ENGR 233: Lecture 16
Surface and flux integrals
Dr. Ali Nazemi
Department of Building, Civil and Environmental Engineering
Tuesday 16th March 2021
Concordia University
Montreal, Quebec
Let’s review
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Let’s review
3
Let’s review
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Why do we need to know about surface
integrals?
• Surface integrals can be seen analogous to line integrals over arc lengths and
extends this concept into two dimensions.
• Applications include those mentioned for double integrals, most importantly
the area, volume and mass of a curved surface as well as fluid and heat fluxes.
 ds = the length of a curve
C
 dS
S
= double integral
= the area of a surface S
B
B
A
A
 ds = 
[ f ' ]2 + [ g ' ]2 dt
 dS = ???
S
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From double integrals to surface integrals
• The double integral is over a flat surface R.
• Now the region moves out of the plane and it becomes
a curved surface S, i.e., part of a sphere, cylinder, cone,
etc.
• We want to compute area and flux but the main
challenge lays in expressing dS
• When the surface has only one z for each (x, y), it is the
graph of a function z(x, y).
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A simple recipe for solving surface integrals
• To integrate a function G(x,y,z) involving z(x,y) = f(x,y) over a curved surface S
I (i.e., mass, volume...) =  G ( x, y, z )dS
S
(
=  G[ x, y, f ( x, y )] 1 + z
R
)
+  z  dA
x
 y 
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R is area down (projection of S) in the xy plane
Then calculate double integral using previous2double integral
2
techniques
surface
area =
dS =
1 + z
+  z  dA

x

y
R
R


Note:
Similar
projection
with
dA expression
= dxdy orfordA
= rdrdon
 other planes


(
2
)
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From surface integrals to flux integrals
• Consider a fluid with field velocity F, the total volume of the fluid
passing through a surface S is called the Flux of F through S

• For a given vector field F, we define Flux through surface= S F • nˆ dS
where n, a unit normal vector, defines the orientation of the surface S
defined by g(x,y,z)=0 and is given by the unit normal vector
g(x,y,z)=0
g
nˆ =
| g |
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Resources
• Read Section 9.13 and go through exercises
• This presentation is available through the moodle
• 4 additional lecture notes are placed in the moodle. All have Q&As with
complete solutions. Note that in this lecture we only discussed situations in
which surface is explicitly represented by x and y. Sup materials also discuss
situations in which x, y and z are represented parametrically with respect to
two independent variables u and v.
• Check out the Oregon State Lecture Notes for surface integrals:
http://math.oregonstate.edu/home/programs/undergrad/CalculusQuestStud
yGuides/vcalc/surface/surface.html
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