Math 2415 – Calculus III
Section 16.6 Parametric Surfaces and Their Areas
”I wish I had paid more attention in calculus class when we were studying parametric surfaces. It sure would have
helped me today.”
• We can describe a surface by a vector function~r(u, v) = x(u, v)î + y(u, v) jˆ + z(u, v)k̂
• The set of all points (x, y, z) in R3 such that x = x(u, v), y = y(u, v), z = z(u, v) and (u, v) varies throughout D is
called a parametric surface S.
Ex: Identify and sketch the surface with vector equation~r(u, v) = 2 cos uî + v jˆ + 2 sin uk̂.
• We can get different surfaces if we restrict u and v.
π
Ex: In previous example, let 0 ≤ u ≤ and 0 ≤ v ≤ 3. This gives
2
• There are two useful families of curves that lie on S, one with u constant, the other with v constant.
Math 2415
Section 16.6 Continued
• If u = u0 is constant, ~r(u0 , v) is a vector function of a single parameter v and defines curve C1 lying on S.
Likewise, if v = v0 . These are called grid curves. (Computers graph like this).
• It may be difficult to find a vector function to represent a surface.
Ex: Find a vector function that represents the plane passing through the point P0 with position vector ~r0 and
contains two nonparallel vectors ~a and ~b.
Ex: Find a parametric representation of the sphere x2 + y2 + z2 = a2 .
Ex: Find a parametric representation for the cylinder x2 + y2 = 4, 0 ≤ z ≤ 1.
2
Math 2415
Section 16.6 Continued
• In general, a surface given as a graph of a function x and y (z = f (x, y)) can be regarded as a parametric surface
with equations x = x, y = y, z = f (x, y). Parameterizations are not unique.
p
Ex: Find a parametric representation for z = 2 x2 + y2 , i.e. the top half of the cone z2 = 4x2 + 4y2 .
Surfaces of Revolution Can be represented parametrically.
• Consider the surface S obtained by rotating y = f (x), a ≤ x ≤ b where f (x) ≥ 0 about the x−axis. Let θ be the
angle of rotation.
Ex: Find parametric equations for the surface generated by rotating the curve y = sin x, 0 ≤ x ≤ 2π about the
x−axis. Graph the surface of revolution.
3
Math 2415
Section 16.6 Continued
Tangent Planes Let~r(u, v) = x(u, v)î + y(u, v) jˆ + z(u, v)k̂ be a vector function at point P0 with position vector
~r(u0 , v0 ). Keep u constant by letting u = u0 : ~r(u0 , v) is a vector function of a single parameter v and defines a
grid curve C1 lying on S.
• Tangent vector to C1 at P0 : ~rv =
∂x
∂y
∂z
(u0 , v0 )î + (u0 , v0 ) jˆ + (u0 , v0 )k̂
∂v
∂v
∂v
• Let v = v0 . We get a grid curve C2 given by~r(u, v) that lies on S and its tangent vector at P0
∂x
∂y
∂z
is ~ru = (u0 , v0 )î + (u0 , v0 ) jˆ + (u0 , v0 )k̂.
∂u
∂u
∂u
• If a surface is smooth, the tangent plane exists at all points. Since ~ru × ~rv is the normal vector, if ~ru ×~rv = ~0,
there can be no tangent plane at that point. Thus the surface is not smooth at that point. (if ~ru × ~rv 6= ~0, the
surface is smooth.)
Then write the equation using the normal vector and a point.
Ex: Find the tangent plane to the surface with parametric equations x = u2 , y = v2 , z = u + 2v at the point
(1, 1, 3).
4
Math 2415
Section 16.6 Continued
s
• Surface area: A(S) =
Z Z
1+
D
∂z
∂x
2
+
∂z
∂y

2
Z b
dA. Very similar to arc length L =
Ex: Find the area of the part of the paraboloid z = x2 + y2 that lies under the plane z = 9.
5
s
1+
a
dy
dx
2

dx