Level Curves or Contour Curves

Another topic that we should look at is that of level curves or also known as contour curves. The level curves of the function z = f (x, y) are two dimensional curves we acquire by setting z = k , where k is any number.  Thus the equations of the level curves are f (x, y) = k .  

Notice: occasionally the equation will be in the form f (x, y, z) = 0 and in these cases the equations of the level curves are f (x, y, k) = 0.

You have probably seen level curves (or contour curves, doesn't matter whatever you want to call them) before.  If you have ever observed the elevation map for a piece of land, this is not anything more than the contour curves for the function that provides the elevation of the land in that area.  Actually, we probably do not have the function which gives the elevation, although we can at least graph the contour curves.