电磁场与电磁波英文版
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Divergence Theorem
or
V
divA dV A dS
S
V
Ad V A dS
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.
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
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.
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
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.
In rectangular coordinates, the divergence can be expressed as
Ax Ay Az divA x y z
Using the operator , the divergence can be written as
divA A
Δl
P
P
of scalar l P at point P in the direction of l is defined as
The directional derivative
l
lim
P
( P) ( P)
Δl
Δl 0
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
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.
divA 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.
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.
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
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.
A dS
S源自文库
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.
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.
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
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
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.
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.
or
V
divA dV A dS
S
V
Ad V A dS
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.
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
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.
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
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.
In rectangular coordinates, the divergence can be expressed as
Ax Ay Az divA x y z
Using the operator , the divergence can be written as
divA A
Δl
P
P
of scalar l P at point P in the direction of l is defined as
The directional derivative
l
lim
P
( P) ( P)
Δl
Δl 0
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
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.
divA 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.
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.
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
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.
A dS
S源自文库
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.
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.
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
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
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.
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.