Difficulty: Intermediate | Prerequisites: Chapter 12 (vectors, dot/cross product)
Chapter 13 brings calculus into the vector world. Where Chapter 12 gave you the static tools (points, vectors, planes), this chapter makes things move. You will learn to describe curves in 3D with vector-valued functions, take their derivatives to get velocity and acceleration, and measure how sharply a curve bends. If you are comfortable with single-variable derivatives and integrals and with the vector operations from Chapter 12, you are ready.
This chapter teaches you to describe curves in 3D using vector-valued functions r(t), differentiate them to get velocity and acceleration, and measure geometric properties like arc length and curvature. The TNB frame (tangent, normal, binormal) gives you a moving coordinate system attached to the curve, and the tangential/normal decomposition of acceleration tells you how much of the acceleration changes speed versus changes direction.
Vector-valued function
A function r(t) = ⟨x(t), y(t), z(t)⟩ that outputs a vector for each value of t. Think of it as a set of instructions that traces out a curve in 3D as t varies.
Unit tangent vector T(t)
T(t) = r'(t) / |r'(t)|. A unit-length vector pointing in the direction the curve is heading at time t.
Principal unit normal vector N(t)
N(t) = T'(t) / |T'(t)|. A unit vector perpendicular to T that points toward the centre of the curve's turning. Think of it as the direction the curve is bending into.
Binormal vector B(t)
B(t) = T(t) × N(t). Perpendicular to both T and N. Together, T, N, and B form the TNB frame, a moving right-handed coordinate system that rides along the curve.
Osculating plane
The plane spanned by T and N at a point on the curve. This is the plane that best approximates the curve's local behaviour, like holding a flat sheet against a bending wire.
Arc length
The total distance travelled along a curve from t = a to t = b. Computed as the integral of |r'(t)| dt.
Arc length function s(t)
s(t) = ∫₀ᵗ |r'(u)| du. The cumulative distance from a starting point. Its derivative ds/dt = |r'(t)| is the speed.
Curvature κ
A measure of how sharply a curve changes direction per unit distance. κ = |dT/ds| = |r'(t) × r''(t)| / |r'(t)|³. A straight line has curvature 0; a tight circle has high curvature.
Radius of curvature
1/κ. The radius of the circle that best fits the curve at a given point.
Tangential acceleration aT
The component of acceleration along T. It measures how fast the speed is changing: aT = d|v|/dt.
Normal acceleration aN
The component of acceleration along N. It measures how fast the direction is changing: aN = κ|v|² = |r' × r''| / |r'|.
To visualise a 3D curve given parametrically, eliminate the parameter from the equations to see the shape
To parameterise a circle or ellipse x²/a² + y²/b² = 1, use x = a cos(t), y = b sin(t)
Intersections:
Curve and surface: substitute the curve's parametric equations into the surface equation and solve for t
Two curves: use different parameters (t and s), set the coordinates equal, and solve the resulting system
Two objects collide only if the intersection happens at the same parameter value (same time)
Two surfaces: to parameterise their intersection curve, let one variable be t and solve for the others
For r(t) = ⟨x(t), y(t), z(t)⟩, the derivative is r'(t) = ⟨x'(t), y'(t), z'(t)⟩
This is a tangent vector to the curve at time t
Unit tangent vector: T(t) = r'(t) / |r'(t)|
Antiderivative: ∫r(t) dt = ⟨∫x(t) dt, ∫y(t) dt, ∫z(t) dt⟩
Integrate each component separately, just as in single-variable calculus
Distance from t = a to t = b: ∫ₐᵇ |r'(t)| dt = ∫ₐᵇ √(x'(t)² + y'(t)² + z'(t)²) dt
If the curve is traversed exactly once over the interval, this integral gives the arc length
Arc length function: s(t) = ∫₀ᵗ |r'(u)| du
Speed = ds/dt = |r'(t)|
T and T' are always orthogonal: T · T' = 0
Principal unit normal: N(t) = T'(t) / |T'(t)|
Curvature measures how fast the curve changes direction per unit distance:
κ = |dT/ds| = |T'(t)| / |r'(t)| = |r'(t) × r''(t)| / |r'(t)|³
Radius of curvature = 1/κ (the radius of the best-fit circle)
For a 2D function y = f(x): κ = |f''(x)| / (1 + (f'(x))²)^(3/2)
Binormal: B(t) = T(t) × N(t), orthogonal to both T and N
T, N, r'(t), and r''(t) are all parallel to the osculating plane; B is orthogonal to it
r(t) = position, r'(t) = velocity, |r'(t)| = speed, r''(t) = acceleration
Newton's second law in vector form: a = (1/m) F
Decomposing acceleration into tangential and normal components:
Tangential acceleration: aT = comp_T(a) = a · T = (r' · r'') / |r'|
This equals v' (the derivative of the speed), so it tells you whether the object is speeding up or slowing down
Normal acceleration: aN = comp_N(a) = a · N = |r' × r''| / |r'|
This equals κ v² (curvature times speed squared), so it measures the centripetal pull toward the centre of the curve
The full acceleration is a = aT T + aN N
Topic | Formula |
|---|---|
Derivative of r(t) | r'(t) = ⟨x'(t), y'(t), z'(t)⟩ |
Unit tangent | T(t) = r'(t) / |
Principal unit normal | N(t) = T'(t) / |
Binormal | B(t) = T(t) × N(t) |
Arc length | ∫ₐᵇ |
Speed | ds/dt = |
Curvature (general) | κ = |
Curvature (2D, y = f(x)) | κ = |
Radius of curvature | 1/κ |
Tangential acceleration | aT = (r' · r'') / |
Normal acceleration | aN = |
Circle/ellipse param. | x = a cos t, y = b sin t |
Students often confuse speed with velocity. Velocity r'(t) is a vector (it has direction). Speed |r'(t)| is a scalar (just a number). You can have zero velocity but never negative speed.
Students sometimes think curvature measures how fast a particle is moving. It does not. Curvature measures how fast the direction is changing per unit distance. A car driving slowly around a tight bend and a car driving fast around the same bend experience the same curvature; the fast car has higher normal acceleration because aN = κv².
Students forget the cross-product formula for curvature (κ = |r' × r''| / |r'|³) and try to compute T, then T', then |T'|/|r'| instead. Both work, but the cross-product route is usually faster on exams.
When finding where two curves intersect, students often use the same parameter for both curves. You must use different parameters (say t and s) unless the question specifically asks when the objects collide (same parameter value = same time).
⚠️ The curvature formula κ = |r' × r''| / |r'|³ appears on nearly every exam. Memorise it and practise computing cross products quickly.
⚠️ Arc length integrals are common. Set up ∫ |r'(t)| dt carefully, and simplify the expression under the square root before integrating.
⚠️ Tangential vs. normal acceleration: know that aT = v' (rate of change of speed) and aN = κv². Exam questions often ask you to decompose a given acceleration vector.
⚠️ Distinguishing intersection from collision: two curves that meet at a point are intersecting. They collide only if they are at that point at the same parameter value.
True or false: the unit tangent vector T always has magnitude 1.
True. That is what "unit" means.
Fill in the blank: ds/dt = ______.
|r'(t)| (the speed).
True or false: curvature can be negative.
False. Curvature κ is always ≥ 0.
Fill in the blank: the binormal vector B is defined as ______ × ______.
T × N.
True or false: if two curves intersect at a point, the objects on those curves must collide.
False. They collide only if they reach the intersection point at the same parameter value.
Q: Given r(t) = ⟨cos t, sin t, t⟩, find the speed at any time t.
A: r'(t) = ⟨-sin t, cos t, 1⟩. Speed = |r'(t)| = √(sin²t + cos²t + 1) = √2. The speed is constant.
Q: Find the arc length of r(t) = ⟨cos t, sin t, t⟩ from t = 0 to t = 2π.
A: Since the speed is √2 (constant), the arc length = √2 × 2π = 2π√2.
Q: For the curve r(t) = ⟨t, t², 0⟩, compute the curvature at t = 0.
A: r'(t) = ⟨1, 2t, 0⟩, r''(t) = ⟨0, 2, 0⟩. At t = 0: r' = ⟨1, 0, 0⟩, r'' = ⟨0, 2, 0⟩. r' × r'' = ⟨0, 0, 2⟩, |r' × r''| = 2, |r'| = 1. κ = 2/1 = 2.
Q: An object moves along a curve with speed 3 and curvature 4 at a certain point. What is the normal acceleration?
A: aN = κv² = 4 × 9 = 36.
The derivative and integral of vector-valued functions from this chapter are the conceptual bridge between single-variable calculus (MATH 124/125) and the multivariable calculus of Chapters 14 and 15. Curvature returns when you study tangent planes (Chapter 14), because the shape of a surface near a point is described by curvatures in different directions.
The TNB frame and acceleration decomposition connect directly to physics: orbital mechanics, roller-coaster design, and any problem where you need to separate "how fast" from "which way."
Related Terms / Search Tags
vector-valued function, parametric curve, 3D curve, unit tangent vector, principal unit normal, binormal, TNB frame, Frenet frame, osculating plane, arc length, arc length function, speed, curvature, radius of curvature, tangential acceleration, normal acceleration, centripetal acceleration, MATH 126, Calculus III, UW, multivariable calculus