Level Curves

Introduction

A lot of mathematics is spent in the two-dimensional plane, which is perfectly suited to a variety of applications. However, by graduating to working with functions in the three-dimensional realm, a mathematician gains the ability to handle many more real-world scenarios. To acclimate ourselves to thinking and working in 3D space, we are going to discuss the topic of level curves.

Level Curves

If I were to give you a function of two variables, for example \( z= \sqrt{4-x^2-y^2} \), would you be able to imagine what its graph looked like? If you haven’t had a lot of experience working in 3 dimensions, then probably not (by the way, it’s a sphere). But even for those more experienced with 3 dimensions, it can be difficult to envision the surface formed by less standard functions. For this reason, level curves are sometimes used to help us imagine what a surface would look like. A level curve for some value \( k \) is simply the line where a function takes on the value \( k \) , and can be found by setting \( f(x,y)=k \). Once we have found the level curves for a few values of \( k \), we can graph these equations in 2 dimensions to see how our surface is behaving. Let’s take a look at an example.

An Example

Imagine there is a wildlife preserve consisting of rolling hills that can be modeled by the following function: \[ f(x,y)=\sin(4xy) \] You have a good friend, Bob, a botanist, who frequently explores this preserve in search of endangered plant species. Bob knows that certain plants will only grow within certain altitude ranges, but poor Bob is directionally challenged and has difficulty keeping track of his location and altitude. Since you are an awesome friend, you decide to make Bob a map that will show him the changing altitudes so he can find these plant species more efficiently. We’ll calculate some level curves! For any value \( k \), we can calculate our level curve like below: \[ \sin(4xy)=k \] Let’s make this easier to graph: \[ y= \frac{\arcsin(k)}{4x} \] So, if we look at the values for which \( k=-0.8,-0.6,-0.4, -0.2, 0, 0.2, 0.4, 0.6, 0.8 \), we will get the following graph of the level curves.

Now, how can you explain to Bob how to use this map? (Note that the cooler colors represent a positive \( k \) value while the warmer colors represent the negative \( k \) values. Meanwhile, the function is 0 on the green line, which lines up with the \( x \) and \( y \) axes.). Looking at the locations of each level curve, we can see that our altitude is increasing in the first and third quadrants but decreasing in the second and fourth quadrants. Furthermore, the closeness of the lines indicates how steep the altitude of these regions is changing. For example, if Bob were to walk from \( (0,0) \) to \( (1,1) \), he would still be between the level curves for 0 and 0.2. However, if he again walked the same distance from \( (1,1) \) to \( (2,2) \), he would pass two level curves and be very close to our level curve for 0.6. This means we can imagine that the incline here is increasing, or, in other words, the hill is becoming steeper. Now take a look at the graph for our equation and compare it to our level curves.

The model below, on which the level curves we discussed above are highlight, demonstrate how you can think of a level curve as finding the cross section of our surface at height \( k \).

Why this example?

This example was chosen for several reasons, foremost among them its interesting shape. The surface contains a saddle point at its center, which allow the observer to see the function both increase and decrease on either side of this point. The presence of the sine function also contributes to making this function interesting by causing the slope to vary widely across the surface. This better illustrates the concepts of level curves because it allows an observer to observe how evenly spaced level curves are closer together when the function is steeper and farther apart when it is more gradual.

Author: Sarah Bombrys