vector calculus

Vector-Valued Functions

Calculus III30 minDifficulty

You should know: vectors, derivative, arc length and curvature

Overview

A vector-valued function r(t) = (x(t), y(t), z(t)) maps a scalar parameter t to a vector in R^2 or R^3. The derivative r'(t) is the tangent vector, and |r'(t)| is the speed. The unit tangent, normal, and binormal vectors (TNB frame) describe the geometry of the curve. These functions naturally describe trajectories in space.

Intuition

Think of r(t) as the position of a particle at time t. r'(t) is the velocity vector (direction of motion, magnitude = speed). r''(t) is the acceleration. The curve traced by r(t) has a natural frame: T = r'/|r'| (direction you're moving), N = T'/|T'| (direction you're turning toward), B = T x N (perpendicular to both -- like the axis of a spiral).

Formal Definition

Definition

r(t) = (x(t), y(t), z(t)). Derivative: r'(t) = (x'(t), y'(t), z'(t)). Speed: |r'(t)|. Arc length from a to b: integral_a^b |r'(t)| dt. Unit tangent: T = r'/|r'|. Principal normal: N = T'/|T'|. Binormal: B = T x N. Curvature: kappa = |T'|/|r'| = |r' x r''| / |r'|^3.

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

Unit tangent vector

\kappa = \frac{|\mathbf{r}' \times \mathbf{r}''|}{|\mathbf{r}'|^3}

Curvature

L = \int_a^b |\mathbf{r}'(t)|\,dt

Arc length

Notation

Notation	Meaning
$\mathbf{r}(t)$	Vector-valued function (position curve)
$\mathbf{T}, \mathbf{N}, \mathbf{B}$	Tangent, Normal, Binormal (TNB frame)
$\kappa$	Curvature of the curve

Theorems

Theorem 1: Frenet-Serret Formulas

T' = kappa*|r'|*N, N' = |r'|(-kappa*T + tau*B), B' = -tau*|r'|*N, where tau is the torsion (how fast the curve twists out of its osculating plane). These three equations describe all geometric properties of a space curve.

Worked Examples

1
r'(t) = (-sin t, cos t, 1).
$\mathbf{r}'(0) = (0, 1, 1)$
2
|r'(0)| = sqrt(0+1+1) = sqrt(2).
$\mathbf{T}(0) = \frac{(0,1,1)}{\sqrt{2}} = \left(0, \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)$

✓ Answer

T(0) = (0, 1/sqrt(2), 1/sqrt(2)).

Practice Problems

Mediumapplication

Find the arc length of r(t) = (3t, 4t, 0) from t=0 to t=2.

Common Mistakes

Common Mistake

Confusing the unit tangent T with the unit normal N.

T = r'/|r'|: direction of motion. N = T'/|T'|: direction the curve is turning (perpendicular to T). N is derived from T, not directly from r.

Quiz

The unit tangent vector T(t) is:

r(t) / |r(t)|r'(t) / |r'(t)|r''(t) / |r''(t)||r'(t)|

Historical Background

Vector calculus developed in the 19th century, largely through the work of Gibbs and Heaviside who introduced modern vector notation from Hamilton's quaternions. The TNB frame (Frenet-Serret formulas) was developed by Frenet (1847) and Serret (1851). Vector-valued functions are essential in physics for describing motion, electromagnetic fields, and fluid flow.

1847
Frenet publishes his doctoral thesis introducing the moving frame
Jean Frederic Frenet
1881
Gibbs introduces modern vector calculus notation
Josiah Willard Gibbs

Summary

r(t) = (x(t), y(t), z(t)): position curve. r'(t): velocity/tangent. |r'(t)|: speed.
Unit tangent T = r'/|r'|. Principal normal N = T'/|T'|. Binormal B = T x N.
Curvature kappa = |T'|/|r'| = |r' x r''|/|r'|^3.
Arc length L = integral |r'(t)| dt.

References

BookStewart, J. Multivariable Calculus. 8th ed. Brooks/Cole, 2015.