Vector Valued Functions

Recall that for a particle moving in a straight line, for every time t we have a real number $x(t)$ 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 $(x(t),y(t))$ . We may say that the pair is the value of a function whose range is the set of points in the plane $\Re^{2}$ $\equiv \Re \times \Re$ . Thus, the function representing the position of the particle at different times is a function

$\overline{\gamma}: \Re \rightarrow \Re^{2}$

such that $\overline{\gamma} (t)=(x(t),y(t)) \in \Re^{2}$ , or if we write $\overline{i}$ and $\overline{j}$ for the unit vector along the x and y axis respectively, then

$\overline{\gamma}(t)=\overline{i}x(t)+\overline{j}y(t)$ .

We can similarly write the position of a particle in space as a function $\overline{\gamma}(t): \Re \rightarrow \Re^{2}$ such that

$\overline{\gamma}(t)=\overline{i}x(t)+\overline{j}y(t)+\overline{k}z(t)$

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 $\theta$ and longitude $\phi$ , we have a definite temperature

$T(\theta, \phi)$ at any instant. Thus,

$T: (-\pi/2,\pi/2) \times (0,2\pi)$ 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 $\theta$ , longitude $\phi$ and altitude h, at a a given instant, we have a unique real number $p(\theta, \phi, h)$ called the atmospheric pressure at that point. Thus, p can be deemed as a function whose domain is a part of $\Re^{3}$ and the range $\Re$ .

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 $(x,y,z) \in \Re^{3}$ is associated a vector $\overline{E}(x,y,z) \in \Re^{3}$ which is called the electric field strength at the point $(x,y,z)$ . Thus, we may think of electric field strength as a map $\overline{E}:\Re^{3} \rightarrow \Re^{3}$ . Similarly, magnetic field strength is a function

$\overline{H}: \Re^{3} \rightarrow \Re^{3}$ and the velocity of a fluid is again a function (or map) $\overline{q}: \Re^{3} \rightarrow \Re^{3}$ .

Exercises:

a) Give five more examples of vector valued functions.

b) Give five more examples of functions of many variables.

More later,

Nalin Pithwa

Mathematics Hothouse