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We introduce level sets.

### Level sets

It was Descartes who said “Je pense, donc je suis.” He also developed our rectangular coordinate system, the $(x,y)$-plane. This is also known as the Cartesian coordinate system. This coordinate system allows us to consider the graph of a function. First, recall that the graph of a function of a single variable, $y=f(x)$ is a curve in a two-dimensional plane. In the same sense, the graph of a function of two variables, $z = F(x,y)$ is a surface in three-dimensional space. The graph of a function of three variables, $w=F(x,y,z)$ is a surface in four-dimensional space. A surface in higher than three dimensions is often called a hypersurface. How can we visualize such functions? For visualizing functions $f:\R \to \R$, a graphing utility like Desmos is really great. For visualizing functions $F:\R ^2\to \R$, GeoGebra is very helpful. However, once we get to functions $F:\R ^3\to \R$ (or $F: \R ^n \to \R$), visualizing the graph of the function as we do in two and three dimensions becomes much more difficult. One powerful technique to help us understand a function $F:\R ^3\to \R$ visually is known as sketching level sets.

When working with functions $F:\R ^2\to \R$ the level sets are known as level curves.

When we are looking at level curves, we can think about choosing a $z$-value, say $z=c$, in the range of the function and ask “at which points $(x,y)$ can we evaluate the function to get $F(x,y)=c$?” Those points form our level curve. If we choose a value $z=c$ that was not in the range of $F$, there would be no points in the $(x,y)$-plane for which $F(x,y)=c$, and hence no level curve associated to $z = c$.

It may be surprising to find that the concept of level sets is familiar to most people, but they don’t realize it. Topographical maps, like the one below represent the surface of Earth by indicating points with the same elevation with contour lines. We also had an example of the contour lines of Meteor Crater as we began this section.

Another example you may know are isotherms, which are curves along which the temperature does not change. We see these in weather maps.

Below we see a surface with level curves drawn beneath the surface. Remember that the level curves are in the domain of the function, not on the surface itself.

We often mark the function value on the corresponding level set. If we choose function values which have a constant difference, then level curves are close together when the function values are changing rapidly, and far apart when the function values are changing slowly.

Suppose you have a differentiable function $F:\R ^2\to \R$ with the following set of level curves. You should interpolate reasonable values of the function $F$ between the level curves which are shown: Consider the points $A$, $B$, and $C$ on the surface $z=F(x,y)$. Order the points from least steep to most steep.

At point $\answer [format=string]{C}$ the surface is less steep than at point $\answer [format=string]{A}$, and the surface is steepest at point $\answer [format=string]{B}$.

Now, let’s see if you can identify some simple surfaces based on their level curves.

Match the following level surfaces to the equations below.    Let’s look at another example.

Let’s see another example.

So far, the level sets we’ve been working with have been curves in $\R ^2$. We can easily generalize to functions $F:\R ^n \to \R$. When working with functions $F:\R ^3\to \R$, our level sets are also called level surfaces.

#### Level sets in higher dimensions

In higher dimensions, we want to try to use what we understand about functions of one and two variables to try to better understand functions of three or more variables.

• A function of one variable can be visualized as a curve drawn in two dimensions.
• A function of two variables can be visualized as a surface drawn in three dimensions.
• A function of three variables can be visualized as what we will call a hypersurface drawn in four dimensions.
• A function of $n$ variables can be visualized as what we will call a hypersurface drawn in $n+1$ dimensions.

We use the term “hypersurface” to refer to an object which is like a surface, but in more than three dimensions. Hypersurfaces are difficult to imagine, and can even be difficult to picture using modern computer utilities.

For a function $F: \R ^3 \to R$ of three variables, one technique we can use is to graph the level surfaces, our three-dimensional analogs of level curves in two dimensions. Given $w=F(x,y,z)$, the level surface at $w=c$ is the surface in space formed by all points $(x,y,z)$ where $F(x,y,z)=c$. It’s time for an example.

We have found that the level surfaces of $F$ in the above example are concentric spheres. If we picture several of these concentric spheres at the same time, we can get some intuition about the graph of $F$ in four dimensions in the same way that a collection of level curves in two dimensions gave us some intuition about the corresponding surface in three dimensions.

#### From explicit surfaces to level surfaces

We turn our attention to an important concept that will arise again in future sections.

Suppose that $F:\R ^2 \to \R$ is a function of two variables. Then, the surface $z = F(x,y)$ is a level surface of the function $G(x,y,z) = F(x,y) - z.$

In fact, if $F: \R ^n \to \R$ is a function of $n$ variables, we can also consider it to be a particular level set for some other function $G: \R ^{n+1} \to \R$ of $n+1$ variables. This idea is very powerful, as it allows us to consider the same function from two different perspectives. Having multiple perspectives gives us extra tools to use when considering our function, as well as allows us to look at the function in whatever manner we find most convenient. Let’s consider a specific example.

Again, it appears that all we did here was some easy algebra. We made a new function of one more variable by simply rearranging the original equation that defined our surface. But having multiple perspectives is always better than having only one. In addition to its other uses, the content of this procedure is vital for

• finding normal vectors for explicitly defined surfaces.
• finding tangent planes for explicitly defined surfaces.

These results will be explored further in later sections. It’s good to become familiar with these ideas now, so that we can make expert use of them later.