Chapter 1 Vector Analysis Gradient 梯度, Divergence 散度, Rotation, Helmholtz’s Theory 1. Directional Derivative 方向导数 & Gradient 2. Flux 通量 & Divergence 3. Circulation 环量 & Curl 3. Circulation 环量 & Curl 旋度 4. Solenoidal 无散 & Irrotational 无旋 Fields 5. Green’s Theorems 6. Uniqueness 唯一性 Theorem for Vector Fields 7. Helmholtz’s Theorem 8. Orthogonal 正交 Curvilinear 曲线 Coordinate 8. 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. The directional derivative of scalar  at point P in the direction of l is defined as P l  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 Where “grad” is the observation of the word “gradient”. In rectangular coordinate system, the operator 算符  is denoted as Then the grad  of scalar field  can be denoted as

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. 2. Flux & Divergence The flux could be positive, negative, or zero. 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 source in the closed surface produces a positive integral, while a sink gives rise to a negative one.

From physics we know that 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.

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 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

Using the operator , the divergence can be written as Divergence Theorem or 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.

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 环量 & Curl 旋度 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. 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.

Where e n 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 maximum circulation intensity about its direction, i.e.

In rectangular coordinates, the curl can be expressed by the matrix as or by using the operator  as Stokes’ Theorem or

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. They describe the change of the field about a point, and may be different at different points. 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 无散 & Irrotational 无旋 Fields The divergence of the curl of any vector field A must be zero, i.e. 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.

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 格林定理 The first scalar Green’s theorem: S V ,, or 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, the outward normal to the surface S.

The first vector Green’s theorem: 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 scalar Green’s theorem:

The second vector Green’s theorem: 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

7. Helmholtz’s Theorem where A vector field can be expressed in terms of the sum of an irrotational 无旋 field and a solenoidal 无散 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 as 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 x y z = z 0 x = x 0 y = y 0 P0P0 O

Cylindrical coordinates (r, , z) 柱坐标系 y z x P0P0  0 0  = 0 = 0 r = r 0 z = z 0 O The relationships between the variables r, , z and the variables x, y, z are and

Spherical coordinates (r, ,  ) 球坐标系 x z y  =  0  0 0  0 0 r =r0r =r0  =  0 P0P0 O r0r0 The relationships between the variables r, ,  and the variables x, y, z are and

The relationships among the coordinate components of the vector A in the three coordinate systems are

* In rectangular coordinate system, a vector 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? * In cylindrical coordinate system, a vector * In spherical coordinate system, a vector Questions