.
Chapter 17
Vector analysis: Surface integrals, Divergence,
Curl, and Potential
------------------- .
5 x 11
5 x 11
1
9
x 2 6 x 8 x 2 ˜ x 4 2x 2 2 x 4
1
1
1
f 2 x 9( x 1) 3( x 2) 9( x 2) 2
4
2x 3
f3 x 2
x x 4x 5
……………………………………………………………………….
f1 x 1
60
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
If you succeeded, you know the basic rules for finding partial fractions and you may proceed to the
next section.
------------------- 61
If, however, you encountered difficulties, it makes sense to return to the textbook and study the
relevant pages 170–174, work through the relevant frames in the study guide and do some examples.
You will be happier knowing the basics well when entering subsequent sections.
Afterwards
------------------- 1
61
Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential
.
Laplace transforms
1
Objective: In the first section the definition of Laplace transforms will be explained and
developed. It will prove useful, to copy all definitions and results into your notebook when going
through this and subsequent sections. Then they will be readily accessible to you and you will not
have to take recourse to the textbook for each and every single computation.
READ
11.1 Introduction
11.2 The Laplace transform definition
Textbook pages 321–322
When done
.
------------------- 11.3 Solution of linear differential equations with constant coefficients
2
61
After having successfully gone through all preliminary steps, which might have been somewhat
depletive in places, you now encounter applications and you will reap the benefits. Follow all the
examples arduously and if necessary, consult the table on page 275.
You will learn to transform a set of linear differential equations into a set of algebraic equations and
after solving these to use the inverse transform, thus obtaining a solution of the original DEs.
.
READ
11.4 Solution of linear differential equations
with constant coefficients
Textbook pages 328–329
When done
------------------- 2
62
Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential
.
In the section you studied some new concepts have been introduced and defined.
Write down at least three of them:
2
1. …………………………………………………..
2. ………………………………………..
3………………………………………….
------------------- 3
.
The tendency to overlook things we do not understand is natural and even necessary
to survive. Nobody can understand everything. But if we even do not notice that we do
not understand something, this may be dangerous. And may have consequences. Try to
develop competence in noticing things you do not understand and develop a tendency
to use indexes and encyclopedia. At least a few times per week.
------------------- 3
62
63
Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential
.
G
Flow density j
3
G
Surface element vector A
G
G
Flow of a vector field F through a surface A
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
Try to write down on a separate sheet the definitions and meanings first using your memory but in
case of difficulties by consulting your notes and if you do not succeed the textbook.
------------------- 4
.
17.5 Divergence of a vector field and Gauss’s theorem
63
The introduction of the concept of divergence is closely related to the last section on surface
integrals.
Study in the textbook
17.5 Divergence of a vector field
17.6 Gauss’s theorem
Pages 475–479
Having concluded go to
------------------- 4
64
Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential
.
G
The flow density j is the quantity which passes through a unit area per time unit.
The area is assumed to be perpendicular to the flow.
4
G
The surface element vector is a vector A whose direction is perpendicular to the surface and whose
magnitude is equal to the area A.
Flow of a vector field through a surface element is given by the
G G
dot product of the two vectors F ˜ A .
------------------- 5
.
G
F represents a vector field. Complete the definition
64
G
div F = ………………………
G
div F is a ………………….
------------------- 5
65
Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential
.
We want the flow of a vector field through an even square surface whose magnitude is A.
The surface belongs to the y-z plane. The flow hits the surface at an angle E
G
The flow vector is constant and given by j
Ed
S
5
2
( j x , j y ,0 )
We decompose the task:
G
1. We determine A
G G
2. We calculate the flow j ˜ A
The flow is: …………………
Further explanations wanted
Solution found
------------------- 6
------------------- 11
.
G GFx GFy GFz
div F
Gx Gy
Gz
G
div F is a scalar field.
65
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
Calculate the divergence for the given vector field:
G
F
( x , y b, z 2 )
G
div F
…
Determine the location of sources and sinks.
------------------- 6
66
Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential
.
G
We first determine the surface element vector A . Since the surface
element belongs
G
to the y-z plane its direction is parallel to the x-axis. Thus, A
also has the direction of the x-axis.
6
G
G
A (1,0,0) or A(1,0,0) Since the flow vector is to hit
the surface element at an angle less
G
S
we obtain A ………………
than
2
G
Hint: remember the definition of the given j .
------------------- 7
.
G
div F
(1 1 2 z )
(2 2 z )
2(1 z )
66
For the plane z 1 there are no sources or sinks.
The space above this plane, z ! 1 , consists of sinks.
The space beneath this plane, z d 1 , consists
of sources
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
All correct, go
Another exercise wanted
7
------------------- 69
------------------- 67
Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential
.
G
A
A(1,0,0)
7
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
No more problem
More explication wanted
------------------- 9
------------------- 8
.
G
We have no sink or sources if div F 0 .
With the preceding example this was the case for
G
divF 21 z 0 .
The equation 1 z 0 or z 1 represents a plane
G
which is parallel to the x-y-plane. For it div F
is negative. This means the space above the plane
z 1 consists of sinks.
G
Beneath the plane z 1 divF is positive and thus
the space consists of sources.
67
Calculate the divergence for the following
vector field:
G
F
xG
2
div F
1, y, z 5 :
…………………
Find the spaces where we have sources or sinks and where the field is free of sources and sinks.
------------------- 8
68
Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential
.
The surface element belongs to the y-z plane. Thus, its surface element vector is directed
in the x-direction. But there are two possibilities. It may point in the positive or in
the negative x-direction. We have to decide which of these directions is the right one.
In our exercise is given the condition that the angle between both given vectors is to be
less than 90° or
8
S
.
2
G
G
In our exercise the x-component of j is negative. Thus, A
has to point in the negative direction of the x-axis. From this
G
we obtain: A = (–1,0,0,)
------------------- 9
.
G
divF 2 x 1 1 2 x 1
For the plane x 1 the field is free of sinks and sources.
The space to the left of this plane, defined by x 1 , consists of sinks.
The space to the right of this plane, defined by x ! 1 ,
consists of sources.
68
Go on to
------------------- 9
69
Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential
.
G
G
We determined the surface element vector A to be A
A(1,0,0)
9
G
Now we can determine the flow I of j through the surface:
G
j ( jx , j y , 0 )
G G
I = j ˜ A …………………………
Solution found
Further help wanted
------------------- 11
------------------- 10
.
Calculate the divergence for the
vector field
G
F
x ,
G
divF
3
y 3 , 3z
69
………………..
Where do you find sinks and sources?
………………………………………………………………………………………………………....
………………………………………………………………………………………………………....
………………………………………………………………………………………………………....
------------------- 10
70
Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential
.
G
We remember the dot product of two vectors a
is defined by
G G
a ˜ b a x bx a y b y a z bz
G
(a x , a y , , a z ) and b
(bx , b y , bz )
10
Now you should be able to calculate using the results obtained before:
G G
j ˜ A = ……………………………………………………..
------------------- 11
.
G
div F 3x 2 3 y 2 3 3 x 2 y 2 1
Sources and sinks vanish for x 2 y 2 1 .
This is a circle with radius 1 which does not depend on the value of z.
70
Thus it is a cylinder.
G
The space inside the cylinder consists of sinks since div F is negative.
G
The space outside the cylinder consists of sources since div F is positive.
------------------- 11
71
Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential
.
I
jx ˜ A
11
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
Give the surface element vector for the
four given surfaces the magnitude
of which is always A
------------------- 12
.
In the textbook the Nabla operator has been introduced. It is a new notation which seems
to be merely formal. But it will prove quite useful because it is a short-hand notation that
simplifies notations once you have familiarity with it.
G
’
§G
G
G ·
¨¨ ,
¸
,
G
G
G
x
y
z ¸¹
©
G
’
………………….
------------------- 12
71
72
Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential
.
In each case we have two answers. The difference is the sign.
This time a vector field that determines the direction is not given.
G
1. A
A(0,0,1) .
G
2. A
A
2
A(0,0,1)
G
(1,0,1) ) or A
G
3. A
A
G
4. A
A
3
2
G
or A
(1,1,1)
or
(1,0,1)
or
12
A
2
G
A
G
A
(1,0,1)
A
( 1,1,1)
3
A
2
(1,0,1)
All correct
Errors or explanation wanted
------------------- 16
------------------- 13
.
G
’
§G
G
G ·
¨¨ Gx , Gy , Gz ¸¸
©
¹
72
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
Using the Nabla operator we obtain a short-hand notation to represent
the gradient of a scalar field f x, y, z and
G
the divergence of a vector field F x, y, z .
We start with the calculation of the gradient:
G
grad f x, y, z ’ ˜ f x, y, z =……………………………..
------------------- 13
73
Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential
.
G
The surface element vector A is perpendicular
to the surface. Thus, we look at first for a vector
perpendicular to the surface, the magnitude of
which does not matter for the time being. We
give the first two solutions then obtain the next
two solutions.
a has not been defined yet.
a will be defined later.
1.
G
A
a(0,0,1)
2
G
.A
a(1,0,0)
3.
G
A
a(..............)
4,
G
A
a(.................)
13
------------------- 14
.
G
§ Gf G Gf G Gf G ·
gradf x, y, z ’ ˜ f x, y, z ¨¨ ˜ ex ˜ e y ˜ ez ¸¸
Gy
Gz
© Gx
¹
73
G
’ is a vector. In this case we compute the product of a vector with a scalar
since the function f x, y, z is a scalar.
The result is a vector since the product of a vector with a scalar is a vector.
Now we calculate the divergence of a vector field
G
div F
G G
’ ˜ F x, y , z GFx G GFy G GFz G
˜ ez
ex ey Gx
Gy
Gz
G
Again ’ is a vector. But this time we compute
G
the dot product of the vector Nabla with the vector F . The result is a scalar. We remember well from
vector algebra the dot product of two vectors results in a scalar.
G
Calculate ’ ˜ M x, y, z for M ( x 2 y 2 z 2 ) :
G
grad M ’ ˜ M ……………………………..
G
G G
Calculate ’ ˜ F for F x 2 , y 2 , z 2 :
G G G
div F ’ ˜ F =…………………………………
------------------- 14
74
Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential
.
G
1. A
a(0,0,1)
G
2. A
a (1,0,1)
G
3. A
a(1,1,1)
G
4. A
a (1,0,1)
14
G
Now we will determine a. A must have the magnitude A. To obtain this we have to choose a.
For the first surface it is evident that:
G
A A(0,0,1) In this case a = A
For the second exercise we have
G
A
A2
A
(1,0,1) We may prove it: A 2
(1 1)
2
2
G
Now you may try to calculate a for the third exercise A .......(1,1,1)
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
------------------- 15
.
grad M
G
’ ˜M
2 x,2 y,2 z 2 xeGx 2 yeGy 2 zeGz 74
(The vector grad M may be written down in short-hand or extensively)
G
div F
G G
’˜F
2 x 2 y 2 z ------------------- 15
75
Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential
.
G
. A
A
3
(1,1,1)
Prove:
G
( A) 2
G
We give the systematic solution. Given: A
A2
(1 1 1)
3
15
A2
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
a(a x , a y , a z ).
G
G
If A is to be of magnitude A this results in ( A) 2
A2
A 2 this results in
a 2 (a x2 a y2 a z2 )
Thus, finally we obtain
a
A
a a y2 a z2
2
x
In our case no direction of the vector field is given.
Thus, the sign of the surface element vector may be inverted.
------------------- 16
.
In the textbook page 470 the electrical field of a sphere is discussed.
Given a sphere with homogeneous charge distribution U , the total charge Q ,
and radius R. Then outside the surface of the sphere the electrical field is given by
G
E ( x, y , z )
Q
4SH 0
x, y , z x y2 z2
2
75
3
G
Now calculate the divergence E outside the surface of the sphere.
G
div E =…………………………………………………
------------------- 16
76
Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential
.
17.2 Surface integral
Study in the textbook
16
17.2 Surface integral
Textbook pages 464–466
This done go to
------------------- 17
.
G
div E
0
76
In case of difficulties consult the textbook again.
………………………………………………………………………………..
Inside the sphere the electrical field is given by
G
E ( x, y , z )
Q
( x, y , z )
4SH 0 R 3
Calculate the divergence inside the sphere:
:
G
div E ……………………
------------------- 17
77
Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential
.
The following integral is named ……….
I
G
17
G
³ F ˜ dA
The circle in the integral I
G
G
³ F ˜ dA
significates the integral has to be taken over a ……………………….surface.
------------------- 18
.
G
div E
3Q
4SH 0 R 3
U
H0
77
Q
3 3
SR V and
U.
V
4
…………………………………………………………………………………………
Write down from memory the Gauss theorem. If possible do not consult the textbook. You should
memorize this theorem.
In case of difficulties remember
………………………………………………..
………………………………………………..
………………………………………………..
------------------- 18
78
Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential
.
Surface integral
18
Closed surface
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
Give at least three examples for a closed surface ….
1.: ………………………………………………………..
2: ………………………………………………
3:………………………………………………..………………………………………………...……
……………………………………..
------------------- 19
.
³ div
V
G
F ˜ dV
G
³ F ˜ dA
78
AV
……………………………………………………………………………………….
In the following we will use Gauss’s theorem to calculate the given electrical field inside and outside
of the surface of the sphere.
First calculate the electrical field outside the surface of the sphere:
G
E …………………………………..................
Solution found
Help and explanations wanted
19
------------------- 82
------------------- 79
Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential
.
Control your examples using the definitions given in the textbook
19
This control is necessary if there are no right answers given.
In your further study and in practice this is the normal case.
------------------- 20
.
In the textbook page 470 the electric field of a sphere is given with radius R and
charge density U . Outside of the sphere the electrical field is:
G
E ( x, y , z )
Q
4SH 0
x, y , z x y2 z2
2
79
3
We can obtain this result using Gauss’s theorem. The center of the sphere is to coincide with the
origin of the coordinate system.
First we calculate the total charge Q of the sphere. Given radius R and charge density inside the
sphere U :
Q …………………………………
------------------- 20
80
Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential
.
The direction of the surface element vectors for closed
surfaces is defined
unambiguous
ambiguous
20
------------------- 21
------------------- 22
.
Q
³U
4S
UR 3
3
dv
VKugel
80
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
Secondly we calculate the electrical field outside the sphere using Gauss’s theorem.
G
³ divF ˜ dV
V
G
³ F ˜ dA
A (V )
From electrodynamics we know that the flow of the electrical field through a closed surface is
proportional to the enclosed total charge Q.
G
Q
³ E ˜ dA
Surfase
H0
Calculate the flow through the surfaces outside the sphere with the given radius Routside :
G
Q
³ E dA ...........................
H0
------------------- 21
81
Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential
.
You are right. With closed surfaces the sign is defined unambiguously.
21
With closed surfaces the surface element vector always points to …………
………………………………………………………….……………………
------------------- 23
.
G
³ E ˜ dA
2
E ˜ 4SRaußen
Q
81
H0
From this result we obtain:
Q
E
2
H 0 ˜ 4SRaußen
The field has spherical symmetry. The field vector points to the outside and its direction is given by
the unit vector:
G
r
( x, y , z )
2
r
x y2 z2
Thus we obtain:
G
E
r
R
x2 y2 z2
G
Q
r
=…………………………......
˜
H 0 4SR 2 r
------------------- 22
82
Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential
.
Unfortunately you are wrong.
22
Look again into the textbook and read again the definition or better the convention
of the direction of surface elements.
The surface element vectors are
a) perpendicular to the surface
b) they point with closed surfaces always to…………………………..
------------------- 23
.
G
E ( x, y , z )
Q
4SH 0
x, y , z x y2 z2
2
82
3
Now we are to calculate the electrical field inside the sphere.
Let us regard a surface of a sphere inside with radius Rinside .
We are to calculate the enclosed charge. Use Gauss’s theorem again and use the total charge Q of the
original sphere.
G
Einnen =………………………………………….
Further help and detailed calculation
Solution happily found
23
------------------- 83
------------------- 86
Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential
.
The surface element vector for closed surfaces always point to the outer space.
23
This is a convention, but it is worth memorizing.
------------------- 24
.
The surface of an inner sphere encloses a part of the total charge given by:
83
Qinnen =………………
------------------- 24
84
Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential
.
17.3 Special cases of surface integrals
Study in the textbook
24
17.3.1 Flow of a homogeneous vector field through a cuboid
Pages 466–468
------------------- 25
.
Qinnen
4S 3
˜ Rinnen ˜ U
3
³ UdV
84
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
G
Now we apply Gauss’s theorem. Given: div E
G
³ E ˜ dA
surface
³ div
U
.
H0
G
E dV
volume
We calculate both integrals:
a)
G
³ E ˜ dA
=………………
Surface
b)
³ div
G
EdV =……………….
Volume
Inserting into Gauss’s theorem we obtain:……………=………………….
------------------- 25
85
Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential
.
Decide whether the following vector fields are homogeneous or not
25
Vector field is homogeneous
G
1. F
G
2. F
yes
no
( x, y , z )
x2 y2 z2
(1,0, x)
3.
G
F
( y , x, z )
4.
G
F
(6,3,5)
5.
G
F
(2,0,0)
------------------- 26
.
G
a)
³ E ˜ dA
b)
³ divEdV
2
E ˜ 4SRinnen
G
85
U 4S 3
˜
˜ Rinnen
H0 3
Inserting into Gauss’s theorem we obtain
G
U 4S 3
2
E ˜ 4SRinnen
˜
˜ Rinnen
H0 3
G U Rinnen
E
H0 3
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
G
K
To obtain E we have to regard the direction of E :
G
E
E˜
G
r
=………………………
r
G
Remember: r
Rinside
------------------- 26
86
Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential
.
Homogeneous vector fields are
4. 5.
26
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
All correct
Not all answers correct
------------------- 29
------------------- 27
.
G
Einnen
U x, y , z ˜
H0
3
86
Finally, we substitute U by the total charge Q of the original sphere using the known equation
Q
U˜
G
Einnen
4S 3
R
3
…………………..
------------------- 27
87
Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential
.
In this case we suggest you read again the section you just studied.
Theses questions did not imply calculations. Thus, you have problems with
the understanding of the definition of a homogeneous vector field.
Read again to try and find the correct answers.
Vector field
27
homogeneous
not homogeneous
G
1. F
(1,2,3)
x2 y2 z2
G
2. F
1 1 1
( , ; )
x y z
3.
G
F
(1;0;0)
4.
G
F
(x;0;0)
5.
G
F
(21;1; )
------------------- 28
.
G
Einnen
Q
˜ ( x, y , z )
4SH 0 ˜ R 3
87
In case of remaining difficulties repeat this section of the study guide and consult the textbook again.
------------------- Go on to
28
88
Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential
.
G
1. F
(1,2,3)
x y2 z2
G
2. F
1 1 1
( , ; )
x y z
not homogeneous
2
28
not homogeneous
3.
G
F
(1;0;0)
homogeneous
4.
G
F
(x;0;0)
not homogeneous
5.
G
F
(21;1; )
homogeneous
------------------- 29
.
17.6 Curl of a vector field and Stoke’s theorem
88
The concept of curl will be introduced. After completing section 17.7 it is suggested
you have a break and a cup of coffee or tea. But do not forget to take notes of new
definitions on a separate sheet.
Study
17.7 Curl of a vector field.
17.8 Stoke’s theorem
Textbook pages 480–485
Having studied
------------------- 29
89
Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential
.
G
Determine the flow of the field F ( x, y, z ) (1,4,3) through a cube whose faces
are parallel to the axis of the coordinate system.
29
G
The flow I of the vector field F is
I =………………………..
Solution found
One more hint wanted
------------------- 31
------------------- 30
.
G
If a vector field F1 is curl free we have: …………=………..
G
If a vector field F2 has curl we have gilt……...…..=………..
89
------------------- 30
90
Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential
.
G
Hint: F (1,4,3) is a homogeneous vector field.
Thus, you can apply rule 17.7 given in the textbook.
30
G
Give the flow of the vector field F (1;4;3) through
a cube whose faces are not parallel to the axes of the
coordinate system.
I =………………………………..
------------------- 31
.
Field is curl free
Field has curl
G
³ FG ˜ds 0
³ F ˜ ds z 0
90
1
2
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
It is not that easy, but it is worthwhile to memorize the definition of curl or to reconstruct it using the
G G G
formula rot F = ’ u F .
G
rot F = ………………
------------------- 31
91
Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential
.
I=0
31
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
G
For which of the given surfaces below does the flow of the homogeneous field F
not vanish? The surfaces are closed.
cuboid
sphere
ellipsoid
(0,2,0)
dumbbell-shaped
------------------- 32
.
G
rot F
§ GFz GFy ·
¸
¨
Gz ¸
¨ Gy
¨ GF GF ¸
¨ x z¸
Gx ¸
¨ Gz
¨ GFy GF ¸
x ¸¸
¨¨
G
Gy ¹
x
©
91
Write down as a determinant:
G
rot F
…………………………
------------------- 32
92
Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential
.
None of these. All are closed. A homogeneous vector field vanishes for all closed surfaces.
32
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
G
Given the vector field F
( x, y, z )
. Does its flow through a sphere vanish if the sphere’s
x y2 z2
center coincides with the origin of the coordinate system?
2
Answer found
Help or more explanation wanted
------------------- 35
------------------- 33
.
G G G
ex e y ez
G
rotF
92
G G G
Gx Gy Gz
Fx Fx Fz
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
Expand the determinant and give:
G
rotF
…………………………
------------------- 33
93
Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential
.
The surface of the sphere is closed.
G
Given the vector field F
( x, y, z )
x y2 z2
2
33
. This vector field is not homogeneous.
Sketch some vectors of the field along the axes of
the coordinate system
------------------- 34
.
G
rot F
§ GFz GFy ·
¸
¨
Gz ¸
¨ Gy
¨ GF GF ¸
¨ x z¸
Gx ¸
¨ Gz
¨ GFy GF ¸
x ¸¸
¨¨
G
Gy ¹
x
©
93
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
Regard a flow of water. The field of velocities may be given by:
G
v
(1, ln z , 0)
Calculate:
G
div v ………………………
G
rot v
………………………
------------------- 34
94
Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential
.
If vectors in all directions are sketched, the field
34
is represented as shown to the left.
Imagine a sphere with its center at the origin of the
coordinate system. The field passes through the surface
always from the inner side to the outer side.
Does the flow through the surface vanish?
------------------- 35
.
G
divv 0
G
1
rotv ( ,0,0)
z
94
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
Calculate the line integral for the rectangle with sides a and b for a complete circulation.
G
G
Given F to be F ( x, y, z ) (5,0, z 2 )
G
³ F ds
………………………
------------------- 35
95
Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential
.
G
No. The flow of F
( x, y, z )
does not vanish.
x y2 z2
The field passes at all points of the surface from the inner
space to the outer space.
2
35
G
A vector field F has radial symmetry if
1…………………………………………………
2………………………………………….
In case of doubt look again at textbook chapter 13 (section “coordinate systems”).
------------------- 36
.
The field of velocities is free of curl. Thus:
G
³ F ds
95
0
------------------- 36
96
Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential
.
A vector field has radial symmetry if
1. All vectors point in a radial direction and;
2. Its amount depends only on the radius r.
36
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
Decide whether the following field possesses radial symmetry:
G
2. Its amount depends only on r. F
( x, y, z )
( x 2 y 2 z 2 )3
=
G
r
r3
Solution found
Further explanation or help wanted
------------------- 39
------------------- 37
.
17.7 Potential of a vector field
Study
96
17.9 Potential of a vector field
Textbook pages 485–487
------------------- 37
97
Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential
.
G
Firstly sketch some of the vectors of the field F
G
r
r3
37
------------------- 38
.
In the textbook we treated the example of the gravitational field of a sphere, the mass of
97
which is M. The case most familiar to us is the earth. At the surface of the earth, however,
we simplify the situation by calculating with a homogeneous field of gravitation. In this
simplification the x-y plane is parallel to the earth’s surface. The z-axis points upwards and
the origin of the coordinate system coincides with the earth’s surface. This approximation holds
for a lot of calculations. In this case the gravitational force acting on a mass m is given:
G
F
……………………………….
------------------- 38
98
Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential
.
Your sketch may be similar to the sketch to the left.
The vectors point outwards. Because of r 3
in the denominator the amount of the vectors decreases
with the distance from the origin.
The vectors at other positions point in a radial direction
as well.
Since ( x, y, z ) = r we obtain:
F
( x, y, z )
( x y 2 z 2 )3
2
=
r
r3
38
1
r2
F only depends on r.
Thus, this vector field has ……………………………………
------------------- 39
.
G
F
0,
0, m ˜ g 98
G
In the case of a gravitational field the force F is the product of the mass m of the body in question
G
and the gravitational field vector Fg of the gravitational field. This field vector is in this case
G
G
Fg g ˜ ez 0, 0, g In the following we discuss fields and their field vectors. This holds as well for electrical fields and
field vectors. In the case of a static electrical field the force acting on a charge Q is the product of Q
G
with the electrical field vector E :
G
F
G
Q˜E
G
The field is completely represented by the electrical field vector E .
G
Check gravitational field Fg .
Is it free of curl?
G
rot Fg
………………….
G
Fg is……………………..
------------------- 39
99
Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential
.
G
The vector field F has radial symmetry because
G
( x, y, z )
F
points in a radial direction and its amount depends only on r.
2
( x y 2 z 2 )3
39
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
------------------- 40
.
rot 0,0, g 0
99
G
Fg is curl-free, it is a conservative field.
G
Calculate the potential of F
G
m ˜ Fg .
Remember: The following convention holds in physics. For a given force field
G
Fg
the potential
M is the work done against the force field
M x, y, z …………………..
------------------- 40
100
Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential
.
17.3.1 Flow of a spherically symmetrical field through a sphere
40
In this section we show the calculation of surface integrals through a surface of a sphere.
In physics this is an important special case.
Read in the textbook
17.3.2 Flow of a spherically symmetrical field
through a sphere
Textbook pages 468–469
------------------- 41
.
M x, y , z m ˜ g ˜ z C
100
The potential is thus determined except the integration constant C:
Determine the potential given above for three different situations:
1) M1 0 for the ground floor with z0 0
M1 ……………….
2) M 2
M2
3) M 3
M3
0 for the underground of a high building with z0
10
………………..
0 for the roof of a high building with z0
90
………………..
------------------- 41
101
Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential
.
G eGr
G G
Calculate the surface integral ³ F ˜ dA of the field F
r2
G
G
r
for a surface of a sphere with radius R.
with er
r
G
Wanted is the flow of F through the surface of the sphere.
G
41
G
³ F ˜ dA = ………………………………………..
------------------- 42
.
M1
0,0, mgz M2
0,0, mg z 10
M3
0,0, mg z 90
101
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
G
Give the gravitational field Fg for these three cases:
G
Fg1
…………………
G
Fg 2
…………………
G
Fg 3
…………………
------------------- 42
102
Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential
.
G
G
dA
³ F ˜ dA = ³ r
4SR 2 ˜
2
1
R2
4S
42
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
G
Given a field of a force with radial symmetry: F (r )
a G
er
r3
with
G
G r
er
r
G
Compute the flow of the field of the force F (r ) through the surface of a sphere which has a
distance R from the origin of the field. The center of the field is defined by r = 0.
Solution found
Explanation or help wanted
------------------- 44
------------------- 43
.
M1 M 2
M3
0,0, mgz 102
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
In the textbook you studied the gravitational field of a mass M. This mass is assumed to be distributed
homogeneously in the inner space of a sphere with radius R. Outside of the sphere the gravitational
field is:
G
x, y , z
Fg x, y, z J ˜ M
3
x2 y2 z2
G
Simplify using r
x, y , z G
Fg x, y, z …………………..
------------------- 43
103
Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential
.
G
F (r )
The field of the force is given by
Wanted: the flow given by
G
a G
er
r3
with
G
G r
er
r
43
G
³ F ˜ dA
G
First hint: The field has radial symmetry of the following form: F f (r ) ˜ er
G G
Second hint: ³ F ˜ dA is calculated for the general case in the section of the textbook you just studied.
Repeat your study and try again:
G
G
³ F ˜ dA
= …………………………………………………………………..
44
------------------- .
G
Fg
JM ˜
G
r
r3
JM
G
1 r
˜
r2 r
103
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
G
Now we calculate the potential M ³ F dr for a path of integration in the direction of a radius.
G
r
Then we have: dr dr .
r
r
1 r
M JM ³ ˜ dr =…………………………….
r r
r0 2
------------------- 44
104
Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential
.
³
G G
a G
er dA
r3
³ f (r )e dA
r
4SR 2 ˜ f ( R)
4SR 2 ˜
a
R3
4S ˜ a
R
44
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
Have a short break
------------------- 45
.
r
M r JM ˜ ³
r0
ª1
r
ª1 º
JM « »
¬ r ¼r
dr
r2
104
0
1º
M r JM « »
¬ r0 r ¼
We can scale the potential of the gravitational field of the mass M in a way that it vanishes
for r o f , but the potential may be scaled as well to vanish for the surface for which holds
r rsurface
In the textbook we explained the first case.
In the study guide we calculated the second case.
Calculate the potential for the first case in order to vanish for infinity.
M1 …………………..
------------------- 45
105
Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential
.
17.3.2 Application: The electrical field of a point charge.
45
Here we apply the new knowledge
Study in the textbook
17.3.3 Application: The electrical field of a point charge
Textbook page 470
------------------- 46
.
M1
JM
1
r
105
All correct
Further explanation wanted
46
------------------- 108
------------------- 106
Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential
.
The calculations in the foregoing section are obvious. In this case using physical units
facilitates the calculations.
46
Thus, we can proceed to the next section without further exercises.
Go to
------------------- 47
.
The potential of the gravitational field was represented by:
r
dr
r2
r0
M r JM ³
106
ª 1 1º
»
¬ r0 r ¼
JM «
The condition was M r
f 0
1
0 or r 0 f .
r0
A difficulty in understanding this may arise because r0 is the lower limit of integration and the lower
limit can not be f while the upper limit of integration is finite. But this problem is solved; we invert
the direction of integration. We integrate from r to r0 .
In this case the bracket must be zero. This is obtained by letting
r
Thus we obtain
dr
r2
r0
M r JM ³
r0
JM ³
r
dr
r2
…………………
------------------- 47
107
Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential
.
17.4 General case of computing surface integrals
47
In the following section we compute the surface integral for a general case. This section is slightly
formal and more difficult. It is worth studying this section if you are not in a hurry and did not have
difficulty with the preceding subject matter. Decide according to your preferences.
I prefer to skip section 17.4 for the time being and want to proceed
------------------- 55
I prefer to study section 17.4.
Then study in the textbook
17.4 General case of computing surface
integrals
Textbook pages 470–474
------------------- 48
.
ª 1 1º
»
¬ r0 r ¼
M r JM «
107
Now we let r0 grow to infinity to obtain the result of the foregoing frame.
M1
JM
1
r
Further problems may arise if we change signs. Do not underestimate the problems related to signs. It
is always advisable to calculate meticulously.
------------------- 48
108
Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential
.
A rectangular plane A is defined by the points
P0 (0,0,0) , P1 (4,0,0) , and P3 (4,0,0) .
Thus it lies in the x-z plane.
G
An inhomogeneous vector field is given by F
Compute
48
(0,2 x,0).
G G
³ FdA = ……..
Solution found
Explanation or hints wanted
------------------- 54
------------------- 49
.
Now we will treat the second case and let the potential of the gravitational field vanish
for the surface of the earth.
ª 1 1º
We calculated before: M r JM « »
¬ r0 r ¼
The height z above the surface is given by r
Thus: M z 108
r0 z
M z =……………………….
------------------- 49
109
Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential
.
In this case we have an inhomogeneous field. The field is quite simple, because it has
G
only one component in the y-direction. Remember the field was given by F (0,2x,0)
G G
To obtain I = ³ F ˜ dA
49
G
G
we determine F and dA
A
F
………..
G
dA …….
I = …………..
Solutions found
Further help and explanations
------------------- 52
------------------- 50
.
ª1
1 º
»
¬ r0 r0 z ¼
M r JM «
109
For places near the surface we state z r0 . Applying this approximation we write:
1
1
1
| ...........................
r0 z
§
z · r0
r0 ¨¨1 ¸¸
© r0 ¹
We insert this into the equation above and obtain:
M z JM >............................@
------------------- 50
110
Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential
.
G
G
We have to determine F and dA .
G
Difficulties may arise in determining dA . Plane A lies in the x-z plane, as was pointed out
in frame 48. Thus, the surface element vector points in the y-direction.
50
The amount of a differential surface element for this plane
is given by dA = dx ˜ dz.
The surface element vector in the y-direction with amount
dxdz is given by
G
dA
G
G
Remember: F is given to be F
(..............)
(.....................)
51
------------------- .
Approximation:
M z JM
1
§
z·
r0 ¨¨1 ¸¸
r
0 ¹
©
|
1ª
zº
«1 »
r0 ¬ r0 ¼
110
z
2
r0
For the case of the earth with mass M and radius rE this corresponds to the potential we used in
previous frames. There we used the expression
M z 0,0
g ˜ z
g˜z
Both expressions are identical if we define g appropriately. Calculate and define:
g
……………………
------------------- 51
111
Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential
.
G
dA (0, dx ˜ dz ,0)
G
F (0,2x,0)
51
G G
We want to determine the flow I = ³ F ˜ dA
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
a
A
Now we can calculate the dot product in the integral using the results obtained:
G G
I = ³ F ˜ dA = ³ (0,2 x,0) ˜ (0, dxdz,0)
A
A
³ .. …………….….
A
------------------- 52
.
g
JM
r0
111
2
------------------- 52
112
Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential
.
G G
³ FdA ³ 2 xdxdz
A
52
A
This integral must be calculated for plane A.
Written correctly this is a double integral. Write this integral as a double integral and insert the limits
given by our plane A:
A³ 2 x ˜ dx ˜ dz
........... ........
³ ³ 2 x ˜ dx ˜ dz
x ...... z .....
------------------- 53
.
At the end of this chapter we recapitulate:
112
G
Definition of divergence of a vector field F :
G
div F
……………………=………
------------------- 53
113
Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential
.
4
I
3
53
³ 2 xdx ˜ dz ³ ³ 2 xdx ˜ dz
x 0z 0
A
You can calculate the double integral. It was explained in chapter 13 “Multiple integrals.”
In case of difficulties you should repeat that chapter, at least section 13.2 “multiple integrals with
constant limits.”
I
³ 2 xdx ˜ dz
A
4
3
³ ³ 2 xdx ˜ dz =……………………………………
x 0z 0
------------------- 54
.
G
div F
wFx wFy wFz
wx
wy
wz
G G
’˜F
113
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
G
Calculate the divergence of the given vector field F :
G
F
§ x3 y3 z 2 ·
¨¨ , , ¸¸
2¹
© 3 3
G
div F =………………….
Distribution of sources and sinks:
…………………………………………………………………………………………………………
…………………………………………………………………………………………………………
…………………………………………………………………………………………………………
------------------- 54
114
Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential
.
I
G
G
³ F ˜ dA
48
54
A
Congratulations. You have mastered some difficult reasoning.
------------------- 55
.
G
div F
x2 y2 z
114
No sinks or sources for:
G
div F 0 :
Thus, we obtain for this condition 0 x 2 y 2 z
or z x 2 y 2 . This represents a paraboloid
of revolution around the z-axis.
The inner space consists of sinks. The outer space
consists of sources.
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
Give Gauss’s theorem:
……………………………………………..=………………………………
------------------- 55
115
Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential
.
Before we finish this section we should recapitulate.
G
In a vector field F we have a quadratic plane with an area of 2.
The position is sketched below.
G
Give the surface element vector A .
55
G
A =……….
------------------- 56
.
³ div
G
FdV
V
Volume integral
G
G
³ F ˜ dA
115
Surface integral. l
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
G
Give the definition of the curl of a vector field F :
G
rot F
G G
’u F
…………………………….
------------------- 56
116
Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential
.
G
A (0, 2 , 2)
G
You may have factorized the root: A
2 (0,1,1)
56
Calculate for this plane A the flow of three
vector fields:
F1
(0,6,0)
I1
................
F2
(0,2,1)
I2
.............
F3
(6,0,0)
I3
....................
------------------- 57
.
G
rot F
§ GFz GFy ·
¸
¨
Gz ¸
¨ Gy
¨ GF GF ¸
¨ x z¸
Gx ¸
¨ Gz
¨ GFy GF ¸
x ¸¸
¨¨
G
x
Gy ¹
©
116
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
Given a flow of water
G
V
z
2
G
rot V
, 0, 0
………………..
------------------- 57
117
Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential
.
I1
6˜ 2
I2
3˜ 2
I3
0
57
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
G
Given a vector field with radial symmetry j . The origin coincides with the origin of the coordinate
system.
G
The amount of j is constant.
G
Calculate the flow I of j through a sphere with radius R. The center of the sphere lies in the origin of
the coordinate system.
I =…………….
------------------- 58
.
rot z 2 ,0,0
0,2 z,0
117
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
Give Stoke’s theorem:
………………………..=……………………….
------------------- 58
118
Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential
.
I
G
G
³ j ˜ dA
G
4SR 2 j
58
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
Calculate the flow ) of a vector field
G
F (1,0,0)
Through the sketched cuboid with the edges
a = 6, b = 1, c = 3
) =…………………………………
------------------- 59
.
G
³ divFdV
V
Volume integral
G
G
G
³ F ˜ dA
³ rotF ˜ dA
A
Surface integral
Surface integral
G
³ F ˜ ds
C A
Line integral
The theorems of Gauss and Stokes are worth understanding and memorizing in order to avoid
difficulties when applying them in physics.
You have reached the end of chapter 17.
You have now made considerable progress.
59
118
Chapter 17 Vector analysis: Surface integrals, Divergence, Curl, and Potential
.
)
0
59
____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
G
Given F
(0.5, 0, 0.5)
Calculate the flow for three planes:
G
A1 = (1, 1, 0)
I 1 =…………
A2 = (1, 0, 1)
I 2 =……………….
G
A3 = (1, 1, 1,)
I 3 =………………
------------------- 60
Please continue on page 1
(bottom half)
60