MATH 23
Lecture Notes
April, 2010
NAME:
B. Dodson
We have the definition of the surface area of a surface
S
when
S
is given as a
graph
z
=
g
(
x, y
) for (
x, y
) in a region
D
in the
xy
plane, using the element of
surface area
dS.
To match the treatment given in Chapter 16, we take
r
(
x, y
) =
< x, y, g
(
x, y
)
>,
and recall that
dS
was given as the product of the length of
the cross product
r
x
×
r
y
, with
dxdy
since these vectors (scaled by
dx
and
dy
,
respectively) span a parallelogram in the tangent plane whose area approximates
the area of the small piece of the surface over a rectangle
dxdy
.
Then the area
of the (parameterized) surface is
ZZ
D

r
x
×
r
y

dA.
See 16.6 formulas 8 and 9, and
example 1 below.
We use
dS
to define surface integrals
RR
S
f dS
. Next, we specify a unit normal
n
on
S
, and a vector field
F
=
F
(
x, y, z
), then the integral of
F
across
S
is given
by
ZZ
S
F dS
=
ZZ
S
F
·
n dS,
which is the integral that appears in the two main formulas, Stokes’ Theorem and
the Divergence formula. Two examples illustrate the integrals.
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 Spring '06
 YUKICH
 Math, Calculus, Surface, Stokes' theorem, Surface integral, surface area ds

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