Calculus 2nd Edition Rogawski

SECTION 17.4 Parametrized Surfaces and Surface Integrals 991
17.4 SUMMARY
• A parametrized surface is a surface S whose points are described in the form
G(u, v) = (x(u, v), y(u, v), z(u, v))
where the parameters u and v vary in a domain D in the uv-plane.
• Tangent and normal vectors:
T u = ∂G 〈 ∂x
∂u = ∂u , ∂y
∂u , ∂z 〉
, T v = ∂G 〈 ∂x
∂u
∂v = ∂v , ∂y
∂v , ∂z 〉
∂v
n = n(u, v) = T u × T v
The parametrization is regular at (u, v) if n(u, v) ̸= 0.
• The quantity ∥n∥ is an “area distortion factor.” If D is a small region in the uv-plane
and S = G(D), then
where (u 0 ,v 0 ) is any sample point in D.
• Formulas:
∫∫
Area(S) =
D
∫∫
∫∫
f (x, y, z) dS =
S
D
• Some standard parametrizations:
Area(S) ≈∥n(u 0 ,v 0 )∥Area(D)
– Cylinder of radius R (z-axis as central axis):
G(θ,z)= (R cos θ,Rsin θ,z)
∥n(u, v)∥ dudv
f (G(u, v)) ∥n(u, v)∥ dudv
Outward normal: n = T θ × T z = R ⟨cos θ, sin θ, 0⟩
dS = ∥n∥ dθ dz = Rdθ dz
– Sphere of radius R, centered at the origin:
G(θ, φ) = (R cos θ sin φ,Rsin θ sin φ,Rcos φ)
Unit radial vector: e r = ⟨cos θ sin φ, sin θ sin φ, cos φ⟩
Outward normal:
n = T φ × T θ = (R 2 sin φ) e r
– Graph of z = g(x, y):
dS = ∥n∥ dφ dθ = R 2 sin φ dφ dθ
G(x, y) = (x, y, g(x, y))
n = T x × T y = 〈 −g x , −g y , 1 〉
dS = ∥n∥ dx dy =
√
1 + gx 2 + g2 y dx dy