数学物理方法Chapt 345_review.pptxVIP

下周一（11月25）小测验2（三、四、五章讲过的内容）Chapt 12: Reviewshi.xq@sustc.edu.cnChapters 3, 4, 5: Review主要内容1：矢量分析主要内容2：ODE、PDE复习的内容仅为最基本的内容和必须掌握的内容Chapt 12: Reviewshi.xq@sustc.edu.cn主要内容1：矢量分析1）Basic Vector Algebra: 直角坐标表示，点乘、叉乘2）Differentiation of Vectors：直角坐标、极坐标（对坐标矢量微分），多变量矢量微分公式3） Vector Differential OperatorsGradient; Divergence; CurlVector Operator formulae：Cylindrical and Spherical polar coordinatesChapt 12: Reviewshi.xq@sustc.edu.cnVector in Cartesian basis set??unit vector投影大小、模?模的点乘定义Chapt 12: Reviewshi.xq@sustc.edu.cn点乘、叉乘1) Scalar product可用来求矢量的模、可用来证明极坐标基矢的正交归一性垂直点乘为0，正交平行点乘为1，归一2) Vector productChapt 12: Reviewshi.xq@sustc.edu.cn3) Scalar triple product体积轮换性x?Chapt 12: Reviewshi.xq@sustc.edu.cnExample:Cartesian Polar Coordinatesare fixed (constant), ?Polar coordinates are time-dependent (variable of t), the basis vectors themselves must also be differentiated.Polar coordinates:constant magnitude 1 Chapt 12: Reviewshi.xq@sustc.edu.cnDifferentiation of VectorsVector a(u) with constant magnitude If a vector a(u) has a constant magnitude as u varies, then it is perpendicular to the vector da/du. (compare with polar basis)Chapt 12: Reviewshi.xq@sustc.edu.cnDifferentiation of Vector of several argumentsAn example, the infinitesimal change in an electric field E in moving from a position r to a neighbouring one r + dr is given byChapt 12: Reviewshi.xq@sustc.edu.cnVector Differential OperatorsGradient geometric meaning如：等温线(面)Chain rule…?Chapt 12: Reviewshi.xq@sustc.edu.cnDivergence, Laplacian, CurlDivergence The divergence can be considered as a quantitative measure of how much a vector field diverges (spreads out分散、发散) or converges （聚集）at any given point. Laplacian ? · ?φ = ?2φ? × v is a measure of the angular velocity of the fluid in the neighbourhood of that pointCurl会算：标量场的梯度和Laplacian 、矢量场的散度和旋度Chapt 12: Reviewshi.xq@sustc.edu.cnVector operator formulae?会证明? · ?φ = ?2φ?Chapt 12: Reviewshi.xq@sustc.edu.cn作业讲解 1Chapt 12: Reviewshi.xq@sustc.edu.cn4.1Cylindrical polar coo