电磁波英文汉语课件

1、 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.1-6 Flux & DivergenceS d SAThe flux could be positive, negative, or zero. The direction of a closed surface is defined as the outward n

2、ormal 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 positi

3、ve integral, while a sink gives rise to a negative one. 矢量矢量 A 沿某一有向曲面沿某一有向曲面 S 的的面积分面积分称为矢量称为矢量 A 通过通过该有向曲面该有向曲面 S 的通量，以标量的通量，以标量 表示，即表示，即 1-6 矢量场的通量与散度矢量场的通量与散度S d SA通量可为通量可为正正、负负或或零零。 当矢量穿出某个闭合面时，认为该闭合面中存在产当矢量穿出某个闭合面时，认为该闭合面中存在产生该矢量场的生该矢量场的源源；当矢量进入这个闭合面时，认为该闭；当矢量进入这个闭合面时，认为该闭合面中存在汇聚该矢量场的合面中存在汇聚该矢

4、量场的洞洞（或（或汇汇）。）。 闭合的有向曲面的闭合的有向曲面的方向方向通常规定为闭合面的通常规定为闭合面的外外法线方向。法线方向。 当闭合面中有当闭合面中有源源时，矢量通过该闭合面的通量时，矢量通过该闭合面的通量一定为一定为正正；反之，当闭合面中有；反之，当闭合面中有洞洞时，矢量通过该时，矢量通过该闭合面的通量一定为闭合面的通量一定为负负。前述的前述的源源称为称为正源正源，而，而洞洞称为称为负源负源。S d SAS 1-6 矢量场的通量与散度矢量场的通量与散度 From physics we know thatSq 0dSEIf there is positive electric char

5、ge 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 of the vectors through a closed surface can reveal the properties of the so

6、urces and the presence of sources within the closed surface. The flux only gives the total source in a closed surface, and it cannot describe the distribution of the source. For this reason, the divergence is required.1-6 Flux & Divergence（通量与散度）q：free electric charge 0:permittivity of a free spaceV

7、SVd limdiv 0SAA“div” :observation of the word “divergence; V: 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. We introduce the ratio of the flux of the vector field A

8、at the point through a closed surface to the volume enclosed by that surface, and the limit 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 Definition of DivergenceIn rectangula

9、r coordinates, the divergence can be expressed aszAyAxAzyxAdivDefinition of Divergence (Continued)Using the operator , the divergence can be written asAAdiv 当闭合面当闭合面 S 向某点向某点无限无限收缩时，矢量收缩时，矢量 A 通过该闭通过该闭合面合面 S 的通量与该闭合面包围的体积之比的极限称为的通量与该闭合面包围的体积之比的极限称为矢量场矢量场 A 在该点的在该点的散度散度，以，以 div A 表示，即表示，即 0 ddiv limS

10、VVASA式中，式中，div 是英文字是英文字divergence 的缩写；的缩写； V 为闭合面为闭合面 S 包围的体积。包围的体积。散度的定义 0 ddiv limSVVASA上式表明，上式表明，散度是一个标量散度是一个标量，它可理解为通过包围，它可理解为通过包围单位体积单位体积闭合面的通量。闭合面的通量。 直角直角坐标系中散度可表示为坐标系中散度可表示为 div yxzAAAxyzA因此散度可用算符因此散度可用算符 表示为表示为div AA散度的定义（续）SVV d d div SAADivergence Theorem(散度定理)SVV d d SAAor From viewpoint

11、 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 viewpoint of fields, it gives the relationship betwe

12、en the fields in a region and the fields on the boundary of the region. div d dVSV AAS d dVSVAAS或者写为或者写为 从从数学数学角度可以认为散度定理建立了角度可以认为散度定理建立了面面积分和积分和体体积分的关系。积分的关系。 从从物理物理角度可以理解为散度定理建角度可以理解为散度定理建立了立了区域区域 V 中的场和包围区域中的场和包围区域 V 的边界的边界 S 上的场之上的场之间的关系。因此，如果已知区域间的关系。因此，如果已知区域 V 中的场，中的场， 根据根据散散度度定理即可求出边界定理即可求出边

13、界 S 上的场，反之亦然。上的场，反之亦然。散度定理例例 求空间任一点位置矢量求空间任一点位置矢量 r 的散度的散度 。zzyyxx r求求zyxzyxeeer已知已知解解rOxzyxzy练习zyxzyxeee标量场的标量场的梯度梯度 0 ddiv limSVVASAzAyAxAzyx A矢量场的矢量场的散度散度zyxzyxeee算子算子Summary The line integral of a vector field A evaluated along a closed curve is called the circulation of the vector field A aroun

14、d the curve, and it is denoted by , i.e.1-7 Circulation & Curl（环量与旋度）l d lAIf 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；若处；若处处处相反相反，则，则 0 。可见，环量可以用来描述矢量。可见，环量可以用来描述矢量场的场

15、的旋涡旋涡特性。特性。lCirculation 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 th

16、e right hand rule. The circulation is therefore an indication of the intensity of a source. Il0 d lB 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.1-7 Circulation & Curl（Example）SlSd lim curlmax

17、 0nlAeAWhere 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 c

18、url 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.AcurlCurl In rectangular coordinates, the curl can be expressed by t

19、he matrix aszyxzyxAAAzyxeeeA curlor by using the operator asAA curlCurl（Continued） 旋度旋度是一个矢量。以符号是一个矢量。以符号 curl A 表示矢量表示矢量 A 的的旋度，其旋度，其方向方向是使矢量是使矢量 A 具有具有最大最大环量强度的方向，环量强度的方向，其其大小大小等于对该矢量方向的最大环量等于对该矢量方向的最大环量强度强度，即，即 maxn0 dcurl limlSSAlAe式中式中 curl 是旋度的英文字是旋度的英文字；en 为为最大环量强度的方最大环量强度的方向上的单位矢量，向上的单位矢量，S

20、为闭合曲线为闭合曲线 l 包围的面积。包围的面积。 矢量场的旋度大小可以认为是包围单位面积的矢量场的旋度大小可以认为是包围单位面积的闭合曲线上的闭合曲线上的最大最大环量。环量。 en1en2en旋度直角直角坐标系中，旋度可用矩阵表示为坐标系中，旋度可用矩阵表示为 curl xyzxyzxyzAAAeeeA或者或者curl AA 无论梯度、散度或旋度都是无论梯度、散度或旋度都是微分运算微分运算，它们表示，它们表示场在场在某点某点附近的变化特性。因此，附近的变化特性。因此，梯度、散度及旋度梯度、散度及旋度描述的是场的描述的是场的点点特性或称为特性或称为微分微分特性特性。 函数的函数的连续性连续性是

21、可微的必要条件。因此在场量发是可微的必要条件。因此在场量发生生不连续不连续处，也就处，也就不存在不存在前述的梯度、散度或旋度。前述的梯度、散度或旋度。 旋度（续）Stokes Theorem （斯托克斯定理） lS d d) curl(lASAlS d d)(lASAor 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 establi

22、shes 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 propert

23、ies 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.Summary例例 试证任何矢量场试证任何矢量场 A 均满足下列等式均满足下列等式 ()d dVSV AAS式中，式中，S 为包围体积为包围体积 V 的闭合表的闭合表面。此式又称为面。此式又称为矢量矢量旋度定理，旋度定理

24、，或或矢量矢量斯托克斯定理。斯托克斯定理。证证ACACCAAC)(设设 C 为任一为任一常常矢量，则矢量，则SVAne练习（见（见P406公式）公式）根据散度定理，上式左端根据散度定理，上式左端VVVV d d)(ACAC ()d ( ) dVSVC ACAS (d ) dSSC ASCAS那么对于任一体积那么对于任一体积 V，得，得 ()d dVSV CACAS求得求得ACACCAAC)(练习（续） ()d dVSV AAS The field with null-divergence is called solenoidal field (or called divergence-free

25、 field 无散场), and the field with null-curl is called irrotational field (or called lamellar field).1-8 Solenoidal & Irrotational Fields The divergence of the curl of any vector field A must be zero, i.e.0)(Awhich shows that a solenoidal field can be expressed in terms of the curl of another vector fi

26、eld, or that a curly field must be a solenoidal field. 1-8 Solenoidal & Irrotational Fields 0)(AProof:S1S21.Integrate the left side(use divergence theorem):SdSAdVA)()(V2.Surface S can be divided into S1 and S2,by using Stokes theorem, we obtain:ldlAdSA1)(S1ldlAdSA2)(S20)(VdVASo,0)(Alenen0)(Which sho

27、ws that an irrotational field（无旋场） 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. 1-8 Solenoidal & Irrotational Fields (Continued) 散度处处为散度处处为零零的矢量场称为的矢量场称为无散场无散场，旋度处

28、处，旋度处处为为零零的矢量场称为的矢量场称为无旋场无旋场。 1-81-8 无散场和无旋场无散场和无旋场可以证明可以证明0)(A 上式表明，上式表明，任一矢量场任一矢量场 A 的旋度的散度一定等的旋度的散度一定等于零于零 。因此，任一。因此，任一无散无散场可以表示为另一矢量场的场可以表示为另一矢量场的旋度旋度，或者说，任何，或者说，任何旋度旋度场一定是场一定是无散无散场。场。 上式表明，上式表明，任一标量场任一标量场 的梯度的旋度一定的梯度的旋度一定等于零等于零。因此，任一。因此，任一无旋无旋场一定可以表示为一个场一定可以表示为一个标量场的标量场的梯度梯度，或者说，任何，或者说，任何梯度梯度场一

29、定是场一定是无旋无旋场场。 0)(又可证明又可证明1-81-8 无散场和无旋场（续）无散场和无旋场（续） The first scalar Greens theorem: SV,neSVSnV 2dd)(SVV 2d)(d)(Sorwhere 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

30、 the direction of , the outward normal to the surface S. nne Two scalar fields and exist in the volume V, They satisfy1-9 Greens Theorems (Scalar)格林定理 SVSnnV 22dd)(SVV 22d d)(S The second scalar Greens theorem: 1-9 Greens Theorems (Scalar)格林定理 or1-9 格林定理 (标量) 设任意两个标量场设任意两个标量场 及及，若在区域若在区域 V 中具有连续的二阶偏

31、中具有连续的二阶偏导数，可以证明该两个标量场导数，可以证明该两个标量场 及及 满足下列等式满足下列等式SV,ne2 ()ddVSVSn式中式中S 为包围为包围V 的闭合曲面；的闭合曲面； 为标量场为标量场 在在 S 表面表面的外法线的外法线 en 方向上的偏导数。方向上的偏导数。n根据方向导数与梯度的关系，上式又可写成根据方向导数与梯度的关系，上式又可写成2 ()d() dVSV S上两式称为上两式称为标量第一格林定理标量第一格林定理。22 ()ddVSVSnn22 ()d dVSV S基于上式还可获得下列两式：基于上式还可获得下列两式：上两式称为上两式称为标量第二格林定理标量第二格林定理 1

32、-9 格林定理 (标量) The first vector Greens theorem:SVV d d )()(SQPQPQPwhere 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 V1-9 Greens Theor

33、ems (Vector) The second vector Greens theorem:SVV dd ()(SPQQPQPPQ 设任意两个矢量场设任意两个矢量场 P 与与 Q ，若在区域，若在区域 V 中具有连中具有连续的二阶偏导数，那么，可以证明该矢量场续的二阶偏导数，那么，可以证明该矢量场 P 及及 Q 满足下列等式：满足下列等式： () ()d dVSV PQPQPQS式中式中S 为包围为包围V 的闭合曲面；面元的闭合曲面；面元 dS 的方向为的方向为S 的外的外法线方向。上式称为法线方向。上式称为矢量第一格林定理矢量第一格林定理。 1-9 格林定理 (矢量) 基于上式还可获得下式：

34、基于上式还可获得下式： ()(d dVSV QPPQPQQPS此式称为此式称为矢量第二格林定理矢量第二格林定理。1-9 格林定理 (矢量) all Greens theorems give the relationship between the fields in the volume V and the fields at its boundary S. By using Greens theorem, the solution of the fields in a region can be expanded in terms of the solution of the fields

35、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. 1-9 Greens Theorems (Summary) 格林定理建立了格林定理建立了区域区域 V 中的场与中的场与边界边界 S 上的场上的场之间的关系。因此

36、，利用格林定理可以将之间的关系。因此，利用格林定理可以将区域区域中场的中场的求解问题转变为求解问题转变为边界边界上场的求解问题。上场的求解问题。 格林定理说明了格林定理说明了两种两种标量场或矢量场之间应该满标量场或矢量场之间应该满足的关系。因此，如果已知其中足的关系。因此，如果已知其中一种一种场的分布特性，场的分布特性，即可利用格林定理求解即可利用格林定理求解另一种另一种场的分布特性。场的分布特性。1-9 格林定理 (小结) 1-10 Uniqueness Theorem for Vector Fields For a vector field in a region, if its dive

37、rgence, rotation(Curl), 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 sources of the field. Therefore, the above uniqueness theorem shows

38、 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)0( ,11R1-10 矢量场的惟一性定理 位于某一区域中的矢量场，当其位于某一区域中的矢量场，当其散度散度、旋度旋度以及边以及边界上场量的界上场量的切向切向分量或分量或法向法向分量给定后，则该区域中的

39、分量给定后，则该区域中的矢量场被惟一地确定。矢量场被惟一地确定。 已知散度和旋度代表产生矢量场的源，可见惟一性已知散度和旋度代表产生矢量场的源，可见惟一性定理表明，矢量场被其定理表明，矢量场被其源源及及边界条件边界条件共同决定。共同决定。VSF(r)tn FFFF和 及或1-11 Helmholtzs Theorem )()()(rArrFVVd)(41)(rrrFrV)()(Vd41rrrFrAwhere 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 asV0)( 1)1R|(|rF The properties of the divergence and the curl of a vector field are among the mos