The gradient
Every level curve is a path of constant height. At any point of the map, the gradient $\nabla f = (f_x, f_y)$ points which way?
On the bowl every field arrow fires straight out from the center, crossing each circle at a right angle; at $(1, 1)$ the stats read $f_x = 2.000$, $f_y = 2.000$, $|\nabla f| = 2.828$. Arrows lengthen toward the rim, exactly where the circles crowd. The walker cuts every contour perpendicularly on its way to the edge — and on the saddle from angle $90$ it glides to the center and stalls: $\nabla f = (0, 0)$ there.
The gradient packs both partials into one vector, $\nabla f = (f_x, f_y)$. It points in the direction of steepest ascent, and its length $|\nabla f| = \sqrt{f_x^2 + f_y^2}$ is that steepest slope — on the bowl at $(1, 1)$, $\sqrt{4 + 4} = 2.83$. Why perpendicular to the level curves: along a level curve the height does not change, so a step's along-curve component gains nothing. The fastest way up keeps no along-curve part — what remains points straight across.
The bowl's $\nabla f = (2x, 2y)$ points radially outward, so the walk uphill is a straight radial line. The saddle's field $\nabla f = (2x, -2y)$ pushes east–west and squeezes north–south; a walker started on the $y$-axis is squeezed into the one point with no uphill direction. An honest note on scope: gradients sit beyond the AP Calculus AB/BC syllabus — this chapter is a preview, built entirely from the single-variable derivative.
About this model
What it represents: gradient ascent — the walker takes many small steps proportional to $\nabla f$, an Euler approximation of the steepest-ascent flow $\dot{p} = \nabla f(p)$, the mirror image of the gradient descent that trains machine-learning models.
What it leaves out: everything that makes real gradient methods hard — choosing a step size, overshooting and zigzagging in narrow valleys, and dimension: practical descent runs on functions of millions of variables, not two. The clean stall at the saddle is also fragile: only walkers starting exactly on the $y$-axis reach it, and any sideways nudge escapes.
Source: S. Boyd and L. Vandenberghe, Convex Optimization, Ch. 9 (Cambridge University Press, free online) — the full treatment of gradient methods, step-size rules included.
Why does the gradient arrow always cross the level curve through its base at a right angle?
On the bowl, what is $|\nabla f|$ at the point $(0.5,\ 0)$?
Transfer: a hiker on a real contour map wants the fastest way down. Relative to the level curves, which way is that?
The gradient is the two partials assembled into one arrow: it points straight across the level curves toward higher ground, its length is the steepest slope, and it vanishes exactly at the flat points — including saddles, neither peaks nor pits.