Chapter 1 Vector Analysis
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Transcript of Chapter 1 Vector Analysis
Chapter 1 Vector Analysis
Gradient, Divergence, Rotation, Helmholtz’s Theory
1. Directional Derivative & Gradient
2. Flux & Divergence
3. Circulation & Curl
4. Solenoidal & Irrotational Fields
5. Green’s Theorems
6. Uniqueness Theorem for Vector Fields
7. Helmholtz’s Theorem
8. Orthogonal Curvilinear Coordinate
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
PP
l lP Δ
)()(lim
0Δ
The directional derivative of scalar
at point P in the direction of l is defined as
Pl
P
l
lΔ P
The gradient is a vector. The magnitude 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.
zyx zy
eeexgrad
zyx zyx
eee
grad
In rectangular coordinate system, the gradient of a scalar field
can be expressed as
Where “grad” is the observation of the word “gradient”.
In rectangular coordinate system, the operator is denoted as
Then the grad of scalar field can be denoted as
The surface integral 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.
2. Flux & Divergence
S
d SA
The flux could be positive, negative, or zero.
The direction of a closed surface is defined as the outward normal o
n the closed surface. Hence, if there is a source in a closed surface, the fl
ux 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.
A source in the closed surface produces a positive integral, while a
sink gives rise to a negative one.
From physics we know that
S
q
0
d
SE
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 a
ny closed surface becomes zero.
The flux of the vectors through a closed surface can reveal the prop
erties of the sources and how the presence of sources within the closed s
urface.
The flux only gives the total source in a closed surface, and it can
not describe the distribution of the source. For this reason, the diverg
ence is required.
VS
V Δ
d limdiv
0Δ
SAA
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
z
A
y
A
x
A zyx
Adiv
We introduce the ratio of the flux of the vector field A at the point th
rough a closed surface to the volume enclosed by that surface, and the li
mit 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, deno
ted by divA, given by
Using the operator , the divergence can be written
as AA div
SV
V
d d div SAA
Divergence Theorem
SV
V
d d SAAor
From the point of view of mathematics, the divergence theorem
states that the surface integral of a vector function over a closed surface
can be transformed into a volume integral 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 and the fields on the boundary of the region.
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.
3. Circulation & Curl
l
d lA
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
in free space, i.e.
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.
Il 0
d lB
However, the circulation only stands for the total source, and it i
s unable to describe the distribution of the source. Hence, the rotatio
n is required.
Sl
S Δ
d lim curl max
0Δn
lAeA
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.
Curl is a vector. If the curl of the vector field A is denoted by
. 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 maxi
mum circulation intensity about its direction, i.e.
Acurl
In rectangular coordinates, the curl can be expressed by the
matrix as
zyx
zyx
AAAzyx
eee
A curl
or by using the operator
as AA curl
Stokes’ Theorem
lS
d d) curl( lASA
lS
d d)( lASAor
A surface integral can be transformed into a line integral by using
Stokes’ theorem, and vise versa.
The gradient, the divergence, or the curl is differential operator. Th
ey describe the change of the field about a point, and may be different a
t different points.
lS
d d)( lASA
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.
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.
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).
4. Solenoidal & Irrotational Fields
The divergence of the curl of any vector field A must be zero, i.e.
0)( A
which shows that a solenoidal field can be expressed in terms of the c
url of another vector field, or that a curly field must be a solenoidal f
ield.
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.
The curl of the gradient of any scalar field must be zero,
i.e.
5. Green’s Theorems
The first scalar Green’s
theorem: S
V
, ne
SVS
nV
2 dd)(
SV
V
2 d)(d)( S
or
where S is the closed surface bounding the volume V, the second order p
artial derivatives of two scalar fields and exist in the volume V, and
is the partial derivative of the scalar in the direction of , the out
ward normal to the surface S.
n
ne
SV
Snn
V
22 dd)(
SV
V
22 d d)( S
The first vector Green’s theorem:
SV
V
d d])()[( SQPQPQP
where S is the closed surface bounding the volume V, the direction of t
he surface element dS is in the outward normal direction, and the seco
nd order partial derivatives of two vector fields P and Q exist in the vo
lume V.
The second scalar Green’s theorem:
The second vector Green’s theorem:
SV
V
d][d]()([ SPQQPQPPQ
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 component or the normal 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 s
ources of the field. Therefore, the above uniqueness theorem shows t
hat 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
)0( ,11
R
7. Helmholtz’s Theorem
)()()( rArrF
V
Vd)(
π4
1)(
rr
rFr V
)()(
V
d
π4
1
rr
rFrAwhere
A vector field can be expressed in terms of the sum of an
irrotational field and a solenoidal field.
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 , then the vector field F(r) can be expressed asV
0)( 1
)1
εR
|(| rF
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
Rectangular coordinates (x, y, z)
z
x
y
z = z0
x = x0
y = y0P0
ze
xeye
O
Cylindrical coordinates (r, , z)
y
z
x
P0
0
= 0
r = r0
z = z0
re
zee
O
The relationships between the
variables r, , z and the variables
x, y, z are
cosrx
sinry
zz
22 yxr
x
yarctan
zz
and
Spherical coordinates (r, , )
x
z
y
= 0
0
0
r = r0
= 0
ere
e
P0
O
r0
The relationships between the
variables r, , and the variables
x, y, z are cossinrx
sinsinry
cosrz
222 zyxr
z
yx 22
arctan
x
yarctan
and
The relationships among the coordinate components of the
vector A in the three coordinate systems are
z
y
x
z
r
A
A
A
A
A
A
100
0cossin
0sincos
z
y
xr
A
A
A
A
A
A
0cossin
sinsincoscoscos
cossinsincossin
z
rr
A
A
A
A
A
A
010
sin0cos
cos0sin
* In rectangular coordinate system, a vector
zyx cba eeeA
where a, b, and c are constants. Is A a constant vector?
where a, b, and c are constants. Is A is a constant vector?
If A is a constant vector, how about a, b, and c?
zr cba eeeA
* In cylindrical coordinate system, a vector
eeeA cba r
* In spherical coordinate system, a vector
Questions