Vector-Valued Functions

• These notes cover the essentials of vector-valued functions of a real variable

Definition

• A vector-valued function maps a real number to a vector in $\mathbb{R}^n$.
• It is expressed as $\overrightarrow{r}(t) = (f_1(t), f_2(t),..., f_n(t))$, where each $f_i(t)$ is a real-valued function.

Example

• $\overrightarrow{r}(t) = (cos(t), sin(t), t)$ is a vector-valued function from $\mathbb{R}$ to $\mathbb{R}^3$.
• Its component functions are $f_1(t) = cos(t)$, $f_2(t) = sin(t)$, and $f_3(t) = t$.

Graphical Representation

• Vector-valued functions are graphed as curves in $\mathbb{R}^n$.

Limit

• The limit of $\overrightarrow{r}(t)$ as $t$ approaches $a$ is found by taking the limit of each component function:
• $\lim_{t \to a} \overrightarrow{r}(t) = (\lim_{t \to a} f_1(t), \lim_{t \to a} f_2(t),..., \lim_{t \to a} f_n(t))$ if the individual limits exist.

Continuity

• $\overrightarrow{r}(t)$ is continuous at $t = a$ if:
• $\overrightarrow{r}(a)$ is defined.
• $\lim_{t \to a} \overrightarrow{r}(t)$ exists.
• $\lim_{t \to a} \overrightarrow{r}(t) = \overrightarrow{r}(a)$.
• Equivalently, $\overrightarrow{r}(t)$ is continuous at $t = a$ if all its component functions are continuous at $t = a$.

Derivative

• The derivative of $\overrightarrow{r}(t)$ is defined as:
• $\overrightarrow{r}'(t) = \lim_{h \to 0} \frac{\overrightarrow{r}(t+h) - \overrightarrow{r}(t)}{h}$, if the limit exists.
• If $\overrightarrow{r}(t) = (f_1(t), f_2(t),..., f_n(t))$, then $\overrightarrow{r}'(t) = (f_1'(t), f_2'(t),..., f_n'(t))$.

Geometric Interpretation of the Derivative

• $\overrightarrow{r}'(t)$ is a tangent vector to the curve at the point $\overrightarrow{r}(t)$.

Integral

• The integral of $\overrightarrow{r}(t)$ is found by integrating each component function separately:
• $\int \overrightarrow{r}(t) dt = (\int f_1(t) dt, \int f_2(t) dt,..., \int f_n(t) dt)$

Arc Length

• The arc length $L$ of the curve from $t = a$ to $t = b$ is:
• $L = \int_a^b ||\overrightarrow{r}'(t)|| dt$, where $||\overrightarrow{r}'(t)||$ is the magnitude of the derivative vector.