Section 3.1 - Vector Functions and Space Curves

Section 3.1 - Vector Functions and Space Curves

Vector Functions

A vector function, or vector-valued function, is a function whose domain is a set of real numbers and whose range is a set of vectors. In calculus, vector functions are commonly used to describe curves and motion in space.

• Definition: A vector function r is typically written as:

$\mathbf{r}(t) = f(t)\mathbf{i} + g(t)\mathbf{j} + h(t)\mathbf{k}$

• Here, $f$, $g$, and $h$ are called the component functions of $\mathbf{r}(t)$.

• The variable $t$ is usually a real number parameter, often representing time.

Limits and Continuity of Vector Functions

The limit of a vector function is defined by taking the limits of its component functions individually. This is analogous to the limit of a real-valued function, but applied to each component.

• Limit Definition: If $\mathbf{r}(t) = f(t)\mathbf{i} + g(t)\mathbf{j} + h(t)\mathbf{k}$, then:

$\lim_{t \to a} \mathbf{r}(t) = \left( \lim_{t \to a} f(t) \right) \mathbf{i} + \left( \lim_{t \to a} g(t) \right) \mathbf{j} + \left( \lim_{t \to a} h(t) \right) \mathbf{k}$

• This limit exists provided the limits of the component functions exist.

Example 1: Computing a Vector Function Limit

Find $\lim_{t \to 0} \mathbf{r}(t)$, where $\mathbf{r}(t) = (1 + t^2)\mathbf{i} + e^{t^2}\mathbf{j} + \sin t\mathbf{k}$.

• Compute each component's limit as $t \to 0$:

  • $\lim_{t \to 0} (1 + t^2) = 1$

  • $\lim_{t \to 0} e^{t^2} = 1$

  • $\lim_{t \to 0} \sin t = 0$

• Therefore, $\lim_{t \to 0} \mathbf{r}(t) = 1\mathbf{i} + 1\mathbf{j} + 0\mathbf{k}$

Space Curves

A space curve is the set of all points $(x, y, z)$ in space traced out by a vector function as the parameter $t$ varies over an interval. If $f$, $g$, and $h$ are continuous real-valued functions on an interval $I$, then the set of all points $(x, y, z)$ where:

• $x = f(t)$

• $y = g(t)$

• $z = h(t)$

as $t$ varies throughout the interval $I$, describes a curve in space.

Example 2: Sketching a Space Curve

Sketch the curve whose vector equation is $\mathbf{r}(t) = \cos t\mathbf{i} + \sin t\mathbf{j} + t\mathbf{k}$.

• This curve traces a helix in three-dimensional space.

• As $t$ increases, the $(x, y)$ coordinates trace a circle, while $z$ increases linearly.

Application: Helices are common in physics and engineering, such as the path of a screw or DNA molecules.

Example 3: Intersection of Surfaces

Find a vector function that represents the curve of intersection of the cylinder $x^2 + y^2 = 1$ and the plane $y + z = 2$.

• Parameterize the cylinder using $x = \cos t$, $y = \sin t$.

• From the plane equation: $z = 2 - y = 2 - \sin t$.

• Therefore, the vector function is:

$\mathbf{r}(t) = \cos t\mathbf{i} + \sin t\mathbf{j} + (2 - \sin t)\mathbf{k}$

Application: Finding intersections is important in geometry, engineering, and computer graphics.