chapter 16 – vector calculus 16.7 surface integrals 1 objectives: understand integration of...
TRANSCRIPT
![Page 1: Chapter 16 – Vector Calculus 16.7 Surface Integrals 1 Objectives: Understand integration of different types of surfaces Dr. Erickson](https://reader031.vdocument.in/reader031/viewer/2022020921/56649cc35503460f9498c96f/html5/thumbnails/1.jpg)
1
Chapter 16 – Vector Calculus16.7 Surface Integrals
16.7 Surface Integrals
Objectives: Understand integration of
different types of surfaces
Dr. Erickson
![Page 2: Chapter 16 – Vector Calculus 16.7 Surface Integrals 1 Objectives: Understand integration of different types of surfaces Dr. Erickson](https://reader031.vdocument.in/reader031/viewer/2022020921/56649cc35503460f9498c96f/html5/thumbnails/2.jpg)
16.7 Surface Integrals 2
Surface IntegralsThe relationship between surface integrals
and surface area is much the same as the relationship between line integrals and arc length.
Dr. Erickson
![Page 3: Chapter 16 – Vector Calculus 16.7 Surface Integrals 1 Objectives: Understand integration of different types of surfaces Dr. Erickson](https://reader031.vdocument.in/reader031/viewer/2022020921/56649cc35503460f9498c96f/html5/thumbnails/3.jpg)
16.7 Surface Integrals 3
Surface IntegralsSuppose 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
Dr. Erickson
![Page 4: Chapter 16 – Vector Calculus 16.7 Surface Integrals 1 Objectives: Understand integration of different types of surfaces Dr. Erickson](https://reader031.vdocument.in/reader031/viewer/2022020921/56649cc35503460f9498c96f/html5/thumbnails/4.jpg)
16.7 Surface Integrals 4
Surface IntegralsIn our discussion of surface area in
Section 16.6, we made the approximation
∆Sij ≈ |ru x rv| ∆u ∆v
where:
are the tangent vectors at a corner
u v
x y z x y z
u u u v v v
r i j k r i j k
Dr. Erickson
![Page 5: Chapter 16 – Vector Calculus 16.7 Surface Integrals 1 Objectives: Understand integration of different types of surfaces Dr. Erickson](https://reader031.vdocument.in/reader031/viewer/2022020921/56649cc35503460f9498c96f/html5/thumbnails/5.jpg)
16.7 Surface Integrals 5
Surface Integrals - Equation 2If the components are continuous and ru and rv
are nonzero and nonparallel in the interior of D, it can be shown that:
( , , ) ( ( , )) | |u v
S D
f x y z dS f u v dA r r r
Dr. Erickson
![Page 6: Chapter 16 – Vector Calculus 16.7 Surface Integrals 1 Objectives: Understand integration of different types of surfaces Dr. Erickson](https://reader031.vdocument.in/reader031/viewer/2022020921/56649cc35503460f9498c96f/html5/thumbnails/6.jpg)
16.7 Surface Integrals 6
Surface IntegralsFormula 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)
Dr. Erickson
![Page 7: Chapter 16 – Vector Calculus 16.7 Surface Integrals 1 Objectives: Understand integration of different types of surfaces Dr. Erickson](https://reader031.vdocument.in/reader031/viewer/2022020921/56649cc35503460f9498c96f/html5/thumbnails/7.jpg)
16.7 Surface Integrals 7
Example 1 Evaluate the surface integral.
2 21 ,
is the helicoid with vector equation
( , ) cos sin , 0 1, 0 .
S
x y dS
S
u v u v u v v u v
r i j k
Dr. Erickson
![Page 8: Chapter 16 – Vector Calculus 16.7 Surface Integrals 1 Objectives: Understand integration of different types of surfaces Dr. Erickson](https://reader031.vdocument.in/reader031/viewer/2022020921/56649cc35503460f9498c96f/html5/thumbnails/8.jpg)
16.7 Surface Integrals 8
GraphsAny 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:
x y
g g
x y
r i k r j k
Dr. Erickson
![Page 9: Chapter 16 – Vector Calculus 16.7 Surface Integrals 1 Objectives: Understand integration of different types of surfaces Dr. Erickson](https://reader031.vdocument.in/reader031/viewer/2022020921/56649cc35503460f9498c96f/html5/thumbnails/9.jpg)
16.7 Surface Integrals 9
GraphsTherefore, Equation 2 becomes:
22
( , , ) ( , , ( , )) 1S D
z zf x y z dS f x y g x y dA
x y
Dr. Erickson
![Page 10: Chapter 16 – Vector Calculus 16.7 Surface Integrals 1 Objectives: Understand integration of different types of surfaces Dr. Erickson](https://reader031.vdocument.in/reader031/viewer/2022020921/56649cc35503460f9498c96f/html5/thumbnails/10.jpg)
16.7 Surface Integrals 10
GraphsSimilar 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
2 2
( , , ) ( , ( , ), ) 1S D
y yf x y z dS f x h x z z dA
x z
Dr. Erickson
![Page 11: Chapter 16 – Vector Calculus 16.7 Surface Integrals 1 Objectives: Understand integration of different types of surfaces Dr. Erickson](https://reader031.vdocument.in/reader031/viewer/2022020921/56649cc35503460f9498c96f/html5/thumbnails/11.jpg)
16.7 Surface Integrals 11
Example 2 – pg. 1145 # 9Evaluate the surface integral.
2 ,
is the part of the plane 1 2 3
that lies above the rectangle 0,3 0,2 .
S
x yz dS
S z x y
Dr. Erickson
![Page 12: Chapter 16 – Vector Calculus 16.7 Surface Integrals 1 Objectives: Understand integration of different types of surfaces Dr. Erickson](https://reader031.vdocument.in/reader031/viewer/2022020921/56649cc35503460f9498c96f/html5/thumbnails/12.jpg)
16.7 Surface Integrals 12
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.
Dr. Erickson
![Page 13: Chapter 16 – Vector Calculus 16.7 Surface Integrals 1 Objectives: Understand integration of different types of surfaces Dr. Erickson](https://reader031.vdocument.in/reader031/viewer/2022020921/56649cc35503460f9498c96f/html5/thumbnails/13.jpg)
16.7 Surface Integrals 13
Possible OrientationsThere are two possible orientations for
any orientable surface.
Dr. Erickson
![Page 14: Chapter 16 – Vector Calculus 16.7 Surface Integrals 1 Objectives: Understand integration of different types of surfaces Dr. Erickson](https://reader031.vdocument.in/reader031/viewer/2022020921/56649cc35503460f9498c96f/html5/thumbnails/14.jpg)
16.7 Surface Integrals 14
Positive OrientationObserve that n points in the same direction as the
position vector—that is, outward from the sphere.
Dr. Erickson
![Page 15: Chapter 16 – Vector Calculus 16.7 Surface Integrals 1 Objectives: Understand integration of different types of surfaces Dr. Erickson](https://reader031.vdocument.in/reader031/viewer/2022020921/56649cc35503460f9498c96f/html5/thumbnails/15.jpg)
16.7 Surface Integrals 15
Negative OrientationThe opposite (inward) orientation would have been
obtained if we had reversed the order of the parameters because rθ x rΦ = –rΦ x rθ
Dr. Erickson
![Page 16: Chapter 16 – Vector Calculus 16.7 Surface Integrals 1 Objectives: Understand integration of different types of surfaces Dr. Erickson](https://reader031.vdocument.in/reader031/viewer/2022020921/56649cc35503460f9498c96f/html5/thumbnails/16.jpg)
16.7 Surface Integrals 16
Closed SurfacesFor 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.
Dr. Erickson
![Page 17: Chapter 16 – Vector Calculus 16.7 Surface Integrals 1 Objectives: Understand integration of different types of surfaces Dr. Erickson](https://reader031.vdocument.in/reader031/viewer/2022020921/56649cc35503460f9498c96f/html5/thumbnails/17.jpg)
16.7 Surface Integrals 17
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:
◦ This integral is also called the flux of F across S.
S S
d dS F S F n
Dr. Erickson
![Page 18: Chapter 16 – Vector Calculus 16.7 Surface Integrals 1 Objectives: Understand integration of different types of surfaces Dr. Erickson](https://reader031.vdocument.in/reader031/viewer/2022020921/56649cc35503460f9498c96f/html5/thumbnails/18.jpg)
16.7 Surface Integrals 18
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).
Dr. Erickson
![Page 19: Chapter 16 – Vector Calculus 16.7 Surface Integrals 1 Objectives: Understand integration of different types of surfaces Dr. Erickson](https://reader031.vdocument.in/reader031/viewer/2022020921/56649cc35503460f9498c96f/html5/thumbnails/19.jpg)
16.7 Surface Integrals 19
Flux Integral If S is given by a vector function r(u, v), then n is
We can rewrite Definition 8 as equation 9:
u v
u v
r r
nr r
( )u v
S D
d dA F S F r r
Dr. Erickson
![Page 20: Chapter 16 – Vector Calculus 16.7 Surface Integrals 1 Objectives: Understand integration of different types of surfaces Dr. Erickson](https://reader031.vdocument.in/reader031/viewer/2022020921/56649cc35503460f9498c96f/html5/thumbnails/20.jpg)
16.7 Surface Integrals 20
Example 3 Evaluate the surface integral 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.
SdF S
2 2 2
( , , ) ,
is the hemisphere 25, 0 oriented
in the direction of the positive -axis.
x y z xz x y
S x y z y
y
F i j k
Dr. Erickson
![Page 21: Chapter 16 – Vector Calculus 16.7 Surface Integrals 1 Objectives: Understand integration of different types of surfaces Dr. Erickson](https://reader031.vdocument.in/reader031/viewer/2022020921/56649cc35503460f9498c96f/html5/thumbnails/21.jpg)
16.7 Surface Integrals 21
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:
From this, formula 9 becomes formula 10:
( ) ( )x y
g gP Q R
x y
F r r i j k i j k
S D
g gd P Q R dA
x y
F S
Dr. Erickson
![Page 22: Chapter 16 – Vector Calculus 16.7 Surface Integrals 1 Objectives: Understand integration of different types of surfaces Dr. Erickson](https://reader031.vdocument.in/reader031/viewer/2022020921/56649cc35503460f9498c96f/html5/thumbnails/22.jpg)
16.7 Surface Integrals 22
Vector Fields
◦ This formula assumes the upward orientation of S.◦ For a downward orientation, we multiply by –1.
S D
g gd P Q R dA
x y
F S
Dr. Erickson
![Page 23: Chapter 16 – Vector Calculus 16.7 Surface Integrals 1 Objectives: Understand integration of different types of surfaces Dr. Erickson](https://reader031.vdocument.in/reader031/viewer/2022020921/56649cc35503460f9498c96f/html5/thumbnails/23.jpg)
16.7 Surface Integrals 23
Example 4 Evaluate the surface integral 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.
SdF S
4
2 2
( , , ) ,
is the part of the cone beneath
the plane 1 with downward directions.
x y z x y z
S z x y
z
F i j k
Dr. Erickson
![Page 24: Chapter 16 – Vector Calculus 16.7 Surface Integrals 1 Objectives: Understand integration of different types of surfaces Dr. Erickson](https://reader031.vdocument.in/reader031/viewer/2022020921/56649cc35503460f9498c96f/html5/thumbnails/24.jpg)
16.7 Surface Integrals 24
Other ExamplesIn groups, please work on the following
problems on page 1145:
#’s 12, 14, and 28.
Dr. Erickson
![Page 25: Chapter 16 – Vector Calculus 16.7 Surface Integrals 1 Objectives: Understand integration of different types of surfaces Dr. Erickson](https://reader031.vdocument.in/reader031/viewer/2022020921/56649cc35503460f9498c96f/html5/thumbnails/25.jpg)
16.7 Surface Integrals 25
Example 5 – pg. 1145 # 12Evaluate the surface integral.
3 3
2 2
,
2 is the surface
3
0 1, 0 1.
S
y dS
S z x y
x y
Dr. Erickson
![Page 26: Chapter 16 – Vector Calculus 16.7 Surface Integrals 1 Objectives: Understand integration of different types of surfaces Dr. Erickson](https://reader031.vdocument.in/reader031/viewer/2022020921/56649cc35503460f9498c96f/html5/thumbnails/26.jpg)
16.7 Surface Integrals 26
Example 6 – pg. 1145 # 14Evaluate the surface integral.
2
,
is the surface 2 ,
0 1, 0 1.
S
z dS
S x y z
y z
Dr. Erickson
![Page 27: Chapter 16 – Vector Calculus 16.7 Surface Integrals 1 Objectives: Understand integration of different types of surfaces Dr. Erickson](https://reader031.vdocument.in/reader031/viewer/2022020921/56649cc35503460f9498c96f/html5/thumbnails/27.jpg)
16.7 Surface Integrals 27
Example 7 – pg. 1145 # 28Evaluate the surface integral 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.
SdF S
2( , , ) 4 ,
is the surface , 0 1,
0 1, with upward orientation.
y
x y z xy x yz
S z xe x
y
F i j k
Dr. Erickson