Changes

Scalar functions which take points in space as the input are not so easily visualized, but not less useful.  For example, while we might visualize the temperature at a point on a hot plate by constructing a rubber surface above it which is raised at the hottest points, visualizing the temperature at points in our atmosphere, which differs at different altitudes, is a horse of another color, but absolutely essential to weather prediction and climatology.   

Scalar functions which take points in space as the input are not so easily visualized, but not less useful.  For example, while we might visualize the temperature at a point on a hot plate by constructing a rubber surface above it which is raised at the hottest points, visualizing the temperature at points in our atmosphere, which differs at different altitudes, is a horse of another color, but absolutely essential to weather prediction and climatology.   

One way of visualizing scalar functions in three dimensions is to construct '''constant surfaces'''.  Not unlike a [[Topographic Map|topographic map]], constant surfaces are a set of surfaces in space which are constructed so that at each point on the surface, the value of the scalar function is precisely the same as it is everywhere else on the surface.  See illustration, right.

One way of visualizing scalar functions in three dimensions is to construct '''level surfaces'''.  Not unlike a [[Topographic Map|topographic map]], level surfaces are a set of surfaces in space which are constructed so that at each point on the surface, the value of the scalar function is precisely the same as it is everywhere else on the surface.  See illustration, right.

=Problems=

=Problems=