chain rule,

g'(0)=f_xu_1+f_yu_2

Equation of tangent plane via gradient?

z=f(\mathbf p)+\nabla f(\mathbf p)\cdot(\mathbf x-\mathbf p)

Direction of steepest descent?

-\nabla f

, rate

-|\nabla f|

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Socho tum ek pahaadi (hill) par khade ho jahan height ko function $f(x,y)$ describe karta hai. Ek hi point par alag-alag directions mein chalo to slope alag aata hai. Toh ek single number se kaam nahi chalega — humein ek vector chahiye jo saari directions ki slope ko ek saath store kar le. Wahi hai gradient $\nabla f = (f_x, f_y)$ , yaani partial derivatives ka vector.

Magic ye hai: kisi bhi unit direction $\hat u$ mein slope nikalne ke liye bas dot product lo — $D_{\hat u}f = \nabla f \cdot \hat u$ . Aur dot product tab sabse bada hota hai jab dono vector ek hi taraf point karein. Isliye sabse steep uphill direction hamesha $\nabla f$ ki taraf hoti hai, aur uski steepness $|\nabla f|$ hoti hai. Agar tum $\nabla f$ ke perpendicular chaloge, slope zero — yaani height constant — isliye gradient hamesha level curve ke normal (perpendicular) hota hai.

Do common galtiyan yaad rakho: pehli, direction vector ko hamesha unit banao warna answer scale ho jaata hai. Doosri, gradient curve ke saath nahi, curve ke across (perpendicular) point karta hai. Yeh concept gradient descent (machine learning), Lagrange multipliers, aur tangent plane sab mein use hota hai — isliye iski intuition pakki kar lo, ratte se kuch nahi banega.

Gradient vector ∇f — definition, properties

1. Building it from scratch (HOW)

1.1 Partial derivatives — slope along one axis

1.2 The directional derivative (WHAT we really want)

2. The three big properties (with WHY)

2.1 Direction of steepest ascent

2.2 Gradient ⟂ level sets

2.3 Linearity & tangent plane

3. Worked examples

Connections

Flashcards

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Multivariable Calculus

Connections