电磁场与电磁波第讲旋度和旋度定理零恒等式亥姆霍兹定理

1、Field and Wave Electromagnetic电磁场与电磁波2014. 31作业情况1班：人合计：人情况:2P.2-153Review1. Gradient of a Scalar Field 标量场的梯度2. Divergence of a Vector Field 矢量场的散度3. Divergence Theorem 散度定理456电子科技大学2014考研复试分数线已公布 7重庆大学2014考研复试分数线公布 8东南大学2014考研复试分数线公布 910Main topic1. Curl of a Vector Field 矢量场的旋度2. Stokess Theorem

2、斯托克斯定理3. Two Null Identities 两零恒等式4. Helmholtzs Theorem 亥姆霍兹定理11121).矢量场的环量矢量场A沿有向闭合曲线 l 的线积分称为矢量场A沿该曲线的环量，以表示为0, 0,=0如如何显示源的分布特性？矢量场的旋度13SMlen称为矢量A对于方向en的环量强度正最大次之零负最大零注意：在每一点P处都有无穷多的方向环量，且大小可能不等。en1en2141. Curl of a Vector FieldFlow流量 source; vortex漩涡 source; vortex sinkthe (net ) circulation 净环量S

3、ince circulation as defined in Eq. is a line integral of a dot product, its value obviously depends on the orientation(方向） of the contour C relative to the vector A. In order to define a point function, which is a measure of the strength of a vortex source漩涡源, we must make C very small and orient it

4、 in such a way that the circulation is maximum最大.15In words, Eq. states that the curl of a vector field A, denoted by curl A or A, is a vector whose magnitude is the maximum net circulation最大净环量 of A per unit area as the area tends to zero零 and whose direction is the normal direction of the area whe

5、n the area is oriented to make the net circulation maximum. 矢量场A的旋度是一个矢量，其大小为当面积趋于零时单位面积上A的最大净环量，其方向为当面积的取向使得净环量呈最大时，该面积的法线方向。The component of A in any other direction au is au (A), which can be determined from the circulation per unit area normal to au as the area approaches zero.162).旋度概念在某点，旋度矢量的

6、方向是使矢量A具有最大环量强度(环路所围面积的方向)的方向，其大小等于对该矢量方向的最大环量强度，记为式中，en为旋度方向上的单位矢量。此式表明，矢量场的旋度大小可以认为是包围单位面积上的闭合曲线的最大环量，代表了(旋度)源的强度。绕任意方向的方向环量?旋度与该方向单位矢量的点积（投影）其方向为当面积的取向使得环量呈最大时，该面积的法线方向（右手定则）17直角坐标系球坐标系柱坐标系18旋度运算规则19Example 2-21(P57-58)20A curl-free vector field is called an irrotational无旋场 or a conservative fiel

7、d保守场. A divergenceless field is called a solenoidal field无散场.212. Stokess TheoremThe surface integral of the curl of a vector field over an open surface is equal to the closed line integral of the vector along the contour bounding the surface.一矢量场的旋度在一开放表面上的面积分，等于该矢量沿包围该表面的围线的封闭线积分。Stokess theorem c

8、onverts a surface integral of the curl of a vector to a line integral of the vector, and vice versa反之亦然. It always implies an open surface with a rim. We remind ourselves here that the directions of dl and ds(an) follow the right-hand rule.22Example 2-22(P60)23244. Divergence of a Vector Field5. Div

9、ergence Theorem6. Curl of a Vector Field7. Stokess Theorem总结253. Two Null Identities 两零恒等式 The curl of the gradient of any scalar field is identically zero. 梯度的旋度为零(the existence of V and its first derivatives everywhere is implied here.) 3.1 IDENTITY I A converse statement of Identity I can be made

10、 as follows: if a vector field is curl-free, then it can be expressed as the gradient of a scalar field.ifthen An irrotational 无旋(a conservative保守) vector field can always be expressed as the gradient of a scalar field.26 The divergence of the curl of any vector field is identically zero. 旋度的散度为零。 3

11、.2 IDENTITY II A converse statement of Identity II is as follows: if a vector field is divergenceless, then it can be expressed as the curl of another vector field. A divergenceless field is also called a solenoidal 无旋field. Solenoidal fields are not associated with flow sources of sinks. The net ou

12、rward flux of a solenoidal field through any closed surface is zero, and the flux lines close upon themselves.ifthen274. Helmholtzs Theorem 亥姆霍兹定理In previous sections we mentioned that a divergenceless field is solenoidal无散, and a curl-free field is irrotational. We may classify vector fields in acc

13、ordance with their being solenoidal and/or irrotational 1).solenoidal and irrotationalExample: A static electric field静电场 in a charge-free region3).solenoidal and rotationalExample: A steady magnetic field in a current-carrying conductor2). irrotational but not solenoidal Example: A static electric

14、field in a charge region4).Neither solenoidal nor irrotationalExample: An electric field in a charged medium with a time-varying magnetic field.28The most general vector field then has both a nonzero divergence and a nonzero curl, and can be considered as the sum of a solenoidal field and an irrotat

15、ional field. Helmholtzs Theorem: A vector field ( vector point function) is determined to within an additive constant if both its divergence and its curl are specified everywhere.In an unbounded region we assume that both the divergence and the curl of the vector field vanish at infinity. If the vec

16、tor field is confined within a region bounded by a surface, then it is determined if its divergence and curl throughout the region, as well as the normal component of the vector over the bounding surface, are given. Here we assume that the vector function is single-valued and that its derivatives ar

17、e finite and continuous. 29The divergence of a vector is a measure of the strength of the flow source 通量源and that the curl of a vector is a measure of the strength of the vortex source漩涡源. When the strengths of both the flow source and the vortex source are specified, we except that the vector field

18、 will be determined. Thus, we can decompose a general vector field F into an irrotational (conservative) part Fi and a solenoidal part Fs:Because of Two Null Identities:30 位于某一区域中的矢量场，当其散度、旋度以及边界上场量的切向分量或法向分量给定后，则该区域中的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，可见惟一性定理表明，矢量场被其源及边界条件共同决定的。VSF(r)矢量场的惟一性定理31 若矢量场 F(r)

19、 在无限区域中处处是单值的， 且其导数连续有界，源分布在有限区域V 中，则当矢量场的散度及旋度给定后，该矢量场 F(r) 可以表示为 式中V zxyr Or r r F(r)亥姆霍兹定理32Example 2-23 (p65) Given a vector function : F=ax(3y-c1z)+ay(c2x-2z)-az(c3y+z) (a)Determine the constant c1,c2and c3 if F is irrotational; (b)Determine the scalar potential function V whose negative gradient equals to F33341. Products of Vectors2. Orthogonal Coordinate SystemsCartesian CoordinatesPosition vector:Arbitrary Vector A:summ