Recall that for a particle moving in a straight line, for every time t we have a real number representing the distance of the particle measured from a definite point at the time t. But what about a particle moving in planc or in space? We know that every time t, the particle has a position in the plane or the space. But a point in the plane (or the space) is represented by a pair (or a triple) of real numbers. Thus, for a particle moving in a plane its position at time t is represented by a pair
. We may say that the pair is the value of a function whose range is the set of points in the plane
. Thus, the function representing the position of the particle at different times is a function
such that , or if we write
and
for the unit vector along the x and y axis respectively, then
.
We can similarly write the position of a particle in space as a function such that
Function of many variables
We have discussed before that the temperature at a point on Earth, at any instant, is a unique real number. Now every point on Earth is represented by a pair of real numbers depicting its latitude and longitude respectively (one ought to be careful in making this statement when it comes to a point on the date line. Indeed there is a little ambiguity in representing the longitude of a point on the date line. Besides, the poles have unique latitude, but what about their longitudes? Barring such ambiguity, every point on Earth can be represented uniquely by a pair of real numbers.) Thus, for a point with latitude and longitude
, we have a definite temperature
at any instant. Thus,
represents the temperature at a point.
Similarly, if we take any point in the atmosphere, the atmospheric pressure at the point depends on the latitude, the longitude and the altitude of the point. Indeed, for a point with latitude , longitude
and altitude h, at a a given instant, we have a unique real number
called the atmospheric pressure at that point. Thus, p can be deemed as a function whose domain is a part of
and the range
.
Vector Fields
Electric field strength at a point is defined as the force experienced by a unit electric charge at at that point. This means that with every point is associated a vector
which is called the electric field strength at the point
. Thus, we may think of electric field strength as a map
. Similarly, magnetic field strength is a function
and the velocity of a fluid is again a function (or map)
.
Exercises:
a) Give five more examples of vector valued functions.
b) Give five more examples of functions of many variables.
More later,
Nalin Pithwa