Lecture 26 (Nov. 19)
Parametric Surfaces (reading 16.6)
Recall that we parameterize curves (one-dimensional objects) using vector functions r(t) of a single variable (parameter) t.
Similarly, we can parameterize surfaces (two-dimensional objects) using vector
functions
r(u, v) = hx(u, v), y(u, v), z(u, v)i
of two variables (parameters), u and v.
Definition: the set
S = {(x(u, v), y(u, v), z(u, v)) 2 R3 | (u, v) 2 D ⇢ R2 }
is called a parametric surface.
Example: parameterize the ellipsoid x2 /a2 + y 2 /b2 + z 2 /c2 = 1.
Example: identify the surface parameterized by r(x, ✓) = hx, cos(✓), sin(✓)i, 0  ✓ 
2⇡, 0  x  1.
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Example: parameterize a surface of revolution:
Tangent Planes
Di↵erentiating r(u, v) with respect to u and v yields vectors ru and rv which are
tangent to the surface S:
So ru ⇥ rv is normal to the tangent plane.
Definition: we say a parametric surface S is smooth if ru ⇥ rv is never 0.
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Example: find the tangent plane to the ellipsoid given by
r(u, v) = ha sin(u) cos(v), b sin(u) sin(v), c cos(u)i,
at (a, 0, 0).
Surface Area
Let’s find the area of a parametric surface:
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Definition: Let S be a smooth parametric surface parameterized by r(u, v), (u, v) 2 D
(with S covered just once as (u, v) range over D). Then the surface area of S is
A(S) :=
ZZ
D
|ru ⇥ rv |dA.
Example: find the surface area of the ellipsoid x2 /a2 + y 2 /a2 + z 2 /c2 = 1.
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Special case (Section 15.6):
S = graph of a function f (x, y):
S = {(x, y, z) | (x, y) 2 D, z = f (x, y)}.
So
A(S) =
ZZ
(1 + [fx (x, y)]2 + [fy (x, y)]2 )1/2 dA
D
(and we recover the formula from Section 15.6).
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