Parametric & Space Curves (Sections 13.1–13.3)

Parametric Equations

Equations that describe x, y, z in terms of a parameter (usually t).

Vector-Valued Function

r(t) = ⟨x(t), y(t), z(t)⟩, describes motion in space.

What Does r(t) Represent?

Position of a particle at parameter value t.

Component Functions of r(t)

x(t), y(t), z(t) give coordinates in space.

Velocity Vector

v(t) = r′(t)

Acceleration Vector

a(t) = r″(t)

Speed

Magnitude of velocity: ‖r′(t)‖

Difference Between Speed and Velocity

Speed is scalar; velocity includes direction.

Direction of Motion

Given by the velocity vector.

Tangent Vector to a Curve

r′(t)

Unit Tangent Vector

T(t) = r′(t) / ‖r′(t)‖

When is the Particle at Rest?

When r′(t) = 0

Arc Length of a Curve

Total distance traveled along the curve.

Arc Length Formula

∫ₐᵇ ‖r′(t)‖ dt

When to Use Arc Length

When asked for total distance along a curve.

Parameterization

Any way of describing a curve using a parameter.

Reparameterization

Describing the same curve with a different parameter.

Does Reparameterization Change the Curve?

No, only how fast it’s traced.

Orientation of a Curve

Direction in which the curve is traced as t increases.

Line as a Vector-Valued Function

r(t) = r₀ + tv

Circle in the xy-Plane

⟨a cos t, a sin t, 0⟩

Helix

⟨a cos t, a sin t, bt⟩

Cycloid (Conceptual)

Path traced by a rolling circle.

Scalar Line Integral (Concept)

Integrating a function along a curve.

Why r′(t) Appears in Line Integrals

It accounts for arc length.

What Determines the Shape of the Curve?

The component functions x(t), y(t), z(t).

What Determines Speed Along the Curve?

Magnitude of r′(t).

Same Curve, Different Speed Means

Same path, different parameterization.