Chapter 28 — Key Takeaways
A compact recap of the vector view of curves. The whole chapter rests on one idea: a vector-valued function is one function of one variable that happens to return a vector, so all of single-variable calculus carries over component-wise — and the only genuinely new content is geometric.
The central object
A vector-valued function sends one real number to one vector:
$$\mathbf{r}(t) = \langle x(t),\, y(t),\, z(t)\rangle = x(t)\,\mathbf{i} + y(t)\,\mathbf{j} + z(t)\,\mathbf{k}.$$
As $t$ runs over its interval, the tip of $\mathbf{r}(t)$ sweeps out a space curve. A planar parametric curve (Chapter 25) is the special case $z(t)=0$.
Do not confuse this with a vector field. $\mathbf{r}(t)$ takes one number and returns one vector — a single curve. A vector field (Chapter 34) attaches a vector to every point of a region. One curve versus a whole field of arrows. (§28.1)
Calculus is component-wise (§28.3)
Limits, derivatives, and integrals all act slot by slot:
| Operation | Formula |
|---|---|
| Derivative | $\mathbf{r}'(t) = \langle x'(t),\, y'(t),\, z'(t)\rangle$ |
| Indefinite integral | $\displaystyle\int \mathbf{r}(t)\,dt = \Big\langle \int x\,dt,\, \int y\,dt,\, \int z\,dt\Big\rangle + \mathbf{C}$ |
| Definite integral | $\displaystyle\int_a^b \mathbf{r}(t)\,dt = \Big\langle \int_a^b x\,dt,\, \int_a^b y\,dt,\, \int_a^b z\,dt\Big\rangle$ |
The constant of integration $\mathbf{C}$ is a constant vector (three constants bundled). FTC holds component-wise. The four product rules mirror the scalar ones:
$$(\mathbf{u}+\mathbf{v})' = \mathbf{u}'+\mathbf{v}', \quad (f\mathbf{u})' = f'\mathbf{u}+f\mathbf{u}', \quad (\mathbf{u}\cdot\mathbf{v})' = \mathbf{u}'\cdot\mathbf{v}+\mathbf{u}\cdot\mathbf{v}', \quad (\mathbf{u}\times\mathbf{v})' = \mathbf{u}'\times\mathbf{v}+\mathbf{u}\times\mathbf{v}'.$$
Constant-length fact: if $|\mathbf{u}(t)|$ is constant, then $\mathbf{u}\cdot\mathbf{u}' = 0$. Used three times in the chapter.
The geometry of the derivative
$\mathbf{r}'(t)$ is the limit of secant (chord) vectors, so it is tangent to the curve, pointing in the direction of increasing $t$. "If the curve is a road, $\mathbf{r}'(t)$ is the direction your headlights face." (§28.3)
Motion: velocity, speed, acceleration (§28.4)
Read $\mathbf{r}(t)$ as the position of a particle:
- Velocity $\mathbf{v}(t) = \mathbf{r}'(t)$ — a vector, tangent to the path.
- Speed $|\mathbf{v}(t)| = \sqrt{x'^2 + y'^2 + z'^2}$ — a non-negative scalar.
- Acceleration $\mathbf{a}(t) = \mathbf{r}''(t)$ — records how velocity changes, whether by speeding up, slowing down, or turning.
The velocity-versus-speed distinction is the conceptual core: a car at steady 30 mph on a bend has constant speed but changing velocity, hence nonzero acceleration. Knowing $\mathbf{a}(t)$ plus initial conditions recovers the trajectory by integrating twice (the projectile $\mathbf{r}'' = \langle 0,0,-g\rangle$ is the model example).
Arc length and unit speed (§28.5–28.6)
$$L = \int_a^b |\mathbf{r}'(t)|\,dt \qquad (\text{integrate the speed}).$$
The arc-length function $s(t) = \int_a^t |\mathbf{r}'(u)|\,du$ satisfies $ds/dt = |\mathbf{r}'(t)|$ by FTC. The unit tangent
$$\mathbf{T}(t) = \frac{\mathbf{r}'(t)}{|\mathbf{r}'(t)|}$$
is the direction of motion with speed scrubbed out. Reparametrizing by arc length gives the unit-speed (natural) parametrization, in which velocity equals $\mathbf{T}$. Most curves cannot be reparametrized in closed form (the ellipse's arc length defines elliptic integrals); the helix can, because its speed is constant.
Curvature (§28.7–28.8)
Curvature measures how sharply the curve bends, per unit distance (so speed drops out):
$$\kappa = \left|\frac{d\mathbf{T}}{ds}\right| = \frac{|\mathbf{T}'(t)|}{|\mathbf{r}'(t)|} = \frac{|\mathbf{r}'(t)\times\mathbf{r}''(t)|}{|\mathbf{r}'(t)|^3}.$$
The cross-product form is the practical one. For a plane curve $y=f(x)$:
$$\kappa(x) = \frac{|f''(x)|}{\big(1+f'(x)^2\big)^{3/2}}.$$
| Curve | Curvature |
|---|---|
| Straight line | $\kappa = 0$ |
| Circle of radius $R$ | $\kappa = 1/R$ (constant) |
| Helix $\langle a\cos t, a\sin t, bt\rangle$ | $\kappa = a/(a^2+b^2)$ (constant) |
The radius of curvature is $\rho = 1/\kappa$, the radius of the osculating circle — the circle that best "kisses" the curve, matching position, tangent, and curvature.
Unit normal and the split of acceleration (§28.9)
The principal unit normal points toward the concave (inner) side:
$$\mathbf{N}(t) = \frac{\mathbf{T}'(t)}{|\mathbf{T}'(t)|}, \qquad \mathbf{T}\perp\mathbf{N} \ \text{(by the constant-length fact)}.$$
Acceleration always lies in the $\mathbf{T}$–$\mathbf{N}$ plane and splits into exactly two pieces — no third component:
$$\mathbf{a} = a_T\,\mathbf{T} + a_N\,\mathbf{N}, \qquad a_T = \frac{d}{dt}|\mathbf{v}| = \frac{\mathbf{v}\cdot\mathbf{a}}{|\mathbf{v}|}, \qquad a_N = \kappa|\mathbf{v}|^2 = \frac{|\mathbf{v}\times\mathbf{a}|}{|\mathbf{v}|}.$$
$a_T$ changes speed; $a_N$ changes direction. Because $a_N = \kappa v^2$ scales with the square of speed, doubling speed through a bend quadruples the sideways force.
Optional (Math Major Sidebar): a third unit vector, the binormal $\mathbf{B} = \mathbf{T}\times\mathbf{N}$, completes the Frenet frame, and torsion $\tau$ measures twisting out of the instantaneous plane. The Fundamental Theorem of Space Curves says $\kappa(s)$ and $\tau(s)$ determine a curve up to rigid motion. These belong to a differential-geometry course; the main text uses only $\mathbf{T}$, $\mathbf{N}$, and $\kappa$.
Common errors to avoid
- Speed vs. velocity: speed is the scalar $|\mathbf{r}'|$; velocity is the vector $\mathbf{r}'$.
- Arc length uses the magnitude $|\mathbf{r}'|$, not $\mathbf{r}'$ itself.
- Cross-product order matters: $(\mathbf{u}\times\mathbf{v})' = \mathbf{u}'\times\mathbf{v}+\mathbf{u}\times\mathbf{v}'$ — never swap factors (the dot-product rule has no such trap).
- Acceleration is not the rate of change of speed in general: $|\mathbf{a}|\ne \frac{d}{dt}|\mathbf{v}|$. Only the tangential part $a_T$ changes speed. (For the helix $a_T=0$ yet $|\mathbf{a}|=1$.)
- Arc length depends on the curve, not the parametrization: re-clocking the curve changes the speed and the time interval together, leaving $L$ fixed.
- The constant of integration for a vector integral is a vector $\mathbf{C}$, not a scalar.
Skills you should have
- Differentiate and integrate vector functions component-wise; apply the four product rules.
- Compute velocity, speed, and acceleration; recover position by integrating $\mathbf{a}(t)$ twice.
- Compute arc length and reparametrize by arc length when feasible.
- Find $\mathbf{T}$, $\mathbf{N}$, and curvature $\kappa$ (cross-product and plane-curve formulas).
- Split acceleration into $a_T\mathbf{T} + a_N\mathbf{N}$ and interpret each part physically.
Connections
- Chapter 25 (parametric curves): the plane ($z=0$) special case; this chapter is its 3D, vector-bundled successor.
- Chapter 27 (conics): orbital paths solving $\mathbf{r}'' = -\frac{GM}{|\mathbf{r}|^3}\mathbf{r}$ are conic sections.
- Chapter 14 (FTC / Net Change): integrating acceleration to position; arc length as accumulated speed.
- Chapter 34 (vector fields): the other vector object — a vector at every point; this chapter's curves return as flow lines and as paths of line integrals (Chapter 35).
- Chapters 29–30: the gradient $\nabla f$ will be perpendicular to level curves just as $\mathbf{N}$ is perpendicular to $\mathbf{T}$.
What's next
Chapter 29 changes the input side: functions of several variables $f(x,y)$ take two numbers in and return one out, so the graph becomes a surface rather than a curve. The component-wise simplicity of this chapter gives way to genuine multivariable structure — but the geometric instincts built here (tangents, normals, perpendicularity) carry straight forward.