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
Canonical name	VectorvaluedFunction
Date of creation	2013-03-22 19:02:19
Last modified on	2013-03-22 19:02:19
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	8
Author	pahio (2872)
Entry type	Definition
Classification	msc 26A36
Classification	msc 26A42
Classification	msc 26A24
Related topic	Component
Related topic	DifferenceOfVectors
Defines	integrable