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1、1,Chapter 1 Vector analysis 2). have continuous first partial derivatives. A is also presented by: A=axAx(x,y,z)+ayAy(x,y,z)+azAz(x,y,z) Vector lines are certain curves: at every point on the lines,vectors of the field lie along its tangential line. E.g.: lines of force of electrostatic field, lines
2、 of force of magnetic field, streamlines of velocity field,21,Assuming: P is any point on point on vector lines of the vector field its radius vector is , Then: In RCS, is represented by Vector lines of the vector field meets the differential equation,do it yourself,22,Ex 1:Assuming electric charge
3、q is on the ordinate origin, the electrical field intensity at any point produced by this charge is Where, & are constant and is the position vector of P. Please calculate the equation for vector lines of .,23,Answer ：,24,1.3.2 Flux and divergence of vector field,1. Flux of vector field :vector fiel
4、d :surface area element :unit vector perpendicular to （normal vector） The vector of surface area element is The flux for passing through,25,1.3.2 Flux and divergence of vector field,1. Flux and of vector field :vector field :surface area element :unit vector perpendicular to （normal vector） The flux
5、 for of passing through the whole surface is If S is a closed surface, the flux is,26,1.3.2 Flux and divergence of vector field,Physical significance of equation,tap,sewer,flow out,flow in,Sink,Source,Positive flow,Negative flow,=Net flow = Positive flow + Negative flow,27,1.3.2 Flux and divergence
6、of vector field,2. Divergence of vector field :vector field :closed surface :point in :volume confined in When , the limit adopted is If the limit of the expression is exit, the limit is called:,Divergence of at,28,1.3.2 Flux and divergence of vector field,2. Divergence of vector field :vector field
7、 :closed surface :point in :volume confined in Physical significance of equation In RCS,29,1.3.2 Flux and divergence of vector field,Hamilton operator Then,30,1.3.2 Flux and divergence of vector field,source point, sink point,non-source If , the field is continuous or solenoidal. Divergence Theorem(
8、Gauss Divergence Theorem) The volume integral of divergence equals to the surface integral of the vectors normal component along the closed surface bonding the volume. Assignment: 1.16,P19,31,1.3.3 Circulation and Curl of vector field,1. Circulation of vector field :vector field :close directing cur
9、ve in The line integral of along , is called,Circulation,Physical significance of equation,swirl source,32,2. Curl(Rotation) of vector field :vector field :close directing curve in :surface area element closed by :point in When , the limit adopted is If the limit of the expression is exit, the limit
10、 is called:,Curl or rotation of at along,1.3.3 Circulation and Curl of vector field,33,2. Curl(Rotation) of vector field :vector field :close directing curve in :surface area element closed by :point in Physical significance of equation In RCS,1.3.3 Circulation and Curl of vector field,34,1.3.2 Flux
11、 and divergence of vector field,Hamilton operator Then,35,1.3.2 Flux and divergence of vector field,Rotational,Irrotational or Conservative Stokes Theorem The surface integral of vector s curl along of equals to the line integral of along the boundary of the surface. Assignment: 1.17,P19,36,1.4 Frad
12、ient of scalar field,1.4.1. Isosurface,A surface composed of all the point with the same scalar value in the field is called isosurface. Scalar function , Equation of isosurface：,二. 标量场的梯度,2、梯度的物理意义,37,1.4 Fradient of scalar field,1.4.2. Directional Derivative & its calculation,Where,cos, cos& cos a
13、re the direction cosines of .,38,1.4 Gradient of scalar field,1.4.3. Gradient of scalar field,Hamilton operator,Laplace operator,CCS,SCS:,39,（1） （2）Gradient is the normal vector of isosurface. （3） u 0 F= 0 F=u u is the potential function of the scalar function.,Properties of gradient,Assignment:1.23
14、,P20,40,Integral of gradient,41,Field Source: Divergence Source, Rotational Source Nonrotational field and nondivergent field are two basic vector fields. Every vector field is generated by both or one of the two kind of sources and can be presented by the summation of Nonrotational field and nondivergent.,1.5 Fradient of scalar field,Divergence Source:Scalar Source,Rotational Source:Vector Source,
