Stewart Calculus · 8th Edition
Chapter 13: Vector Functions
Every key formula from Stewart Calculus Chapter 13, in one searchable page. Click a card to study it — worked examples included.
13.1–13.2 · VECTOR FUNCTIONS
r(t) = ⟨f(t), g(t), h(t)⟩
r′(t) = ⟨f′, g′, h′⟩ = tangent vector
Unit tangent T(t) = r′(t)/|r′(t)|
Limits & continuity
lim r(t) = ⟨lim f, lim g, lim h⟩
Continuous ⟺ every component continuous.
Product rules
d/dt[u·v] = u′·v + u·v′; d/dt[u×v] = u′×v + u×v′
Order matters in the cross-product rule!
Tangent line
At t₀: L(s) = r(t₀) + s·r′(t₀)
WORKED EXAMPLE
∫01⟨t, 2t², 1⟩dt = ⟨1/2, 2/3, 1⟩.
WORKED EXAMPLE
Helix r = ⟨cos t, sin t, t⟩:
r′ = ⟨−sin t, cos t, 1⟩, |r′| = √2;
L(0 → 2π) = 2π√2.
r′ = ⟨−sin t, cos t, 1⟩, |r′| = √2;
L(0 → 2π) = 2π√2.
13.3 · ARC LENGTH & CURVATURE
L = ∫ab |r′(t)| dt
κ = |T′(t)||r′(t)| = |r′ × r″||r′|³
Circle of radius R: κ = 1/R.
WORKED EXAMPLE
Helix: r′ × r″ = ⟨sin t, −cos t, 1⟩, |·| = √2;
κ = √2(√2)³ = 1/2.
κ = √2(√2)³ = 1/2.
Arc length function
s(t) = ∫t₀t|r′(u)| du; ds/dt = |r′(t)|
Torsion
τ = (r′×r″)·r′′′|r′×r″|²
Measures twisting out of the osculating plane.
WORKED EXAMPLE
r = ⟨etcos t, etsin t, 0⟩:
|r′| = √2 et ⇒ L(0→1) = √2(e−1).
|r′| = √2 et ⇒ L(0→1) = √2(e−1).
13.4 · MOTION IN SPACE
v(t) = r′, speed = |v|, a(t) = r″ = v′
a = aTT + aNN, aT = (v · a)/|v|, aN = |v × a|/|v|
aT changes speed; aN changes direction.
WORKED EXAMPLE
r(t) = ⟨t², t⟩: v = ⟨2t, 1⟩, a = ⟨2, 0⟩;
speed at t = 1: √5.
speed at t = 1: √5.
CURVATURE TOOLKIT
Plane curve y = f(x)
κ = |f″|(1 + (f′)²)3/2
Frenet frame
N = T′/|T′|, B = T × N
T: forward, N: turning toward, B: out of the plane.
WORKED EXAMPLE
y = x² at x = 0: f′ = 0, f″ = 2 ⇒ κ = 2.
Planes of the frame
Osculating: (r − r₀)·B = 0; normal: (r − r₀)·T = 0
Frenet–Serret
dT/ds = κN; dN/ds = −κT + τB; dB/ds = −τN
Radius of curvature
ρ = 1/κ — radius of the osculating circle
WORKED EXAMPLE
Helix: N(t) = ⟨−cos t, −sin t, 0⟩;
at t = 0: ⟨−1, 0, 0⟩ (points to the axis).
at t = 0: ⟨−1, 0, 0⟩ (points to the axis).
KEY FORMULAS
Reparametrization by arc length: s(t) = ∫t₀t |r′(u)| du
dr/ds = T (unit speed) · |dT/ds| = κ
dr/ds = T (unit speed) · |dT/ds| = κ
KEY NOTES
- Unit-speed curves make curvature effortless.
- N always points toward the inside of the turn.
Projectile motion
r(t) = ⟨v₀cosα·t, v₀sinα·t − ½gt²⟩
Range = v₀²sin2α/g (max at α = 45°).
Circular motion
Constant speed ⇒ a ⊥ v (centripetal), |a| = v²/R
WORKED EXAMPLE
v₀ = 20 m/s, α = 45°, g = 10:
range = 400·sin90°/10 = 40 m.
range = 400·sin90°/10 = 40 m.
Tangential: aT = v′ (speed change); normal: aN = κv² (direction change).
a = aTT + aNN.
WATCH OUT!
Mistakes that cost points
- Normalize AFTER differentiating for T
- Arc length integrates |r′|, not r′
- κ formula: cross on top, |r′|³ below
- aT uses dot with v; aN uses cross
- Speed is |v|; velocity is the vector v
- T = r′/|r′| — differentiate FIRST, then normalize
- κ has units 1/length: denominator is |r′|³
- aT can be negative (slowing down) — keep the sign
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