Chapter 16 – Vector Calculus 16.7 Surface Integrals 1 Objectives: Understand integration of...

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Chapter 16 – Vector Calculus16.7 Surface Integrals

16.7 Surface Integrals

Objectives: Understand integration of

different types of surfaces

Dr. Erickson

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


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


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


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


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


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


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


GraphsTherefore, Equation 2 becomes:

22

( , , ) ( , , ( , )) 1S D

z zf x y z dS f x y g x y dA

x y

Dr. Erickson


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


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


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


Possible OrientationsThere are two possible orientations for

any orientable surface.

Dr. Erickson


Positive OrientationObserve that n points in the same direction as the

position vector—that is, outward from the sphere.

Dr. Erickson


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


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


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


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


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


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


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


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


Example 4 Evaluate the surface integral for the given vector


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


Other ExamplesIn groups, please work on the following

problems on page 1145:

#’s 12, 14, and 28.

Dr. Erickson



3 3

2 2

,

2 is the surface

3

0 1, 0 1.

S

y dS

S z x y

x y

Dr. Erickson



2

,

is the surface 2 ,

0 1, 0 1.

S

z dS

S x y z

y z

Dr. Erickson


Example 7 – pg. 1145 # 28Evaluate the surface integral for the given vector


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

Chapter 16 – Vector Calculus 16.7 Surface Integrals 1 Objectives: Understand integration of...

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