| 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. | 
