Topographic (also called contour) maps are an effective way to show the elevation in 2-D maps. These maps are marked with contour lines or curves connecting points of equal height.

The same idea can be used to represent a function $z=f(x,y)$ graphically. If the graph of the function $z=f(x,y)$ is cut by the horizontal (or level) plane $z=c$, and if we project this intersection onto the $xy$-plane, then we get a curve that consists of points $(x,y)$ for which $f(x,y)=c$ (Figure 2). Such a curve is called the **level curve of height $c$** or the **level curve with value $c$** and is denoted by $L(c)$ or by $f^{-1}(c)$. By drawing a number of level curves, we get what is called a **contour plot** or** contour map**, which provides a good representation of the function $z=f(x,y)$.

In the next few examples, we will practice how to determine the contour curves. Let’s start with simple examples.

Recall that $(x-x_0)^2+(y-y_0)^2=R^2$ is the equation of a circle of radius $R$ centered at $(x_0,y_0)$

Recall that $\frac{(x-x_0)^2}{a^2}+\frac{(y-y_0)^2}{b^2}=1$ is the equation of an ellipse centered at $(x_0,y_0)$ with semi major axis $a$ and semi minor axis $b$ (if $a\geq b$).

Recall that $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is the equation of a hyperbola with two vertices at $(\pm a,0)$.

We can extend the concept of level curves to functions of three or more variables.

**Definition 1.** Let $f:U\subseteq\mathbb{R}^n\to\mathbb{R}$. Those points $\mathbf{x}$ in $U$ for which $f(\mathbf{x})$ has a fixed value, say $f(\mathbf{x})=c$, form a set denoted by $L(c)$ or by $f^{-1}(c)$, which is called a** level set** of $f$

\[L(c)=\{\mathbf{x}|\ \mathbf{x}\in U \ \text{and}\ f(\mathbf{x})=c\}\]

When $n=3$, the level set is called a **level surface**. As the graph of a function $f(x,y,z)$ of three variables is a set (called *hypersurface*) in $\mathbb{R}^4$— hence, their graphs cannot be represented— the level surfaces are the only way to graphically represent a function of three variables.

** **Remark that the graph of a function $z=f(x,y)$ is the same as the level surface of the function $F(x,y,z)=z-f(x,y)$ with value 0.

If $f(x,y,z)$ gives the temperature at each point of 3-space, the level surfaces (curves of constant temperature) are called **isothermal**. In physics, when $f(x,y,z)$ is a potential function, which gives the value of the potential energy at each point of space, the level surfaces are called **equipotential** or **isopotential**. Figure 11 shows the electrostatic equipotentials between two electric charges.