ENGI 5432
2.
2. Surface Integrals
Page 2.01
Surface Integrals
This chapter introduces the theorems of Green, Gauss and Stokes.
Two different
methods of integrating a function of two variables over a curved surface are developed.
The sections in this chapter are:
2.1
2.2
2.3
2.4
2.5
2.6
Line Integrals
Green’s Theorem
Path Independence
Surface Integrals - Projection Method
Surface Integrals - Surface Method
Theorems of Gauss and Stokes; Potential Functions
ENGI 5432
2.1
2.1 Line Integrals
Page 2.02
Line Integrals
Two applications of line integrals are treated here: the evaluation of work done on a
particle as it travels along a curve in the presence of a [vector field] force; and the
evaluation of the location of the centre of mass of a wire.
Work done:
The work done by a force F in moving an elementary distance r along a curve C is
approximately the product of the component of the force in the direction of r and the
distance | r | travelled:
W  F r  F cos r
Integrating along the curve C yields the total work done by the force F in moving along
the curve C:
W 
 F dr
C

C  f dx  f
1
2
dy  f3 dz  

t1

t0
dx
dy
dz 
 f2
 f3  dt
 f1
dt
dt 
 dt
 W 
 F dr
C


C
F
dr
dt
dt
ENGI 5432
2.1 Line Integrals
Page 2.03
Example 2.1.1
Find the work done by F   y, x, z
in moving around the curve C (defined in
parametric form by x = cos t , y = sin t , z = 0 , 0  t  2 ).
F   y, x, z
C
  sin t , cos t , 0
dr
d

cos t , sin t , 0   sin t , cos t , 0
dt
dt
 F
dr
 sin 2 t  cos 2 t  0  1
dt
 W 

2
F
0
dr
dt 
dt
2
0
1 dt  2
dr
dt  1 everywhere on the curve C, so that
Note that Fv  F vˆ 
dr
dt
W  1 C  2 (the length of the path around the circle).
F
 curl F  2 kˆ everywhere in 3 .
Also note that F   y, x, z
The lesser curvature of the circular lines of force further away from the z axis is balanced
exactly by the increased transverse force, so that curl F is the same in all of 3 .
We shall see later (Stokes’ theorem, page 2.40) that the work done is also the normal
component of the curl integrated over the area enclosed by the closed curve C. In this
case
2
W     F nˆ  A  2 kˆ kˆ  1  2 .


ENGI 5432
2.1 Line Integrals
Page 2.04
Example 2.1.1 (continued)
Example 2.1.2
Find the work done by F  x, y, z
in moving around the curve C (defined in
parametric form by x = cos t , y = sin t , z = 0 , 0  t  2 ).
F  x, y, z
C
 cos t , sin t , 0  r
dr
d

cos t , sin t , 0   sin t , cos t , 0
dt
dt
 F
dr
  cos t sin t  sin t cos t  0  0
dt
 W 
2
0
0 dt  0
In this case, the force is orthogonal to the direction of motion at all times and no work is
done.
ENGI 5432
2.1 Line Integrals
Page 2.05
If the initial and terminal points of a curve C are identical and the curve meets itself
nowhere else, then the curve is said to be a simple closed curve.
Notation:
 F dr as  F dr .
When C is a simple closed curve, write
C
C
F is a conservative vector field if and only if
 F dr
 0 for all simple closed curves
C
C in the domain.
Be careful of where the endpoints are and of the order in which they appear (the
orientation of the curve). The identity
C F dr
 

t1

t
0
dr
dr
F
dt  
F
dt leads to the result
dt
dt
t0
t1
 F dr
 simple closed curves C
C
Another Application of Line Integrals:
The Mass of a Wire
Let C be a segment (t0  t  t1) of wire of line density  (x, y, z). Then
m    x, y, z  s
  ds  
C
 m 
C

ds
 dt 
dt
t1
t0

ds
dt
dt
First moments about the coordinate planes:
M  r m   r s
 M 

t1
t0
r
ds
dt
dt
The location r of the centre of mass of the wire is r 
t1
ds
ds
M
 r dt , m 
 dt and
dt
t0
t0 dt

t1

ds
dr


dt
dt
M
, where
m
2
2
2
 dx   dy   dz 
      .
 dt   dt   dt 
ENGI 5432
2.1 Line Integrals
Page 2.06
Example 2.1.3
Find the mass and centre of mass of a wire C (described in parametric form by
x = cos t , y = sin t , z = t ,   t   ) of line density  = z2 .
Let c = cos t , s = sin t .
r  c, s , t

[The shape of the wire is
one revolution of a helix,
dr
  s , c, 1
dt
aligned along the z axis,
centre the origin.]
ds

dt

 s 
2
 c 2  12 
2
  z2  t2
 m 
  ds  
C
 m 
M 

t1

ds

dt 
dt

2
r
t0
ds
dt 
dt
2
 t c dt  t






t 2c, t 2 s, t 3 dt
Integration by parts.

t dt 
2
2
2 3
3
x component:
2



2

 2 s  2tc 



t 2c dt   t 2  2 s  2tc 


  0  2    0  2    4
 t3 
2 
 3  
ENGI 5432
2.1 Line Integrals
Example 2.1.3 (continued)
y component:
For all integrable functions f (t) and for all constants a note that
a
 a


f  t  dt  
2

0
a
0
if f  t  is an ODD function
f  t  dt if f  t  is an EVEN function
t 2 sin t is an odd function


 t s dt
2
 0
z component:
t 3 is also an odd function


 t
3
dt  0
Therefore M   4 2 ˆi
r 
M
3
6

 4 2 ˆi   2 ˆi
3
m

2 2
 6

The centre of mass is therefore at   2 , 0, 0 
 

Page 2.07
2.2 Green’s Theorem
ENGI 5432
2.2
Page 2.08
Green’s Theorem
Some definitions:
A curve C on 2 (defined in parametric form by r  t   x  t  ˆi  y  t  ˆj , a  t  b) is
closed iff
(x(a), y(a)) = (x(b), y(b)) .
The curve is simple iff r  t1   r  t2  for all t1, t2 such that a < t1 < t2 < b ;
(that is, the curve neither touches nor intersects itself, except possibly at the end points).
Example 2.2.1
Two simple curves:
open
closed
Two non-simple curves:
open
closed
2.2 Green’s Theorem
ENGI 5432
Page 2.09
Orientation of closed curves:
A closed curve C has a positive orientation iff a point r(t) moves around C in an
anticlockwise sense as the value of the parameter t increases.
Example 2.2.2
Positive orientation
Negative orientation
Let D be the finite region of 2 bounded by C. When a particle moves along a curve
with positive orientation, D is always to the left of the particle.
For a simple closed curve C enclosing a finite region D of 2 and for any vector function
F  f1, f 2 that is differentiable everywhere on C and everywhere in D,
Green’s theorem is valid:
 F dr
C


D
  f 2  f1 


 dA
y 
 x
The region D is entirely in the xy-plane, so that the unit normal vector everywhere on D is
k. Let the differential vector dA = dA k , then Green’s theorem can also be written as
C F dr  D   F  kˆ dA  D  curl F  dA
F dr 
f1 , f 2
dx, dy

  f1 dx 
C
f 2 dy  

D
 f 2 f1 


 dA
 x y 
and
 
f 2 f1

 det  x

x y
 f1

y
f2

T
  det     z component of   F
 T 

F 

Green’s theorem is valid if there are no singularities in D.
ENGI 5432
2.2 Green’s Theorem
Example 2.2.3
x
F 
,0 :
r
Green’s theorem is valid for curve C1 but not for curve C2.
There is a singularity at the origin, which curve C2 encloses.
Example 2.2.4
For F  x  y, x  y and C as shown, evaluate
C F dr  PQ F dr  QR F dr  RP F dr
C F dr .
Page 2.10
2.2 Green’s Theorem
ENGI 5432
Page 2.11
Example 2.2.4 (continued)
F  x  y, x  y
Everywhere on the line segment from P to Q, y = 2 – x (and the parameter t is just x)
 r  x, 2  x


PQ
F dr 

dr
 1,  1
dx
0
 2  2   2x  2 dx
and F  2, 2 x  2
0
 2  4  2x  dx  4x  x 

2
= (0 – 0) – (8 – 4) = –4
Everywhere on the line segment from Q to R, y = 2 + x
 r  x, 2  x


QR
F dr 

2
0
dr
 1, 1
dx
  2 x  2  2  dx

and F  2 x  2,  2
2
0
2
2 x dx   x 2 
0
= 4–0 = 4
Everywhere on the line segment from R to P, y = 0
 r  x, 0


dr
 1, 0
dx
2
RP F dr   2  x  0 dx  
= 2–2 = 0

C F dr
 4 4 0  0
and F  x, x
2
 x2 
x dx   
2
 2  2
2
0
2
2.2 Green’s Theorem
ENGI 5432
Example 2.2.4 (continued)
use Green’s theorem!
OR
 
 



det x y  
x  y 

 x  y  1  1  0


x
y
 f1 f 2 
everywhere on D


D
 f 2 f1 


 dA 
 x y 
 0 dA  0
D
By Green’s theorem it then follows that
C F dr
 0
Page 2.12
2.2 Green’s Theorem
ENGI 5432
Page 2.13
Example 2.2.5
Find the work done by the force F  x y , y 2 in one circuit of the unit square.
By Green’s theorem,
W 

F dr 
C

D
  f 2  f1 


 dA
y 
 x
 f 2  f1
 2



y 
 x y  0  x
x
y
x
y
 
The region of integration is the square 0 < x < 1, 0 < y < 1
 W 

 x dA 
1
1
0
0

  x  dy dx
D
 

x
0 
1
 

1 dy  dx  
0

1

1
0
x  y  0 dx  
1

1
0
x 1  0  dx
1
 x2 
1
1
     0  
2
2
 2 0
Therefore
W   12
The alternative method (using line integration instead of Green’s theorem) would involve
four line integrals, each with different integrands!
ENGI 5432
2.3
2.3 Path Independence
Page 2.14
Path Independence
Gradient Vector Fields:
If F   , then F 
 
,
x  y

 f 2  f1

  yx  xy  0
x
y
(provided that the second partial derivatives are all continuous).
It therefore follows, for any closed curve C and twice differentiable potential function 
that
  dr
 0
C
Path Independence
 or
If F  
F     , then  is a potential function for F.
Let the path C travel from point Po to point P1:

F dr 
C

C
 dr 
 

 
dy 
dz  
 dx 

x

y

z


C

 d
C
[chain rule]
   P1    P1     P0 
P
0
which is independent of the path C between the two points.
difference in 
 work done 


Therefore 
  

 by  
 between endpoints of C 

  dr    P     P   0
C
[work done = potential difference]
ENGI 5432
2.3 Path Independence
Page 2.15
Domain
A region  of 2 is a domain if and only if
1)
For all points Po in , there exists a circle, centre Po, all of whose interior points
are inside ; and
2)
For all points Po and P1 in , there exists a piecewise smooth curve C, entirely in
, from Po to P1.
Example 2.3.1
Are these domains?
{ (x, y) | y > 0 }
YES (but not simply connected)
{ (x, y) | x  0 }
NO
If a domain is not specified, then, by default, it is assumed to be all of 2.
ENGI 5432
2.3 Path Independence
Page 2.16
When a vector field F is defined on a simply connected domain , these statements are
all equivalent (that is, all of them are true or all of them are false):
 F   for some scalar field  that is differentiable everywhere in ;
 F is conservative;

 F dr is path-independent (has the same value no matter which path within
C
 is chosen between the two endpoints, for any two endpoints in );

 F dr  
end
  start (for any two endpoints in );
C

 F dr

 f2
 f1

everywhere in ; and
x
y
 0 for all closed curves C lying entirely in ;
C
   F  0 everywhere in  (so that the vector field F is irrotational).
There must be no singularities anywhere in the domain  in order for the above set of
equivalencies to be valid.
Example 2.3.2
Evaluate
  2 x  y  dx   x  3 y  dy  where C
2
is any piecewise-smooth curve from
C
(0, 0) to (1, 2).
F  2 x  y, x  3 y 2
 f2
 f1
 1 
x
y

is continuous everywhere in  
2
 F is conservative and F  

 2 x  y and
x

 x  3y2
y
A potential function that has the correct first partial derivatives is   x 2  xy  y 3
1,2



F dr    
 1  2  8   0  0  0 


 0,0
C

Therefore
  2 x  y  dx   x  3 y  dy  
2
C
11
ENGI 5432
2.3 Path Independence
Page 2.17
Example 2.3.3 (A Counterexample)
 F dr , where F 
Evaluate
C
y
x
, 2
2
x  y x  y2
2
and C is the unit circle, centre at the
origin.
F is continuous everywhere except (0, 0)
  is not simply connected. [  is all of
f 2

x
x2  y 2
 x2  y 
2 2

f1
y
2
except (0, 0).]
everywhere in 
We cannot use Green’s theorem, because F is not continuous everywhere inside C
(there is a singularity at the origin).
Let c = cos t and s = sin t then
r  c, s
s
c
, 2 2
2
c s c s
F 

 0  t  2 
2

F dr 
C
 r  s, c
 s,  c
2
2
2
0  s  c  dt
 

2
0
2
1 dt   t 
0
Therefore
 F dr 
 2
C
Note:
 F dr
 0 , but
C
x
everywhere on  , F   , where   Arctan    k
 y
The problem is that the arbitrary constant k is ill-defined.
ENGI 5432
Example 2.3.3
2.3 Path Independence
Page 2.18
(continued)
Let us explore the case when k = 0.
x
Contour map of   Arctan    0
 y
We encounter a conflict in the value of the potential function  .
Solution: Change the domain  to the simply connected domain
2

except the 
  

 non-negative x axis 
then the potential function  can be well-defined, but no curve in  can enclose the
origin.