# properties of vector-valued functions

If $F=({f}_{1},\mathrm{\dots},{f}_{n})$ and $G=({g}_{1},\mathrm{\dots},{g}_{n})$ are vector-valued and $u$ a real-valued function of the real variable $t$, one defines the vector-valued functions^{} $F+G$ and $uF$ componentwise as

$$F+G:=({f}_{1}+{g}_{1},\mathrm{\dots},{f}_{n}+{g}_{n}),uF:=(u{f}_{1},\mathrm{\dots},u{f}_{n})$$ |

and the real valued dot product^{} as

$$F\cdot G:={f}_{1}{g}_{1}+\mathrm{\dots}+{f}_{n}{g}_{n}.$$ |

If $n=3$, one my define also the vector-valued cross product^{} function as

$$F\times G:=(\left|\begin{array}{cc}\hfill {f}_{2}\hfill & \hfill {f}_{3}\hfill \\ \hfill {g}_{2}\hfill & \hfill {g}_{3}\hfill \end{array}\right|,\left|\begin{array}{cc}\hfill {f}_{3}\hfill & \hfill {f}_{1}\hfill \\ \hfill {g}_{3}\hfill & \hfill {g}_{1}\hfill \end{array}\right|,\left|\begin{array}{cc}\hfill {f}_{1}\hfill & \hfill {f}_{2}\hfill \\ \hfill {g}_{1}\hfill & \hfill {g}_{2}\hfill \end{array}\right|).$$ |

It’s not hard to verify, that if $F$, $G$ and $u$ are differentiable^{} on an interval, so are also
$F+G$, $uF$ and $F\cdot G$, and the formulae

$${(F+G)}^{\prime}={F}^{\prime}+{G}^{\prime},{(uF)}^{\prime}={u}^{\prime}F+u{F}^{\prime},{(F\cdot G)}^{\prime}={F}^{\prime}\cdot G+F\cdot {G}^{\prime}$$ |

are valid, in ${\mathbb{R}}^{3}$ additionally

$${(F\times G)}^{\prime}={F}^{\prime}\times G+F\times {G}^{\prime}.$$ |

Likewise one can verify the following theorems.

Theorem 1. If $u$ is continuous^{} in the point $t$ and $F$ in the point $u(t)$, then

$$H=F\circ u:=({f}_{1}\circ u,\mathrm{\dots},{f}_{n}\circ u)$$ |

is continuous in the point $t$. If $u$ is differentiable in the point $t$ and $F$ in the point $u(t)$, then the composite function $H$ is differentiable in $t$ and the chain rule^{}

$${H}^{\prime}(t)={F}^{\prime}(u(t)){u}^{\prime}(t)$$ |

is in .

Theorem 2. If $F$ and $G$ are integrable on $[a,b]$, so is also ${c}_{1}F+{c}_{2}G$, where ${c}_{1},{c}_{2}$ are real constants, and

$${\int}_{a}^{b}({c}_{1}F+{c}_{2}G)\mathit{d}t={c}_{1}{\int}_{a}^{b}F\mathit{d}t+{c}_{2}{\int}_{a}^{b}G\mathit{d}t.$$ |

Theorem 3. Suppose that $F$ is continuous on the interval $I$ and $c\in I$. Then the vector-valued function

$$t\mapsto {\int}_{c}^{t}F(\tau )\mathit{d}\tau :=G(t)\mathit{\hspace{1em}}\forall t\in I$$ |

is differentiable on $I$ and satisfies ${G}^{\prime}=F$.

Theorem 4. Suppose that $F$ is continuous on the interval $[a,b]$ and $G$ is an arbitrary function such that ${G}^{\prime}=F$ on this interval. Then

$${\int}_{a}^{b}F(t)\mathit{d}t=G(b)-G(a).$$ |

Theorem 2 may be generalised to

Theorem 5. If $F$ is integrable on $[a,b]$ and $C=({c}_{1},\mathrm{\dots},{c}_{n})$ is an arbitrary vector of ${\mathbb{R}}^{n}$, then dot product $C\cdot F$ is integrable on this interval and

$${\int}_{a}^{b}C\cdot F(t)\mathit{d}t=C\cdot {\int}_{a}^{b}F(t)\mathit{d}t.$$ |

Title | properties of vector-valued functions |
---|---|

Canonical name | PropertiesOfVectorvaluedFunctions |

Date of creation | 2013-03-22 19:02:42 |

Last modified on | 2013-03-22 19:02:42 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 9 |

Author | pahio (2872) |

Entry type | Topic |

Classification | msc 26A42 |

Classification | msc 26A36 |

Classification | msc 26A24 |

Related topic | ProductAndQuotientOfFunctionsSum |