16.7 Surface Integrals
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Transcript 16.7 Surface Integrals
Chapter 16 – Vector Calculus
16.7 Surface Integrals
Objectives:
Understand integration of
different types of surfaces
Dr. Erickson
16.7 Surface Integrals
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Surface Integrals
The relationship between surface integrals
and surface area is much the same as the
relationship between line integrals and arc
length.
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Surface Integrals
Suppose a surface S has a vector equation
r(u, v) = x(u, v) i + y(u, v) j + z(u, v) k
(u, v) D
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Surface Integrals
In our discussion of surface area in
Section 16.6, we made the approximation
∆Sij ≈ |ru x rv| ∆u ∆v
where:
x
y
z
ru
i
j k
u
u
u
x y
z
rv i j k
v
v
v
are the tangent vectors at a corner
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Surface Integrals - Equation 2
If the components are continuous and ru and rv
are nonzero and nonparallel in the interior of D,
it can be shown that:
f ( x, y, z ) dS f (r(u, v)) | r
u
S
rv | dA
D
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Surface Integrals
Formula 2 allows us to compute a surface integral by
converting it into a double integral over the parameter
domain D.
◦ When using this formula, remember that
f(r(u, v) is evaluated by writing
x = x(u, v), y = y(u, v), z = z(u, v)
in the formula for f(x, y, z)
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Example 1
Evaluate the surface integral.
1 x 2 y 2 dS ,
S
S is the helicoid with vector equation
r (u, v) u cos v i u sin v j v k , 0 u 1, 0 v .
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Graphs
Any surface S with equation z = g(x, y)
can be regarded as a parametric surface
with parametric equations
x=x
y=y
z = g(x, y)
◦ So, we have:
g
rx i k
x
g
ry j k
y
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Graphs
Therefore, Equation 2 becomes:
f ( x, y, z) dS
S
D
2
z z
f ( x, y, g ( x, y )) 1 dA
x y
2
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Graphs
Similar formulas apply when it is more convenient to
project S onto the yz-plane or xy-plane.
For instance, if S is a surface with equation y = h(x, z)
and D is its projection on the xz-plane, then
f ( x, y, z) dS
S
D
y y
f ( x, h( x, z ), z ) 1 dA
x z
2
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10
Example 2 – pg. 1145 # 9
Evaluate the surface integral.
2
x
yz dS ,
S
S is the part of the plane z 1 2 x 3 y
that lies above the rectangle 0,3 0, 2.
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Oriented Surface
If it is possible to choose a unit normal vector n at every
such point (x, y, z) so that n varies continuously over S,
then
◦ S is called an oriented surface.
◦ The given choice of n provides S with an orientation.
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Possible Orientations
There are two possible orientations for
any orientable surface.
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Positive Orientation
Observe that n points in the same direction as the
position vector—that is, outward from the sphere.
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Negative Orientation
The opposite (inward) orientation would have been
obtained if we had reversed the order of the parameters
because rθ x rΦ = –rΦ x rθ
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Closed Surfaces
For a closed surface—a surface that is the boundary of a
solid region E—the convention is that:
◦ The positive orientation is the one for which
the normal vectors point outward from E.
◦ Inward-pointing normals give the negative orientation.
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Flux Integral (Def. 8)
If F is a continuous vector field defined on an oriented
surface S with unit normal vector n, then the surface
integral of F over S is:
F dS F n dS
S
S
◦ This integral is also called the flux of F across S.
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Flux Integral
In words, Definition 8 says that:
◦ The surface integral of a vector field over S is equal to
the surface integral of its normal component over S (as
previously defined).
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Flux Integral
If S is given by a vector function r(u, v), then n is
ru rv
n
ru rv
We can rewrite Definition 8 as equation 9:
F
d
S
F
(
r
r
)
dA
u
v
S
D
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Example 3
Evaluate the surface integralS F dS for the given vector
field F and the oriented surface S. In other words, find
the flux of F across S. For closed surfaces, use the
positive (outward) orientation.
F( x, y, z ) xz i x j y k ,
S is the hemisphere x 2 y 2 z 2 25, y 0 oriented
in the direction of the positive y-axis.
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Vector Fields
In the case of a surface S given by a graph
z = g(x, y), we can think of x and y as parameters and
write:
g g
F (rx ry ) ( P i Q j R k ) i
jk
y
x
From this, formula 9 becomes formula 10:
g
g
F
d
S
P
Q
R
dA
S
D x y
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Vector Fields
g
g
F
d
S
P
Q
R
dA
S
D x y
◦ This formula assumes the upward orientation of S.
◦ For a downward orientation, we multiply by –1.
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Example 4
Evaluate the surface integralS F dS for the given vector
field F and the oriented surface S. In other words, find
the flux of F across S. For closed surfaces, use the
positive (outward) orientation.
F ( x, y , z ) x i y j z 4 k ,
S is the part of the cone z x 2 y 2 beneath
the plane z 1 with downward directions.
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Other Examples
In groups, please work on the following
problems on page 1145:
#’s 12, 14, and 28.
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Example 5 – pg. 1145 # 12
Evaluate the surface integral.
y dS ,
S
3
2 32
2
S is the surface z x y
3
0 x 1, 0 y 1.
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Example 6 – pg. 1145 # 14
Evaluate the surface integral.
z dS ,
S
S is the surface x y 2 z 2 ,
0 y 1, 0 z 1.
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Example 7 – pg. 1145 # 28
Evaluate the surface integralS F dS for the given vector
field F and the oriented surface S. In other words, find
the flux of F across S. For closed surfaces, use the
positive (outward) orientation.
F ( x, y, z ) xy i 4 x 2 j yz k ,
S is the surface z xe y , 0 x 1,
0 y 1, with upward orientation.
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