Level Curves

 Civil engineers building roads in mountains use a topographical map that shows the different elevations of the mountain. Hikers walking through rugged trails use also a topographic map to show how steeply the trail changes.  A topographical map contains curved lines called contour lines. Each contour line corresponds to the points of the map that have the same elevation. A level curve of a function of two variables f(x,y) is similar to a contour line in a topographic map.

The photo on the left is a topographical map of the Devil's Tower, Wyoming. USA. Lines that are close together indicate very steep terrain. The picture on the right was taken in perspective. It shows how steep the sides of the tower are. Notice that the top of the tower has the same shape as the center of the topographical map. 

Definition.

Given a function f(x,y) and a number c of its range, a level curve of the function with two variables f(x, y) for the value c is the set of points that satisfy the equation f(x, y) = c. A graph of the various level curves of a function with two variables is called a contour map.

Example 

Let's consider a previous function:

Let's determine the level curves of this function, The range  of this function is the closed interval [0.3]. let's choose any number in this interval, for example c = 2. The corresponding level curve is given by the following equation:

Let's square both sides of the equation. We have:

Let's multiply both sides by -1 and add 9 to both sides:

This represents the equation of a circle of radius ⎷5 and centered at the origin. This is one level curve of the function. Choosing different values of c lead to other level curves. For c =3 and substituting this value in the function, we obtain x₂² + y² = 0, which is the origin. For c = 0, we have  x₂² + y² =9, which is the circle centered at the origin and radius 3, For c = 1, we have  x² + y² = 8, which is the circle centered at the origin and of radius 2⎷2. The contour map of the function is shown below: