Vector Calculus: An Introduction

Quite often, surfaces are given as the set of points $\mathit{x}=(x_1,x_2,x_3)\in {\mathbb{R}}^3$ such that $f(\mathit{x})=0$ for some given function $\mathit{f}:{\mathbb{R}}^3\to \mathbb{R}\text{.}$ We call this an implicit representation of a surface. In some cases it is possible to solve $f(\mathit{x})=0$ for one of the variables. For instance we can solve for $x_3\text{.}$ If we can solve with one single function then $S$ is the graph of a function. Graphs of a function $h$ defined on a subset $D\subset {\mathbb{R}}^3$ are the simplest possible surfaces. We call $x_3=h(x_1,x_2)\text{,}$ $(x_1,x_2)\in D$ an explicit representation of the surface. In many cases, however, solving for one variable will only be possible locally, even for very simple surfaces like the sphere.

Not for every function $f$ does the equation $f(\mathit{x})=0$ describe a smooth surface. For instance the only solution to $x_1^2+x_2^2+x_3^2=0$ is $(0,0,0)\text{,}$ that is, a single point. Hence we need to find criteria which guarantee that the solutions of $f(\mathit{x})=0$ form a surface. We have the following theorem, which is a generalised form of the implicit function Theorem 4.33, where also the ideas for a more general result are indicated.

In the above example of the sphere we have $\mathrm{grad}f(\mathit{x})=(2x_1,2x_2,2x_3)\text{,}$ which is clearly nonzero if $R>0$ and $x_1^2+x_2^2+x_3^2=R\text{.}$

As mentioned already smooth surfaces, in principle, can be represented by all three means. We consider some examples.