Vector Functions of Scalar Variables

• A vector function of a scalar variable assigns a vector to each real number in an interval.
• It is expressed as $\overrightarrow{r}: I \subset \mathbb{R} \longrightarrow \mathbb{R}^{n}$, where each $t \in I$ corresponds to a vector $\overrightarrow{r}(t)=\left(f_{1}(t), f_{2}(t), \ldots, f_{n}(t)\right)$.
• Each $f_{i}: I \subset \mathbb{R} \longrightarrow \mathbb{R}$ is a real function of a real variable.
• An example is: $\vec{r}(t)=(\cos (t), \operatorname{sen}(t)), t \in[0,2 \pi]$.

Trace or graph of a vector function

• The trace or graph of a vector function $\overrightarrow{r}(t)$ is the set of points in $\mathbb{R}^{n}$ defined by $\left{\left(f_{1}(t), f_{2}(t), \ldots, f_{n}(t)\right) \mid t \in I\right}$.
• It is the set of all vectors resulting from evaluating the function $\overrightarrow{r}(t)$ at each point in its domain.
• For instance, the graph of the vector function $\overrightarrow{r}(t)=(\cos (t), \operatorname{sen}(t)), t \in[0,2 \pi]$ is a circle with a radius of 1, centered at the origin.

Derivative of a vector function

• The derivative of a vector function $\overrightarrow{r}(t)=\left(f_{1}(t), f_{2}(t), \ldots, f_{n}(t)\right)$ is the vector function defined by $\overrightarrow{r}^{\prime}(t)=\left(f_{1}^{\prime}(t), f_{2}^{\prime}(t), \ldots, f_{n}^{\prime}(t)\right)$.
• The derivative exists if the derivatives of the component functions exist.
• For example: If $\overrightarrow{\mathrm{r}}(\mathrm{t})=\left(\mathrm{t}, \mathrm{t}^{2}, \mathrm{t}^{3}\right)$ then $\overrightarrow{\mathrm{r}}^{\prime}(\mathrm{t})=\left(1,2 \mathrm{t}, 3 \mathrm{t}^{2}\right)$.

Geometric interpretation of the derivative

• The derivative of a vector function $\overrightarrow{r}^{\prime}(t)$ represents a tangent vector to the curve described by the vector function at the point $\overrightarrow{r}(t)$.
• The direction of this vector indicates the direction in which an object moving along the curve is moving at time $t$.

Integrals of vector functions

• The integral of a vector function $\overrightarrow{r}(t)=\left(f_{1}(t), f_{2}(t), \ldots, f_{n}(t)\right)$ is defined by integrating each component function: $\int \overrightarrow{\mathrm{r}}(\mathrm{t}) \mathrm{dt}=\left(\int \mathrm{f}{1}(\mathrm{t}) \mathrm{dt}, \int \mathrm{f}{2}(\mathrm{t}) \mathrm{dt}, \ldots, \int \mathrm{f}_{\mathrm{n}}(\mathrm{t}) \mathrm{dt}\right)$.
• To find the integral of a vector function, each component is integrated separately.
• For example: $\int\left(t, t^{2}, t^{3}\right) d t=\left(\int t d t, \int t^{2} d t, \int t^{3} d t\right)=\left(\frac{t^{2}}{2}+C_{1}, \frac{t^{3}}{3}+C_{2}, \frac{t^{4}}{4}+C_{3}\right)$.