# vector-valued function

Let $n$ be a positive integer greater than 1.  A function $F$ from a subset $T$ of $\mathbb{R}$ to the Cartesian product $\mathbb{R}^{n}$ is called a vector-valued function of one real variable.  Such a function to any real number $t$ of $T$ a coordinate vector

 $F(t)\;=\;(f_{1}(t),\,\ldots,\,f_{n}(t)).$

Hence one may say that the vector-valued function $F$ is composed of $n$ real functions$t\mapsto f_{i}(t)$,  the values of which at $t$ are the components of $F(t)$.  Therefore the function $F$ itself may be written in the component form

 $\displaystyle F\;=\;(f_{1},\,\ldots,\,f_{n}).$ (1)

Example.  The ellipse

 $\{(a\cos{t},\,b\sin{t})\,\vdots\;\;t\in\mathbb{R}\}$

is the value set of a vector-valued function  $\mathbb{R}\to\mathbb{R}^{2}$  ($t$ is the eccentric anomaly).

Limit, derivative and integral of the function (1) are defined componentwise through the equations

• $\displaystyle\lim_{t\to t_{0}}F(t)\;:=\;\left(\lim_{t\to t_{0}}f_{1}(t),\,% \ldots,\,\lim_{t\to t_{0}}f_{n}(t)\right)$

• $\displaystyle F^{\prime}(t)\;:=\;\left(f_{1}^{\prime}(t),\,\ldots,\,f_{n}^{% \prime}(t)\right)$

• $\displaystyle\int_{a}^{b}\!F(t)\,dt\;:=\;\left(\int_{a}^{b}\!f_{1}(t)\,dt,\,% \ldots,\,\int_{a}^{b}\!f_{n}(t)\,dt\right)$

The function $F$ is said to be continuous, differentiable or integrable on an interval$[a,\,b]$  if every component of $F$ has such a property.

Example.  If $F$ is continuous on  $[a,\,b]$,  the set

 $\displaystyle\gamma\;:=\;\{F(t)\,\vdots\;\;\;t\in[a,\,b]\}$ (2)

is a (continuous) curve in $\mathbb{R}^{n}$.  It follows from the above definition of the derivative $F^{\prime}(t)$ that $F^{\prime}(t)$ is the limit of the expression

 $\displaystyle\frac{1}{h}[F(t\!+\!h)-F(t)]$ (3)

as  $h\to 0$.  Geometrically, the vector (3) is parallel to the line segment connecting (the end points of the position vectors of) the points $F(t\!+\!h)$ and $F(t)$.  If $F$ is differentiable in $t$, the direction of this line segment then tends infinitely the direction of the tangent line of $\gamma$ in the point $F(t)$.  Accordingly, the direction of the tangent line is determined by the derivative vector $F^{\prime}(t)$.

Title vector-valued function VectorvaluedFunction 2013-03-22 19:02:19 2013-03-22 19:02:19 pahio (2872) pahio (2872) 8 pahio (2872) Definition msc 26A36 msc 26A42 msc 26A24 Component DifferenceOfVectors integrable