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

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 ${ℝ}^3$ 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

$${\int }_a^b(c_{1}F+c_{2}G)𝑑t=c_1{\int }_a^{b}F𝑑t+c_2{\int }_a^{b}G𝑑t.$$

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

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

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

$${\int }_a^{b}F(t)𝑑t=G(b)-G(a).$$

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 ${ℝ}^n$, then dot product $C\cdot F$ is integrable on this interval and

$${\int }_a^{b}C\cdot F(t)𝑑t=C\cdot {\int }_a^{b}F(t)𝑑t.$$