Level curve

The idea is that of a topographic map: each line joins the points of the same height. For \(f(x,y)=x^2+y^2\) the level curve \(f=c\) is the circle of radius \(\sqrt c\) if \(c>0\), the origin if \(c=0\) and the empty set if \(c<0\). It is also called a contour line, and in meteorology an isotherm or isobar when \(f\) is temperature or pressure.

Each level curve is the section of the graph \(z=f(x,y)\) by the horizontal plane \(z=c\), projected onto the \(xy\)-plane. Two different level curves never cross, since each point has a single value of \(f\). When drawn with equally spaced values, curves close together indicate a steep graph and curves far apart a nearly flat one.

A level curve may be a line, a circle, an ellipse, a hyperbola or even two lines, as in \(f(x,y)=xy\) with \(c=0\). For functions of three variables the analogue is the level surface \(f(x,y,z)=c\). These curves prepare the gradient: later you will see that it is perpendicular to them.