Given a surface in three-dimensional space described by a function z = f(x,y), a level curve of the surface is a plot of those points in the x-y plane satisfying f(x,y) = k for some constant k. This curve traces the "cut" made by the surface as it intersects the plane z = k; it is a plot of those points lying at the same level above the x-y plane, hence the name "level curve".

Plotting various level curves of a surface on the same axes, each labelled with its associated z-coordinate, can give a good sense of the surface's shape. As an example, think of the circular paraboloid z = x2 + y2, obtained by rotating the parabola z = x2 about the z-axis. A set of level curves for this surface would be a single point (the origin) corresponding to z = 0, a circle of radius 1 corresponding to z = 1, a circle of radius 2 for z = 4, a circle of radius 3 for z = 9, and so on. If one imagines lifting each level curve z units off the page (where z is the z-coordinate associated with that level curve), a sequence of cross-sections of the paraboloid is obtained which helps in its visualisation.

It is a theorem that the gradient vector, which for each point of a surface indicates the direction in which the surface is "steepest" (that is, the direction in which the corresponding function values change most quickly) is perpendicular to the level curve at that point.

Log in or register to write something here or to contact authors.