1. The Object: Mappings of the Plane to the Plane

1.1 What Is a Mapping?

Until now you have studied functions that take one or more inputs and produce a single output — for example, $f(x,y) = x^2 + y^2$. Now we generalise: a mapping (or transformation) takes a pair of inputs and produces a pair of outputs:

This is a function $\mathbf{F}: \mathbb{R}^2 \to \mathbb{R}^2$. In three dimensions the same idea gives $\mathbf{F}: \mathbb{R}^3 \to \mathbb{R}^3$. The single-output function you already know is the special case $m = 1$: the mapping has only one row.

1.2 Standing Example 1 — Polar Coordinates

The transformation from polar coordinates $(r,\theta)$ to Cartesian coordinates $(x,y)$ is:

This is a mapping $\mathbb{R}^2 \to \mathbb{R}^2$: two inputs $(r,\theta)$, two outputs $(x,y)$. It will be our most important worked example throughout this lesson.

1.3 Standing Example 2 — Linear Mappings (The Matrix Case)

A linear mapping is one of the form

where $A$ is a constant matrix. For instance, with $A = \begin{pmatrix} 2 & 1 \\ 0 & 3 \end{pmatrix}$:

Recall from linear algebra that a matrix is a linear transformation. This is the thread we now pick up by name: the Jacobian will connect all smooth mappings back to the matrix world.