Section 2.1 Graphs and Level Sets

A scalar-valued function of several variables assigns one number to each point in its domain. We can visualize this number as a height or examine the points where it has a fixed value. These viewpoints lead to graphs and level sets.

Subsection 2.1.1 Graphs and Contours

Definition 2.1.1. Graph of a Function.

Let \(A\subseteq \R^n\) and \(f\colon A\to \R\text{.}\) The graph of \(f\) is

\begin{equation*} \operatorname{graph}(f) :=\{\langle x_1,\ldots,x_n,t\rangle\in A\times\R: t=f(\langle x_1,\ldots,x_n\rangle)\}. \end{equation*}

Thus \(\operatorname{graph}(f)\) is a subset of \(\R^{n+1}\text{.}\) For \(n=1\text{,}\) it is the familiar curve \(y=f(x)\text{.}\) For \(n=2\text{,}\) it is the surface \(z=f(x,y)\) in \(\R^3\text{.}\)

When \(n\ge 3\text{,}\) the graph lies in at least four dimensions and cannot be drawn directly in three-dimensional space. Level sets give a useful lower-dimensional picture that works in any dimension.

Definition 2.1.2. Level Set.

Let \(f\colon A\to \R\) and let \(c\in\R\text{.}\) The level set of \(f\) at level \(c\) is

\begin{equation*} \mathcal L_c(f) :=\{\vx\in A:f(\vx)=c\}=f^{-1}(\{c\}). \end{equation*}

For a function on \(\R^2\text{,}\) a level set is also called a level curve or contour line. A collection of labeled contour lines is a contour map. For a function on \(\R^3\text{,}\) a level set is often called a level surface.

If \(f\colon A\subseteq\R^2\to\R\text{,}\) then the horizontal plane \(z=c\) intersects the graph \(z=f(x,y)\) above the contour \(f(x,y)=c\text{.}\) Projecting this intersection onto the \(xy\)-plane produces the contour line. Closely spaced contours indicate that the height changes rapidly, as on a topographic map.

Example 2.1.3. A Paraboloid and Its Contours.

Describe the graph and all level sets of \(f(x,y)=x^2+y^2\text{.}\)

Solution.

The graph \(z=x^2+y^2\) is an upward-opening paraboloid. At level \(c\text{,}\) the equation is

\begin{equation*} x^2+y^2=c. \end{equation*}

The level set is empty when \(c\lt 0\text{,}\) is the single point \(\langle 0,0\rangle\) when \(c=0\text{,}\) and is a circle of radius \(\sqrt{c}\) when \(c\gt 0\text{.}\) The circular contours record the heights of horizontal slices through the paraboloid.

Upward paraboloid with three horizontal circular slices. — (a) Horizontal slices of the graph.

Concentric circular contours labeled one half, one, and two. — (a) Horizontal slices of the graph.

Example 2.1.5. Level Curves of a Saddle.

Describe the level curves of \(g(x,y)=x^2-y^2\text{.}\)

Solution.

The level equation is \(x^2-y^2=c\text{.}\) For \(c\gt 0\text{,}\) it is a hyperbola opening left and right. For \(c\lt 0\text{,}\) it is a hyperbola opening up and down. At \(c=0\text{,}\) it becomes the two lines \(y=x\) and \(y=-x\text{.}\) The change in the contours reflects the saddle shape of the graph.

Hyperbolic contours and the two diagonal lines at level zero. — Figure 2.1.6. Selected level curves of \(g(x,y)=x^2-y^2\text{.}\)

Subsection 2.1.2 Graphs as Zero Level Sets

A graph is itself a level set of a function with one additional input variable. Indeed, let \(f\colon A\subseteq\R^n\to\R\) and define \(F\colon A\times\R\to\R\) by

\begin{equation*} F(\vx,t)=f(\vx)-t. \end{equation*}

Then

\begin{equation*} \operatorname{graph}(f) =\{\langle\vx,t\rangle:F(\vx,t)=0\} =\mathcal L_0(F). \end{equation*}

Thus the graph of an \(n\)-variable function can be viewed as the level \(0\) set of a function of \(n+1\) variables.

Example 2.1.7.

Express the graph of \(f(x,y)=x^2+y^2\) as a zero level set.

Solution.

Define \(F\colon\R^3\to\R\) by

\begin{equation*} F(x,y,z)=x^2+y^2-z. \end{equation*}

Then \(F(x,y,z)=0\) exactly when \(z=x^2+y^2\text{.}\) Therefore \(\operatorname{graph}(f)=\mathcal L_0(F)\text{.}\)

The converse is not always true globally: a level set need not be the graph of a function in a chosen coordinate direction. For example, the level set \(x^2+y^2-1=0\) is the unit circle. It is not the graph of a single function \(y=f(x)\text{,}\) because most vertical lines meet it twice. It can, however, be split into the two graphs \(y=\sqrt{1-x^2}\) and \(y=-\sqrt{1-x^2}\text{.}\) This local graph viewpoint will return in the implicit function theorem.

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