电磁场与电磁波课件(电子科大)_FWEJ-ch.01.ppt.pptVIP

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. Directional Derivative GradientThe directional derivative of a scalar at a point indicates the spatial rate of change of the scalar at the point in a certain direction.The directional derivativeof scalar ? at point P in the direction of l is defined as P lThe 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 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 asThe 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 DivergenceThe 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.From physics we know that 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