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