電磁場(chǎng)與電磁波第講旋度和旋度定理零恒等式亥姆霍茲定理

1、Field and Wave Electromagnetic電磁場(chǎng)與電磁波2014. 31作業(yè)情況1班：人合計(jì)：人情況:2P.2-153Review1. Gradient of a Scalar Field 標(biāo)量場(chǎng)的梯度2. Divergence of a Vector Field 矢量場(chǎng)的散度3. Divergence Theorem 散度定理456電子科技大學(xué)2014考研復(fù)試分?jǐn)?shù)線已公布 7重慶大學(xué)2014考研復(fù)試分?jǐn)?shù)線公布 8東南大學(xué)2014考研復(fù)試分?jǐn)?shù)線公布 910Main topic1. Curl of a Vector Field 矢量場(chǎng)的旋度2. Stokess Theorem

2、斯托克斯定理3. Two Null Identities 兩零恒等式4. Helmholtzs Theorem 亥姆霍茲定理11121).矢量場(chǎng)的環(huán)量矢量場(chǎng)A沿有向閉合曲線 l 的線積分稱為矢量場(chǎng)A沿該曲線的環(huán)量，以表示為0, 0,=0如如何顯示源的分布特性？矢量場(chǎng)的旋度13SMlen稱為矢量A對(duì)于方向en的環(huán)量強(qiáng)度正最大次之零負(fù)最大零注意：在每一點(diǎn)P處都有無窮多的方向環(huán)量，且大小可能不等。en1en2141. Curl of a Vector FieldFlow流量 source; vortex漩渦 source; vortex sinkthe (net ) circulation 凈環(huán)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漩渦源, 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最大凈環(huán)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. 矢量場(chǎng)A的旋度是一個(gè)矢量，其大小為當(dāng)面積趨于零時(shí)單位面積上A的最大凈環(huán)量，其方向?yàn)楫?dāng)面積的取向使得凈環(huán)量呈最大時(shí)，該面積的法線方向。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).旋度概念在某點(diǎn)，旋度矢量的

6、方向是使矢量A具有最大環(huán)量強(qiáng)度(環(huán)路所圍面積的方向)的方向，其大小等于對(duì)該矢量方向的最大環(huán)量強(qiáng)度，記為式中，en為旋度方向上的單位矢量。此式表明，矢量場(chǎng)的旋度大小可以認(rèn)為是包圍單位面積上的閉合曲線的最大環(huán)量，代表了(旋度)源的強(qiáng)度。繞任意方向的方向環(huán)量?旋度與該方向單位矢量的點(diǎn)積（投影）其方向?yàn)楫?dāng)面積的取向使得環(huán)量呈最大時(shí)，該面積的法線方向（右手定則）17直角坐標(biāo)系球坐標(biāo)系柱坐標(biāo)系18旋度運(yùn)算規(guī)則19Example 2-21(P57-58)20A curl-free vector field is called an irrotational無旋場(chǎng) or a conservative fiel

7、d保守場(chǎng). A divergenceless field is called a solenoidal field無散場(chǎng).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.一矢量場(chǎng)的旋度在一開放表面上的面積分，等于該矢量沿包圍該表面的圍線的封閉線積分。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總結(jié)253. Two Null Identities 兩零恒等式 The curl of the gradient of any scalar field is identically zero. 梯度的旋度為零(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 無旋(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. 旋度的散度為零。 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 無旋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 亥姆霍茲定理In previous sections we mentioned that a divergenceless field is solenoidal無散, 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靜電場(chǎng) 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漩渦源. 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 位于某一區(qū)域中的矢量場(chǎng)，當(dāng)其散度、旋度以及邊界上場(chǎng)量的切向分量或法向分量給定后，則該區(qū)域中的矢量場(chǎng)被惟一地確定。 已知散度和旋度代表產(chǎn)生矢量場(chǎng)的源，可見惟一性定理表明，矢量場(chǎng)被其源及邊界條件共同決定的。VSF(r)矢量場(chǎng)的惟一性定理31 若矢量場(chǎng) F(r)

19、 在無限區(qū)域中處處是單值的， 且其導(dǎo)數(shù)連續(xù)有界，源分布在有限區(qū)域V 中，則當(dāng)矢量場(chǎng)的散度及旋度給定后，該矢量場(chǎng) F(r) 可以表示為 式中V zxyr Or r r F(r)亥姆霍茲定理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