vector-valued function

Let $n$ be a positive integer greater than 1.  A function $F$ from a subset $T$ of $ℝ$ to the Cartesian product ${ℝ}^n$ is called a vector-valued 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 vector-valued 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

$$\{(a\cos t,b\sin t)\mathrm{⋮}t\in ℝ\}$$

is the value set of a vector-valued function  $ℝ\to {ℝ}^2$  ($t$ is the eccentric anomaly).

Limit, derivative and integral of the function (1) are defined componentwise through the equations

• $\underset{t\to t_0}{lim}F(t):=(\underset{t\to t_0}{lim}{f}_1(t),\mathrm{\dots },\underset{t\to t_0}{lim}{f}_n(t))$

• $F^{\prime }(t):=(f_1^{\prime }(t),\mathrm{\dots },f_n^{\prime }(t))$

• ${\displaystyle {\int }_a^b}F(t)𝑑t:=({\displaystyle {\int }_a^b}{f}_1(t)𝑑t,\mathrm{\dots },{\displaystyle {\int }_a^b}{f}_n(t)𝑑t)$

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

$\gamma :=\{F(t)\mathrm{⋮}t\in [a,b]\}$(2)

is a (continuous) curve in ${ℝ}^n$.  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 $F^{\prime }(t)$.