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→Scalar Functions: better
=Scalar Functions=
=Scalar Functions=
Scalar functions are the kinds of functions you're used to, but in more dimensions. The example given earlier of a
Scalar functions are the kinds of functions you're used to, but in more dimensions. The key property of a scalar function is that it returns a number. The example given earlier of temperature at a point on the Earth is a scalar function, because even though the inputs are not numbers (they are locations, e.g., New York), the output is a temperature (e.g., 35).
Frequently in multivariate calculus, we encounter functions which take as their inputs points in the plane or points in three dimensional space, but return numbers. These are scalar functions. A typical example would be a function which describes the pressure at any point in a fluid. However, a function which takes a point in a fluid and returns the direction of the flow at that point would NOT be scalar function because the information returned it not a number - we will examine these kinds of functions in the next lecture.
Scalar functions on the plane are easily visualized. For example, a scalar function on the plane which takes a point <math>(x,y) \ </math> and returns a number <math>z \ </math> can be visualized by constructing a surface consisting of points in three dimensional space <math>(x,y,z) \ </math>. See illustration at right.
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, 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.
=Problems=
=Problems=