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vectorvalued 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 vectorvalued function of one real variable. Such a function joins 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 vectorvalued 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 vectorvalued 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 $F^{{\prime}}(t)$.
Mathematics Subject Classification
26A36 no label found26A42 no label found26A24 no label found Forums
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