| | 2025.01.28

3: Vector Functions and Space Curves

Vector-valued functions describe curves and motion in space.

Vector Functions

A vector function maps a scalar to a vector:

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

As $t$ varies, $\mathbf{r}(t)$ traces out a space curve.

Examples

Helix:

\mathbf{r}(t) = \langle \cos t, \sin t, t \rangle

Line:

\mathbf{r}(t) = \mathbf{r}_0 + t\mathbf{v}

Circle in the $xy$ -plane:

\mathbf{r}(t) = \langle r\cos t, r\sin t, 0 \rangle

Calculus of Vector Functions

Limits and Continuity

Take limits component-wise:

\lim_{t \to a} \mathbf{r}(t) = \left\langle \lim_{t \to a} f(t), \lim_{t \to a} g(t), \lim_{t \to a} h(t) \right\rangle

Derivatives

\mathbf{r}'(t) = \langle f'(t), g'(t), h'(t) \rangle

Geometric meaning: $\mathbf{r}'(t)$ is the tangent vector to the curve at $\mathbf{r}(t)$ .

Unit tangent vector:

\mathbf{T}(t) = \frac{\mathbf{r}'(t)}{|\mathbf{r}'(t)|}

Differentiation Rules

$(c\mathbf{r})' = c\mathbf{r}'$
$(\mathbf{u} + \mathbf{v})' = \mathbf{u}' + \mathbf{v}'$
$(f\mathbf{r})' = f'\mathbf{r} + f\mathbf{r}'$ (scalar times vector)
$(\mathbf{u} \cdot \mathbf{v})' = \mathbf{u}' \cdot \mathbf{v} + \mathbf{u} \cdot \mathbf{v}'$
$(\mathbf{u} \times \mathbf{v})' = \mathbf{u}' \times \mathbf{v} + \mathbf{u} \times \mathbf{v}'$

Integrals

\int_a^b \mathbf{r}(t) \, dt = \left\langle \int_a^b f(t)\,dt, \int_a^b g(t)\,dt, \int_a^b h(t)\,dt \right\rangle

Motion in Space

If $\mathbf{r}(t)$ is the position of a particle:

Quantity	Formula
Position	$\mathbf{r}(t)$
Velocity	$\mathbf{v}(t) = \mathbf{r}'(t)$
Speed	$
Acceleration	$\mathbf{a}(t) = \mathbf{v}'(t) = \mathbf{r}''(t)$

Arc Length

Length of curve from $t = a$ to $t = b$ :

L = \int_a^b |\mathbf{r}'(t)| \, dt = \int_a^b \sqrt{(x')^2 + (y')^2 + (z')^2} \, dt

Arc Length Parameter

The arc length function:

s(t) = \int_a^t |\mathbf{r}'(u)| \, du

Note: $\frac{ds}{dt} = |\mathbf{r}'(t)|$

Curvature

Curvature $\kappa$ measures how fast the curve turns:

\kappa = \left| \frac{d\mathbf{T}}{ds} \right| = \frac{|\mathbf{T}'(t)|}{|\mathbf{r}'(t)|} = \frac{|\mathbf{r}'(t) \times \mathbf{r}''(t)|}{|\mathbf{r}'(t)|^3}

Radius of Curvature

\rho = \frac{1}{\kappa}

The radius of the best-fitting circle (osculating circle) at that point.

Curvature for $y = f(x)$

\kappa = \frac{|f''(x)|}{[1 + (f'(x))^2]^{3/2}}

The TNB Frame (Frenet-Serret Frame)

Three mutually perpendicular unit vectors that move along the curve:

Unit Tangent Vector

\mathbf{T} = \frac{\mathbf{r}'}{|\mathbf{r}'|}

Points in the direction of motion.

Principal Normal Vector

\mathbf{N} = \frac{\mathbf{T}'}{|\mathbf{T}'|}

Points toward the center of curvature (the direction the curve is turning).

Binormal Vector

\mathbf{B} = \mathbf{T} \times \mathbf{N}

Perpendicular to the osculating plane.

Frenet-Serret Formulas

\mathbf{T}' = \kappa |\mathbf{r}'| \mathbf{N}

\mathbf{N}' = -\kappa |\mathbf{r}'| \mathbf{T} + \tau |\mathbf{r}'| \mathbf{B}

\mathbf{B}' = -\tau |\mathbf{r}'| \mathbf{N}

where $\tau$ is the torsion (measures how the curve twists out of the osculating plane).

Tangential and Normal Components of Acceleration

Acceleration can be decomposed:

\mathbf{a} = a_T \mathbf{T} + a_N \mathbf{N}

where:

Tangential component (changes speed):

a_T = \frac{d}{dt}|\mathbf{v}| = \frac{\mathbf{v} \cdot \mathbf{a}}{|\mathbf{v}|}

Normal component (changes direction):

a_N = \kappa |\mathbf{v}|^2 = \frac{|\mathbf{v} \times \mathbf{a}|}{|\mathbf{v}|}

Summary

Concept	Formula
Tangent vector	$\mathbf{T} = \frac{\mathbf{r}’}{
Arc length	$L = \int
Curvature	$\kappa = \frac{
Normal vector	$\mathbf{N} = \frac{\mathbf{T}’}{
Binormal vector	$\mathbf{B} = \mathbf{T} \times \mathbf{N}$
Tangential acceleration	$a_T = \frac{\mathbf{v} \cdot \mathbf{a}}{
Normal acceleration	$a_N = \frac{