《电磁场》课程教学课件（PPT讲稿，英文）Chapter 1 Vector Analysis

Chapter 1 Vector AnalysisGradient,Divergence,Rotation,Helmholtz's Theory1.Directional Derivative&Gradient2.Flux & Divergence3.Circulation & Curl4. Solenoidal & Irrotational Fields5. Green's Theorems6. Unigueness Theorem for Vector Fields7. Helmholtz's Theorem8. Orthogonal Curvilinear Coordinate

Chapter 1 Vector Analysis Gradient, Divergence, Rotation, Helmholtz’s Theory 1. Directional Derivative & Gradient 2. Flux & Divergence 3. Circulation & Curl 4. Solenoidal & Irrotational Fields 5. Green’s Theorems 6. Uniqueness Theorem for Vector Fields 7. Helmholtz’s Theorem 8. Orthogonal Curvilinear Coordinate

1.DirectionalDerivative&GradientThe directionalderivative of a scalar at a pointindicates the spatialrate of change of the scalar at the pointin a certaindirectionadThedirectional derivativeof scalar @alPat point P in the direction ofl is definedasAadΦ(P) -Φ(P)Φp lim1N1al△/>0The gradient is a vector. The magnitude of the gradient of ascalar field at a point is the maximum directional derivative at thepoint, and its direction is that in which the directional derivative willbemaximumUM

1. Directional Derivative & Gradient The directional derivativeof a scalar at a point indicates the spatial rate of change of the scalar at the point in a certain direction. l P P l l P Δ ( ) ( ) lim Δ 0    − =   → The directional derivative of scalar  at point P in the direction of l is defined as P l  P l Δl P  The gradient is a vector. The magnitude 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, the gradient of a scalar field @can beexpressedasadadadgrad =2+axOzayWhere"grad"is the observation of the word “gradient"In rectangular coordinate system, the operator is denoted asaaaVOzaxayThen the grad@of scalar field @ can be denoted asgrad @ = V@U7

x y z y z   +   +   =     e e e grad x x y z x y z   +   +    = e e e grad =  In rectangular coordinate system, 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  is denoted as Then the grad of scalar field  can be denoted as

2.Flux&DivergenceThe surfaceintegralofthe vectorfield A evaluated overa directedsurface Sis called the flux through the directed surface S, and itisdenoted by scalar ,i.e.yA.dsThe flux could be positive, negative,or zeroA sourcein the closed surface produces a positiveintegral, while asinkgivesriseto a negativeoneThe direction of a closed surfaceis defined as the outward normalonthe closed surface.Hence, if thereis a sourcein a closed surface,the fluxof the vectors must be positive; conversely,if there is a sink, the flux ofthevectorswill benegativeThe sourcea positive source; The sink- a negative source.uV

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. 2. Flux & Divergence  =  S  A dS 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

FromphysicsweknowthatfE.ds-q80If there is positiveelectric chargein the closed surface,the flux willbe positive.If the electric charge is negative, the flux will be negativeIn a source-free region where there is no charge, the flux throughanyclosedsurfacebecomeszeroThe flux of the vectors through a closed surface can revealtheproperties of the sources and how the presence of sources within theclosed surface.The flux only gives the total source in a closed surface, and itcannot describe the distribution ofthe source.For this reason, thedivergence is required.UV

From physics we know that   = S q 0 d  E S 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 of the vectors through a closed surface can reveal the properties of the sources and how the presence of sources within the closed surface. The flux only gives the totalsource in a closed surface, and it cannot describe the distribution of the source. For this reason, the divergence is required

Weintroduce the ratio of the flux of the vector field A at the pointthrough a closed surfaceto thevolumeenclosed by thatsurface.and thelimitofthis ratio,as the surfaceareais madeto becomevanishinglysmallat the point,is called the divergence of the vectorfield at that point.denoted bydivA,givenbyA.dsdivA = limAVAV->0Where“div"is the observation of the word “divergence, and △Vis thevolume closed by the closed surface. It shows that the divergence of avectorfieldis a scalarfield, and it can be considered as the flux throughthe surfaceperunitvolumeIn rectangular coordinates, the divergence can be expressed as0A.aA.OAdivA:OzaxayUEV

V S V Δ d div lim Δ 0   = → A S A Where “div” is the observation of the word “divergence, and Vis 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 z A y A x Ax y z   +   +   divA = We introduce the ratio 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 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 V, the divergence can be written asdivA=V.ADivergenceTheorem[ divAdV=+ A-dsJ,V.AdV=f,A dsorFromthe point of view of mathematics,the divergence theorem statesthat the surfaceintegralof a vectorfunction over a closed surface canbe transformedinto a volumeintegralinvolving the divergence of thevectoroverthe volumeenclosed by the same surface.From the point ofthe view of fields, it gives the relationshipbetween the fieldsin a regionand thefieldsonthe boundaryof the regionUV

Using the operator , the divergence can be written as divA =  A   =  V S V divAd A dS Divergence Theorem    =  V S V or Ad A dS From the point of view 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 the point of the view of fields, it gives the relationship between the fields in a region and the fields on the boundary of the region

3.Circulation&CurlThe lineintegralof a vector field A evaluated along a closed curveis called the circulation ofthe vector field A around the curve,and it isdenoted by I,i.e.F=f,A·dlIf the direction ofthevectorfield Ais the same as that of thelineelement dl everywhere along the curve, then the circulation T > O. Ifthey are in opposite direction, then F < O . Hence, the circulation canprovide a description of the rotationalproperty of a vector field

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 & Curl  =  l  A dl 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 fluxdensity B around a closed curve lis equal to the product of theconduction current I enclosed by the closed curve and the permeabilityin free space, i.e.f B dl = μo!wherethe flowing direction of the current I and the direction of thedirected curve I adhere to the right hand rule.The circulationistherefore an indicationoftheintensityof a sourceHowever, the circulation only stands for the total source, and itisunableto describethe distributionofthe source.Hence,therotationis required.>

From physics, we know that the circulation of the magnetic flux densityB 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 the right hand rule. The circulation is therefore an indication of the intensity of a source. I l 0 d =   B l However, the circulation only stands for the total source, and it is unable to describe the distributionof the source. Hence, the rotation is required

Curl is a vector If the curl of the vector field A is denotedby curlA . The direction is that to which the circulation of the vector Awill be maximum, while the magnitude of the curl vector is equal tothemaximum circulationintensityaboutits direction,i.eA.dimaxlimcurl A=e.AS4S-0Where e. the unit vector at the direction about which the circulation ofthe vector A will be maximum, and ASis the surface closed by the closedline l.The magnitude of the curl vectoris consideredas the maximumcirculationaroundthe closed curve with unitareaU

S l S Δ d curl lim max Δ 0 n   = → A l A e Where 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 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 circulationintensity about its direction,i.e. curlA