Stewart Calculus · 8th Edition
Chapter 16: Vector Calculus
Every key formula from Stewart Calculus Chapter 16, in one searchable page. Click a card to study it — worked examples included.
16.1–16.3 · FIELDS & LINE INTEGRALS
F = ⟨P, Q, R⟩; gradient field F = ∇f is conservative
∫C F · dr = ∫ab F(r(t)) · r′(t) dt
FTC: ∫C ∇f · dr = f(r(b)) − f(r(a)) — path independent!
WORKED EXAMPLE
F = ⟨y, x⟩ = ∇(xy), line (0,0) → (1,1):
∫01 2t dt = 1 = f(1,1) − f(0,0) ✓
∫01 2t dt = 1 = f(1,1) − f(0,0) ✓
Conservative test (2D)
Py = Qx on simply-connected D ⇒ conservative
Find f: fx = P, then match fy = Q.
WORKED EXAMPLE
F = ⟨3x² + y, x + 2y⟩: Py = 1 = Qx ✓;
f = x³ + xy + y² ⇒ ∫C F·dr,
(0,0) → (1,1) = 3 − 0 = 3.
f = x³ + xy + y² ⇒ ∫C F·dr,
(0,0) → (1,1) = 3 − 0 = 3.
16.4 · GREEN’S THEOREM
∮C P dx + Q dy = ∫∫D (Qx − Py) dA
C positive (counterclockwise). Area: A = ½∮C(x dy − y dx).
WORKED EXAMPLE
Unit disk via Green: ½∫02π 1 dt = π.
Flux form of Green
∮C P dy − Q dx = ∫∫D (Px + Qy) dA
Circulation: Qx − Py; flux: Px + Qy.
WORKED EXAMPLE
∮C x dy − y dx, C = unit circle:
∫∫D (1 + 1) dA = 2π.
∫∫D (1 + 1) dA = 2π.
16.5 · CURL & DIVERGENCE
curl F = ∇ × F · div F = ∇ · F = Px + Qy + Rz
curl(∇f) = 0 · div(curl F) = 0.
Curl ≈ rotation, div ≈ expansion.
Curl ≈ rotation, div ≈ expansion.
WORKED EXAMPLE
div⟨x², y², z²⟩ = 2x + 2y + 2z.
WORKED EXAMPLE
div⟨xy, yz, zx⟩ = y + z + x;
at (1,2,3): 6.
at (1,2,3): 6.
Curl as a determinant
curl F = |i j k; ∂/∂x ∂/∂y ∂/∂z; P Q R|
WORKED EXAMPLE
curl⟨yz, xz, xy⟩ = ⟨x−x, y−y, z−z⟩ = 0;
indeed ⟨yz, xz, xy⟩ = ∇(xyz) ✓
indeed ⟨yz, xz, xy⟩ = ∇(xyz) ✓
16.6–16.7 · SURFACES
∫∫S f dS = ∫∫D f(r(u,v)) |ru × rv| dA
Flux: ∫∫S F · dS = ∫∫S F · n dS
|ru × rv| is the area stretch factor.
Graph z = g(x,y)
dS = √(1 + gx² + gy²) dA, S = ∫∫D |ru × rv| dA
Orientation
Upward normal: n = ⟨−gx, −gy, 1⟩/|·|.
WORKED EXAMPLE
Area of z = x² + y² below z = 4:
2π∫02 r√(1+4r²) dr = π(17√17 − 1)/6.
2π∫02 r√(1+4r²) dr = π(17√17 − 1)/6.
16.8–16.9 · STOKES & DIVERGENCE
Stokes’ Theorem
∮C F · dr = ∫∫S (curl F) · dS
Right-hand rule matches C to S.
Divergence Theorem
∫∫S F · dS = ∫∫∫E div F dV
S closed, outward orientation.
KEY NOTES
- Both trade a hard integral for an easier one.
Stokes: ∮C F·dr = ∫∫S (curl F)·n dS.
Divergence: ∫∫S F·n dS = ∫∫∫E div F dV.
Flat special case
S flat in the xy-plane, n = k ⇒ Stokes IS Green.
The big ladder
FTC ∫ab F′ = F(b) − F(a) · Green/Stokes: curve ⇒ region · Divergence: closed surface ⇒ volume. The boundary always drops one dimension.
WORKED EXAMPLE
Flux of ⟨x, y, z⟩ out of the sphere r = 2:
div F = 3; V = 32π/3 ⇒ flux = 32π.
div F = 3; V = 32π/3 ⇒ flux = 32π.
WATCH OUT!
Mistakes that cost points
- Green: C must be CLOSED & counterclockwise
- Stokes: curl matches the surface’s orientation
- Conservative test (Qx = Py) needs simply-connected D
- Flux (through) vs circulation (around) — different integrals!
- Divergence theorem: S must be CLOSED
- ∮ means CLOSED — ∫ does not
- F(r(t)) · r′(t): plug in FIRST, then dot
- Open surface? Close it, use the theorem, subtract the cap
- Stokes: C‘s direction must match S by the right-hand rule
No formulas match your search. Try a different keyword.