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1、,Chapter 1 Vector Analysis,Gradient, Divergence, Rotation, Helmholtzs Theory,1. Directional Derivative 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, whil
2、e 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
3、 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 total source in a closed surface, and it cannot describe the distribution of the source. For this reason, the di
4、vergence is required.,Where “div” is the observation of the word “divergence, and V is 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
5、divergence can be expressed as,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 vec
6、tor field at that point, denoted by divA, given by,Using the operator , the divergence can be written as,Divergence Theorem,or,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 in
7、tegral 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.,The line integral of a vector field A evaluated along a closed curv
8、e is called the circulation of the vector field A around the curve, and it is denoted by , i.e.,3. Circulation & Curl,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 . Henc
9、e, the circulation can provide a description of the rotational property of a vector field.,From physics, we know that the circulation of the magnetic flux density B 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 s
10、pace, 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.,However, the circulation only stands for the total source, and it is unable to describe the distri
11、bution of the source. Hence, the rotation is required.,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
12、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 circulation intensity about its direction, i.e.,In rectangular coor
13、dinates, the curl can be expressed by the matrix as,or by using the operator as,Stokes Theorem,or,A surface integral can be transformed into a line integral by using Stokes theorem, and vise versa.,The gradient, the divergence, or the curl is differential operator. They describe the change of the fi
14、eld about a point, and may be different at different points.,From the point of the view of the field, Stokes theorem establishes the relationship between the field in the region and the field at the boundary of the region.,They describe the differential properties of the vector field. The continuity
15、 of a function is a necessary condition for its differentiability. Hence, all of these operators will be untenable where the function is discontinuous.,The field with null-divergence is called solenoidal field (or called divergence-free field), and the field with null-curl is called irrotational fie
16、ld (or called lamellar field).,4. Solenoidal & Irrotational Fields,The divergence of the curl of any vector field A must be zero, i.e.,which shows that a solenoidal field can be expressed in terms of the curl of another vector field, or that a curly field must be a solenoidal field.,Which shows that an irrotational field can be expressed in terms of the gradient of another scalar field, or a gradient field must be an irrotational field.,The curl of the gradient of any s
