This page assumes nothing. Before you can read the parent topic, you need to recognise every squiggle it writes. Below, each symbol gets three things: what it means in plain words, the picture it stands for, and why the topic can't proceed without it. They are ordered so each one leans only on the ones above it.
Picture. Look at figure s01: a single point sitting in the plane, described by its two coordinates.
Why the topic needs it. The whole problem is "find the special point x∗ in Rn where everything vanishes." If you don't know that x is a point made of n numbers, the boldface below looks like magic.
Picture. In figure s01 the arrow from the origin to the point is the vector x=(x1,x2). Its two numbers are how far right and how far up you walk.
Why the topic needs it. In one dimension we had one unknown x. Here we have n unknowns at once, and bundling them into one bold x lets us write n equations as a single clean line.
Picture. Feed it a point on the paper, get a height. If you plot that height above every point, you get a landscape — a hilly surface floating over the plane. The places where the height is exactly 0 trace out a curve on the floor: the "sea-level line" of that landscape.
Why the topic needs it. Each equation fi=0 is really "walk along the sea-level line of landscape i." Solving the system = finding the point where all the sea-level lines cross.
Picture. Figure s02 draws two curves — a circle (f1=0) and a parabola (f2=0). Each is one landscape's sea-level line. The red dot where they cross is x∗: the only place both are zero together.
Why the topic needs it. This crossing point is the entire goal. Everything Newton does is machinery to hunt this red dot down.
Before the matrix, we need one honest sub-idea: how steep is a landscape if I walk in just one direction?
Why this tool and not an ordinary derivative? An ordinary derivative f′ needs a function of one variable. Here fi depends on many. So we ask the slope one direction at a time, and the curly ∂ is exactly the notation for "slope while everything else is held fixed." It's the smallest honest generalisation of f′.
Picture. Stand on the hill. Face east and feel the tilt under your feet — that's ∂f/∂x. Now face north and feel that tilt — that's ∂f/∂y. Two different steepnesses at the same spot. Figure s03 shows both slope arrows on one surface slice.
Why the topic needs it. A curved rule has no single slope. To pretend it's a flat ramp we must know its tilt in every input direction — and each tilt is one partial derivative.
Picture. Think of J as the "steepness report card" of the whole system at one point: one row per equation, one column per unknown. It is the flat-ramp stand-in for the bumpy landscapes — the Jacobian matrix is literally the n-dimensional version of the number f′.
Why the topic needs it. Newton replaces curved F by a ramp. A ramp in n dimensions is described exactly by this grid of slopes. No Jacobian ⇒ no ramp ⇒ no method.
Why the topic needs it. Every Newton step is one such linear solve. The point of linearizing was precisely to reach a problem this toolbox already crushes.
Picture. Figure s04 plots the error shrinking 10−1→10−2→10−4→10−8: it doesn't just fall, it plummets, because each step feeds on the square of the last.
Why the topic needs it. This is the payoff — the reason we tolerate building Jacobians and solving systems. Compare with plain Fixed-point iteration, which only shrinks the error by a fixed fraction each step (linear). The subscript k on xk, ek simply counts iterations: k=0 is your start, k=1 after one step, and so on.
Read it top to bottom: numbers become vectors, vectors feed F; slopes stack into the Jacobian; Jacobian plus the "hit zero" wish gives a linear model; the linear solver turns that into one step; repeat, and the norm lets us watch the error crash.
