Vector Calculus: An Introduction

We now look at vector valued functions, that is, functions defined on a subset \(D\subset\mathbb R^N\) taking values in \(\mathbb R^k\text{.}\) Such a function assigns every \(\vect x\in D\) a unique vector \(\vect f(\vect x)\text{.}\) As a result \(\vect f(\vect x)\) has \(k\) components, so \(\vect f(\vect x)=(f_1(\vect x),\dots,f_k(\vect x))\text{.}\) Every component is a function of \(N\) variables with values in \(\mathbb R\) as considered before. We call \(f_i\text{,}\) \(i=1,\dots,k\text{,}\) the component functions of \(\vect f\text{.}\) We can visualise functions of one or two variables with values in \(\mathbb R^2\) or \(\mathbb R^3\text{.}\)

Suppose that \(N=1\) and \(k=2\) then \(\vect f(t)=(f_1(t),f_2(t))\) is defined for instance for \(t\) in an interval \(I\subset\mathbb R\text{.}\) Similarly, if \(N=1\) and \(k=3\) then \(\vect f(t)=(f_1(t),f_2(t),f_3(t))\) for \(t\in I\text{.}\) In both cases \(\vect f(t)\) can be thought of as the position of a particle at time \(t\) moving in the plane or in space, respectively. We can represent \(\vect f\) by drawing the path along which the particle travels as shown in Figure 2.14. We talk about a curve.

Now consider a function defined on a subset of \(D\subset\mathbb R^2\) with values in \(\mathbb R^3\text{.}\) Again we can plot the vectors \(\vect f(\vect x)\) for \(\vect x\in D\text{.}\) The resulting set is a surface in \(\mathbb R^3\) as shown in Figure 2.15. We will discuss curves and surfaces later in more detail.