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