Vector Functions: Curves in Space

Vector Functions of a Real Variable

• These functions assign a vector to each real number and are used to describe curves in space.

Definition of Vector Functions

• A vector function has the form: $\vec{r}(t) = f(t)\hat{i} + g(t)\hat{j} + h(t)\hat{k}$
• $f(t)$, $g(t)$, and $h(t)$ are real functions of the real variable $t$.
• $\hat{i}$, $\hat{j}$, and $\hat{k}$ are unit vectors in the $x$, $y$, and $z$ directions.

Domain of Vector Functions

• The domain of $\vec{r}(t)$ is all $t$ values for which $f(t)$, $g(t)$, and $h(t)$ are defined.

Limit of Vector Functions

• The limit of $\vec{r}(t)$ as $t$ approaches $t_0$ is given by: $\lim_{t \to t_0} \vec{r}(t) = \left( \lim_{t \to t_0} f(t) \right) \hat{i} + \left( \lim_{t \to t_0} g(t) \right) \hat{j} + \left( \lim_{t \to t_0} h(t) \right) \hat{k}$
• This requires that the limits of $f(t)$, $g(t)$, and $h(t)$ exist as $t$ approaches $t_0$.

Continuity of Vector Functions

• A vector function $\vec{r}(t)$ is continuous at $t = t_0$ if:
  • $\vec{r}(t_0)$ is defined.
  • $\lim_{t \to t_0} \vec{r}(t)$ exists.
  • $\lim_{t \to t_0} \vec{r}(t) = \vec{r}(t_0)$.

Derivative of Vector Functions

• The derivative of $\vec{r}(t)$ is defined as: $\frac{d\vec{r}}{dt} = \lim_{\Delta t \to 0} \frac{\vec{r}(t + \Delta t) - \vec{r}(t)}{\Delta t}$
• Equivalently: $\vec{r}'(t) = f'(t)\hat{i} + g'(t)\hat{j} + h'(t)\hat{k}$, provided the derivatives of $f(t)$, $g(t)$, and $h(t)$ exist.

Integral of Vector Functions

• The integral of $\vec{r}(t)$ is defined as: $\int \vec{r}(t) dt = \left( \int f(t) dt \right) \hat{i} + \left( \int g(t) dt \right) \hat{j} + \left( \int h(t) dt \right) \hat{k}$

Arc Length

• The arc length of a curve described by $\vec{r}(t)$ from $t = a$ to $t = b$ is: $L = \int_a^b \left| \frac{d\vec{r}}{dt} \right| dt = \int_a^b \sqrt{\left( \frac{df}{dt} \right)^2 + \left( \frac{dg}{dt} \right)^2 + \left( \frac{dh}{dt} \right)^2} dt$

Curvature

• Curvature measures how much the direction of a curve changes at a given point.
• Defined as: $\kappa = \frac{\left| \frac{d\vec{T}}{ds} \right|}{\left| \frac{d\vec{r}}{ds} \right|} = \frac{\left| \vec{T}'(t) \right|}{\left| \vec{r}'(t) \right|}$
• $\vec{T}(t) = \frac{\vec{r}'(t)}{\left| \vec{r}'(t) \right|}$ is the unit tangent vector, and $s$ is the arc length parameter.

Tangential and Normal Components of Acceleration

• Acceleration of a particle moving along a curve can be decomposed into tangential and normal components: $\vec{a} = a_T \vec{T} + a_N \vec{N}$
• Tangential component: $a_T = \frac{d}{dt} \left| \frac{ds}{dt} \right| = \frac{\vec{r}'(t) \cdot \vec{r}''(t)}{\left| \vec{r}'(t) \right|}$
• Normal component: $a_N = \kappa \left( \frac{ds}{dt} \right)^2 = \frac{\left| \vec{r}'(t) \times \vec{r}''(t) \right|}{\left| \vec{r}'(t) \right|}$
• $\vec{N}$ is the unit normal vector.