2012电磁场与电磁波02_矢量场论1_正交坐标系

SchoolofElectronicsandInformationEngineering电磁场与电磁波ElectromagneticFieldsandWaves第一章矢量场论1谢泽明华南理工大学电子与信息学院TEL:13662486310Email:[e-mail]–矢量代数和三种常用的坐标系SchoolofElectronicsandInformationEngineeringSouthChinaUniversityofTechnologySouthChinaUniversityofTechnology内容矢量代数直角坐标系圆柱坐标系球坐标系三种坐标变量的关系三种坐标单位矢量之间的关系SchoolofElectronicsandInformationEngineeringSouthChinaUniversityofTechnologySouthChinaUniversityofTechnology数学是使人类思维走向更高维的桥梁。数学是描述世界的最简洁语言。简洁的语言是深奥理论的源泉。本课程所讨论的矢量是指3维或2维矢量。SchoolofElectronicsandInformationEngineeringSouthChinaUniversityofTechnologySouthChinaUniversityofTechnology矢量代数人类对数的认识过程标量：数字、代数、函数。矢量：2个或3个标量的有序组合。N维矢量：n个标量的有序组合。矩阵：m个n维矢量的有序组合。人的五官感知的世界是三维的，但人的思维是n维。SchoolofElectronicsandInformationEngineeringSouthChinaUniversityofTechnologySouthChinaUniversityofTechnology物理表述：矢量是指既有大小又有方向的量。几何描述：有向线段，即箭头表示方向，长度表示大小。数学表述：单位矢量表示法：坐标表示法：矩阵表示法：矢量表示ˆAAAaˆˆˆxxyyzzAAaAaAa[,,]xyzAAAAˆˆ,1AAAaaAzoyxAxAyAzASchoolofElectronicsandInformationEngineeringSouthChinaUniversityofTechnologySouthChinaUniversityofTechnology矢量运算－线性运算(加减法)加法:矢量加法是矢量的几何和,服从平行四边形规则。CABBACBACˆˆˆxxyyzzAAaAaAaˆˆˆxxyyzzBBaBaBaˆˆˆ()()()xxxyyyzzzCABABaABaABa加法的几何表示：加法的坐标表示：SchoolofElectronicsandInformationEngineeringSouthChinaUniversityofTechnologySouthChinaUniversityofTechnology满足交换律：ABBA满足结合律：()()()()ABCDACBDˆˆˆxxyyzzkAkAakAakAa数乘：SchoolofElectronicsandInformationEngineeringSouthChinaUniversityofTechnologySouthChinaUniversityofTechnology减法：换成加法运算()DABAB逆矢量：和的模相等，方向相反，互为逆矢量。B()BAABCBABDBADBC0推论：任意多个矢量首尾相连组成闭合多边形，其矢量和必为零。SchoolofElectronicsandInformationEngineeringSouthChinaUniversityofTechnologySouthChinaUniversityofTechnology矢量运算－点乘点积或标量积两矢量点积含义一矢量在另一矢量方向上的投影与另一矢量模的乘积，其结果是一标量。如果为单位矢量，则表示矢量在方向的投影。cos(,)(xxyyzTABABABABABABAB矩阵表示）BAˆeˆAeAˆeSchoolofElectronicsandInformationEngineeringSouthChinaUniversityofTechnologySouthChinaUniversityofTechnology运算规律如果与正交，则在直角坐标系中，已知三个坐标轴是相互正交的，即两矢量点积为：(ABBA交换律）)(ABCABAC（分配律）0ABABˆˆˆˆˆˆ0,0,0ˆˆˆˆˆˆ1,1,1xyxzyzxxyyzzaaaaaaaaaaaaˆˆˆˆˆˆ()()xxyyzzxxyyzzxxyyzzABAaAaAaBaBaBaABABABSchoolofElectronicsandInformationEngineeringSouthChinaUniversityofTechnologySouthChinaUniversityofTechnology矢量运算－叉乘叉积或矢量积大小为这两个矢量构成的平行四边形的面积，方向与这两个矢量垂直，且、与符合右手螺旋规则。ˆˆˆxyzxyzxyzaaaABAAABBBsin(,)ABABAB＝ABABABBACABSchoolofElectronicsandInformationEngineeringSouthChinaUniversityofTechnologySouthChinaUniversityofTechnology如果与平行，则运算规律：不服从分配律和结合律AB0AB(ABBA反交换律）)(ABCABAC（分配律）)()()ABCBCACAB（)()()ABCBACCAB（SchoolofElectronicsandInformationEngineeringSouthChinaUniversityofTechnologySouthChinaUniversityofTechnology矢量运算－例题例2-1证明：()()()()()()ABCDACBDADBC证：应用矢量恒等式)()()ABCBCACAB（)()()ABCBACCAB（()()(())(()())()()()()ABCDCDABCADBBDACADBCBDA有得证。SchoolofElectronicsandInformationEngineeringSouthChinaUniversityofTechnologySouthChinaUniversityofTechnology三种常用的坐标系为了描述物理量在空间的位置与分布，必须引入坐标系电磁分析中常用有坐标系有：直角坐标系圆柱坐标系球坐标系根据研究的物体和空间的特点选用不同的坐标系SchoolofElectronicsandInformationEngineeringSouthChinaUniversityofTechnologySouthChinaUniversityofTechnologyyxzxzyzyxaaaaaaaaazyyxxAAAzaaaA直角坐标系三个坐标变量是x，y，zx=常数、y=常数、z=常数的三个曲面（坐标面）为平面正交，为正交坐标系。两个坐标面的交线（坐标曲线）为直线。坐标曲线两两正交（正交坐标系）坐标单位矢量（坐标曲线的切线方向单位矢量）为，指向对应坐标增加的方向。123:::qxqyqz,,xyzaaaSchoolofElectronicsandInformationEngineeringSouthChinaUniversityofTechnologySouthChinaUniversityofTechnology直角坐标系的坐标单位矢量是常矢量，其方向不随点M的位置变化而变化。线元（长度元）矢量线元：带方向的线段dl，大小为长度，方向为线的方向。dl=axdx+aydy+azdz面元矢量面元：带方向的小面积dS=andS，大小为面积，方向为面的法线方向。体元xyzdldxdldydldzdVdxdydzxyzdSdydzdSdxdzdSdxdy0xyzM(x,y,z)azaxaydxdydz0xyzM(x,y,z)azaxaydxdydzdldS=ndSn0xyzM(x,y,z)azaxaydxdydz0xyzM(x,y,z)azaxaydxdydzdxdydzSchoolofElectronicsandInformationEngineeringSouthChinaUniversityofTechnologySouthChinaUniversityofTechnology圆柱坐标系的三个坐标变量r,,zr=常数、=常数、z=常数的三个曲面正交，为正交坐标系坐标曲线为直线和圆圆柱坐标单位矢量为ar,a,az在柱坐标系中，az是常矢量，ar,a都是变矢量，其方向随点M的位置变化而变化。圆柱坐标系aaaaaaaaarzrzzrzzrrAAAaaaA123:0:02:qrqqzSchoolofElectronicsandInformationEngineeringSouthChinaUniversityofTechnologySouthChinaUniversityofTechnology线元（长度元）面元体元rzdldrdlrddldzdVrdrddzrzdSrddzdSdrdzdSrdrdxyzodMaazardrdzxyzodMaazardrdzSchoolofElectronicsandInformationEngineeringSouthChinaUniversityofTechnologySouthChinaUniversityofTechnology球坐标系三个坐标变量是r，，r=常数、=常数、=常数的三个曲面正交，为正交坐标系球坐标单位矢量为ar,a,a坐标单位矢量与位置有关aaaaaaaaarrrAAArraaaA123:0:0:02qrqqSchoolofElectronicsandInformationEngineeringSouthChinaUniversityofTechnologySouthChinaUniversityofTechnology线元（长度元）面元体元sinRdldrdlrddlrd2sindVrdrdd2sinsinRdSrdddSrdrddSrdrdSchoolofElectronicsandInformationEngineeringSouthChinaUniversityofTechnologySouthChinaUniversityofTechnology三种坐标系之间的关系——坐标变量之间的关系直角坐标系与柱坐标系cossinxryrzz222222arctanarcsinarccosrxyyyxxxyxyzzSchoolofElectronicsandInformationEngineeringSouthChinaUniversityofTec