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

This entry was written by Nalin Pithwa, posted on June 10, 2015 at 6:18 pm, filed under IITJEE Mains and tagged vectors. Bookmark the permalink. Follow any comments here with the RSS feed for this post. Post a comment or leave a trackback: Trackback URL.

Mathematics Hothouse