If $F = (f_1,\,\ldots,\,f_n)$ and $G = (g_1,\,\ldots,\,g_n)$ 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 $$F\!+\!G \;:=\; (f_1\!+\!g_1,\,\ldots,\,f_n\!+\!g_n), \quad uF \;:=\; (uf_1,\,\ldots,\,uf_n)$$ and the real valued dot product as $$F \cdot G \;:=\; f_1g_1\!+\ldots+\!f_ng_n.$$ If $n = 3$ , one my define also the vector-valued cross product function as $$F\!\times\!G \;:=\; \left( \left|\begin{matrix} f_2 & f_3 \\ g_2 & g_3 \end{matrix}\right|\!,\, \left|\begin{matrix} f_3 & f_1 \\ g_3 & g_1 \end{matrix}\right|\!,\, \left|\begin{matrix} f_1 & f_2 \\ g_1 & g_2 \end{matrix}\right| \right)\!.$$

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 $$(F\!+\!G)' \;=\; F'\!+\!G', \quad (uF)' \;=\; u'F\!+\!uF', \quad (F\cdot G)' \;=\; F'\cdot G+F\cdot G'$$ are valid, in $\mathbb{R}^3$ additionally $$(F\!\times\!G)' \;=\; F'\!\times\!G+F\!\times\!G'.$$

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 $$H \;=\; F\!\circ\!u \;:=\; (f_1\!\circ\!u,\,\ldots,\,f_n\!\circ\!u)$$ is continuous in the point $t$ . If $u$ is differentiable in the point $t$ and $F$ in the point $u(t)$ , then the compound function $H$ is differentiable in $t$ and the chain rule $$H'(t) \;=\; F'(u(t))\,u'(t)$$ is in force.

Theorem 2. If $F$ and $G$ are integrable on $[a,\,b]$ , so is also $c_1F\!+\!c_2G$ , where $c_1,\,c_2$ are real constants, and $$\int_a^b\!(c_1F\!+\!c_2G)\,dt \;=\; c_1\int_a^b\!F\,dt+c_2\int_a^b\!G\,dt.$$

Theorem 3. Suppose that $F$ is continuous on the interval $I$ and $c \in I$ . Then the vector-valued function $$t\; \mapsto\; \int_c^t\!F(\tau)\,d\tau \;:=\; G(t) \quad \forall\, t \in I$$ is differentiable on $I$ and satisfies $G' = F$ .

Theorem 4. Suppose that $F$ is continuous on the interval $[a,\,b]$ and $G$ is an arbitrary function such that $G' = F$ 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 $C = (c_1,\,\ldots,\,c_n)$ is an arbitrary vector of $\mathbb{R}^n$ , then dot product $C\cdot F$ is integrable on this interval and $$\int_a^b\!C\cdot F(t)\,dt \;=\; C\cdot\!\int_a^b\!F(t)\,dt.$$