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 $\mathit{x}\in D$ a unique vector $\mathit{f}(\mathit{x})\text{.}$ As a result $\mathit{f}(\mathit{x})$ has $k$ components, so $\mathit{f}(\mathit{x})=(f_1(\mathit{x}),\dots ,f_k(\mathit{x}))\text{.}$ 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 $\mathit{f}\text{.}$ 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 $\mathit{f}(t)=(f_1(t),f_2(t))$ is defined for instance for $t$ in an interval $I\subset \mathbb{R}\text{.}$ Similarly, if $N=1$ and $k=3$ then $\mathit{f}(t)=(f_1(t),f_2(t),f_3(t))$ for $t\in I\text{.}$ In both cases $\mathit{f}(t)$ can be thought of as the position of a particle at time $t$ moving in the plane or in space, respectively. We can represent $\mathit{f}$ 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 $\mathit{f}(\mathit{x})$ for $\mathit{x}\in D\text{.}$ 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.