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 to any real number $t$ of $T$ a coordinate vector
$$F(t)=({f}_{1}(t),\mathrm{\dots},{f}_{n}(t)).$$ 
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
$F=({f}_{1},\mathrm{\dots},{f}_{n}).$  (1) 
Example. The ellipse^{}
$$\{(a\mathrm{cos}t,b\mathrm{sin}t)\mathrm{\vdots}t\in \mathbb{R}\}$$ 
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

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$\underset{t\to {t}_{0}}{lim}F(t):=(\underset{t\to {t}_{0}}{lim}{f}_{1}(t),\mathrm{\dots},\underset{t\to {t}_{0}}{lim}{f}_{n}(t))$

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${F}^{\prime}(t):=({f}_{1}^{\prime}(t),\mathrm{\dots},{f}_{n}^{\prime}(t))$

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${\int}_{a}^{b}}F(t)\mathit{d}t:=({\displaystyle {\int}_{a}^{b}}{f}_{1}(t)\mathit{d}t,\mathrm{\dots},{\displaystyle {\int}_{a}^{b}}{f}_{n}(t)\mathit{d}t)$
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
$\gamma :=\{F(t)\mathrm{\vdots}t\in [a,b]\}$  (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
$\frac{1}{h}}[F(t+h)F(t)]$  (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  vectorvalued function 

Canonical name  VectorvaluedFunction 
Date of creation  20130322 19:02:19 
Last modified on  20130322 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 