Transcript Document

Chapter 1 Vector Analysis
Gradient梯度, Divergence散度,
Rotation, Helmholtz’s Theory
1.
2.
3.
4.
5.
6.
7.
8.
Directional Derivative方向导数 & Gradient
Flux通量 & Divergence
Circulation环量 & Curl旋度
Solenoidal无散 & Irrotational无旋 Fields
Green’s Theorems
Uniqueness唯一性 Theorem for Vector Fields
Helmholtz’s Theorem
Orthogonal正交 Curvilinear曲线 Coordinate坐标
1. Directional Derivative & Gradient
The directional derivative of a scalar at a point indicates the spatial
rate of change of the scalar at the point in a certain direction.
l
Δl P 
P


of scalar 
l P
at point P in the direction of l is defined as
The directional derivative

l
 lim
P
Δl 0
 ( P)   ( P)
Δl
The gradient is a vector. The magnitude幅度 of the gradient of a
scalar field at a point is the maximum directional derivative at the
point, and its direction is that in which the directional derivative will
be maximum.
In rectangular coordinate system直角坐标系, the gradient of a
scalar field  can be expressed as
grad   e x



 ey
 ez
x
y
z
Where “grad” is the observation of the word “gradient”.
In rectangular coordinate system, the operator算符  is denoted as



  ex
 ey
 ez
x
y
z
Then the grad of scalar field  can be denoted as
grad   
2. Flux & Divergence
The surface integral面积分 of the vector field A evaluated over a
directed surface S is called the flux through the directed surface S,
and it is denoted by scalar , i.e.
   A  dS
S
The flux could be positive, negative, or zero.
A source in the closed surface produces a positive integral, while a
sink gives rise to a negative one.
The direction of a closed surface is defined as the outward normal on
the closed surface. Hence, if there is a source in a closed surface, the flux
of the vectors must be positive; conversely, if there is a sink, the flux of
the vectors will be negative.
The source  a positive source; The sink  a negative source.
From physics we know that

S
E  dS 
q
0
If there is positive electric charge in the closed surface, the flux will
be positive. If the electric charge is negative, the flux will be negative.
In a source-free region where there is no charge, the flux through
any closed surface becomes zero.
The flux通量 of the vectors through a closed surface can reveal the
properties of the sources and how the sources existed within the closed
surface.
The flux only gives the total source in a closed surface, and it
cannot describe the distribution 分布of the source. For this reason,
the divergence is required.
We introduce the ratio比率 of the flux of the vector field A at the
point through a closed surface to the volume enclosed by that surface,
and the limit极限 of this ratio, as the surface area is made to become
vanishingly small at the point, is called the divergence of the vector field
at that point, denoted by divA, given by
div A  lim
ΔV 0

S
A  dS
ΔV
Where “div” is the observation of the word “divergence, and V is the
volume closed by the closed surface. It shows that the divergence of a
vector field is a scalar field, and it can be considered as the flux through
the surface per unit volume.
In rectangular coordinates, the divergence can be expressed as
Ax Ay Az
div A 


x
y
z
Using the operator , the divergence can be written as
divA    A
Divergence Theorem
or

V
div A dV   A  dS

V
  Ad V   A  dS
S
S
From the point of view of mathematics, the divergence theorem
states that the surface integral面积分 of a vector function over a closed
surface can be transformed into a volume integral体积分 involving the
divergence of the vector over the volume enclosed by the same surface.
From the point of the view of fields, it gives the relationship between the
fields in a region a区域nd the fields on the boundary边界 of the region.
3. Circulation环量 & Curl旋度
The line integral of a vector field A evaluated along a closed curve
is called the circulation of the vector field A around the curve, and it is
denoted by , i.e.
   A  dl
l
If the direction of the vector field A is the same as that of the line
element dl everywhere along the curve, then the circulation  > 0. If
they are in opposite direction, then  < 0 . Hence, the circulation can
provide a description of the rotational property of a vector field.
From physics, we know that the circulation of the magnetic flux
density B around a closed curve l is equal to the product of the
conduction current I enclosed by the closed curve and the permeability
磁导率 in free space, i.e.

l
B  dl   0 I
where the flowing direction of the current I and the direction of the
directed curve l adhere to the right hand rule. The circulation is
therefore an indication of the intensity of a source.
However, the circulation only stands for the total source, and it is
unable to describe the distribution of the source. Hence, the rotation
is required.
Curl is a vector. If the curl of the vector field A is denoted
by curl A . The direction is that to which the circulation of the vector A
will be maximum, while the magnitude of the curl vector is equal to
the maximum circulation intensity about its direction, i.e.
curl A  en lim
ΔS 0

l
A  dl
max
ΔS
Where en the unit vector at the direction about which the circulation of
the vector A will be maximum, and S is the surface closed by the closed
line l.
The magnitude of the curl vector is considered as the maximum
circulation around the closed curve with unit area.
In rectangular coordinates, the curl can be expressed by the
matrix as
ex

curl A 
x
Ax
ey

y
Ay
ez

z
Az
or by using the operator  as
curl A    A
Stokes’ Theorem

or

S
(curl A)  dS   A  dl
S
(  A)  dS   A  dl
l
l

S
(  A)  dS   A  dl
l
A surface integral can be transformed into a line integral by using
Stokes’ theorem, and vise versa.
From the point of the view of the field, Stokes’ theorem establishes
the relationship between the field in the region and the field at the
boundary of the region.
The gradient, the divergence, or the curl is differential operator.
They describe the change of the field about a point, and may be
different at different points.
They describe the differential properties of the vector field. The
continuity of a function is a necessary condition for its differentiability.
Hence, all of these operators will be untenable where the function is
discontinuous.
4.
Solenoidal无散 & Irrotational 无旋Fields
The field with null-divergence is called solenoidal field (or called
divergence-free field), and the field with null-curl is called irrotational
field (or called lamellar field).
The divergence of the curl of any vector field A must be zero, i.e.
  (  A)  0
which shows that a solenoidal field can be expressed in terms of the
curl of another vector field, or that a curly field must be a solenoidal
field.
The curl of the gradient of any scalar field  must be zero, i.e.
  ( )  0
Which shows that an irrotational field can be expressed in terms
of the gradient of another scalar field, or a gradient field must be
an irrotational field.
5. Green’s Theorems格林定理
The first scalar Green’s theorem:

S
,
V
en
V
(    2 )dV   
S

dS
n
or

V
(   2 )dV   ( )  dS
S
where S is the closed surface bounding the volume V, the second order
partial derivatives of two scalar fields  and  exist in the volume V,
and  is the partial derivative of the scalar  in the direction of en , the
n
outward normal to the surface S.
The second scalar Green’s theorem:

V
(2  2 )dV        dS
S
 
 
2
2
(







)
d
V




dS
V
 S  n
n 
The first vector Green’s theorem:

V
[(  P )  (  Q)  P      Q]dV   P    Q  dS
S
where S is the closed surface bounding the volume V, the direction of
the surface element dS is in the outward normal direction, and the
second order partial derivatives of two vector fields P and Q exist in
the volume V.
The second vector Green’s theorem:

V
[Q  (    P )  P  (    Q]dV   [ P    Q  Q    P ]  dS
S
all Green’s theorems give the relationship between the fields in
the volume V and the fields at its boundary S. By using Green’s
theorem, the solution of the fields in a region can be expanded in
terms of the solution of the fields at the boundary of that region.
Green theorem also gives the relationship between two scalar
fields or two vector fields. Consequently, if one field is known, then
another field can be found out based on Green theorems.
6. Uniqueness Theorem for Vector Fields
For a vector field in a region, if its divergence, rotation, and
the tangential切向 component or the normal法向 component at
the boundary are given, then the vector field in the region will be
determined uniquely.
The divergence and the rotation of a vector field represent the
sources of the field. Therefore, the above uniqueness theorem shows
that the field in the region V will be determined uniquely by its
source and boundary condition.
The vector field in an unbounded space is uniquely determined
only by its divergence and rotation if

1
, (  0)
1
R
7. Helmholtz’s Theorem
If the vector F(r) is single valued everywhere in an open space,
its derivatives are continuous, and the source is distributed in a
limited region V  , then the vector field F(r) can be expressed as
F (r )   (r )    A(r )
where
1
  F (r )
 (r ) 
dV 


V
4π
r  r
| F ( r ) |
1
(ε  0)
1
R
1
  F ( r  )
A( r ) 
dV 


V
4π
r  r
A vector field can be expressed in terms of the sum of an
irrotational无旋 field and a solenoidal 无散field.
The properties of the divergence and the curl of a vector field are
among the most essential in the study of a vector field.
8. Orthogonal Curvilinear Coordinates正交曲线坐标系
Rectangular coordinates (x, y, z)
z
z = z0
ez
x = x0
ex
O
x
P0
ey
y = y0
y
Cylindrical coordinates (r,  , z)柱坐标系
The relationships between the
variables r, , z and the variables x,
z
y, z are
x  r cos 
z = z0
y  r sin 
ez
P0
e
zz
er
r = r0
x
O
0
and
 = 0
y
r
x2  y 2
y
  arctan
x
zz
Spherical coordinates (r, ,  )球坐标系
The relationships between the
z
=0
variables r, ,  and the variables x,
y, z are
x  r sin  cos 
0
er
r =
r0
x
P0
r0
O
0
e
=0
y  r sin  sin 
z  r cos
e
and
y
r  x2  y 2  z 2
x2  y 2
  arctan
z
y
  arctan
x
The relationships among the coordinate components of the
vector A in the three coordinate systems are
 Ar   cos 
  
 A    sin 
 A   0
 z
sin 
cos 
0
 Ar   sin  cos 
  
 A   cos  cos 
 A    sin 
 
 Ar   sin 
  
 A   cos 
 A   0
 
0
0
1
 Ax 
 
 Ay 
A 
 z
sin  sin 
cos  sin 
cos 
0 cos  
0  sin  
1
0 
 Ar 
 
 A 
A 
 z
cos  
 sin  
0 
 Ax 
 
 Ay 
A 
 z
Questions
*
In rectangular coordinate system, a vector
A  ae x  be y  ce z
where a, b, and c are constants. Is A a constant vector？
*
In cylindrical coordinate system, a vector
A  ae r  be  ce z
where a, b, and c are constants. Is A is a constant vector?
*
In spherical coordinate system, a vector
A  ae r  be  ce
If A is a constant vector, how about a, b, and c?