1、Chap. 4: Vector Calculus矢量分析簡(jiǎn)介Differentiation of VectorsVector Differential Operatorsshi.xqChap. 4: Vector Calculus1Contents4.1 Basic Vector Algebra(Chapter 7, MMPE)4.2 Differentiation & Integration of Vectors4.3 Space Curves & Surfaces (空間的曲面)(4.2, 4.3: Chapter 10.1-10.5, MMPE)4.4 Vector Op

2、erators(Chapter 10.6-10.9, MMPE)ü Gradient of scalar標(biāo)量; Divergence of vector; Curl of vectorü Vector Operator formulaeü Cylindrical and Spherical polar coordinates4.5 Divergence theoremshi.xqChap. 4: Vector Calculus24.1 Basic Vector AlgebraVectorin Cartesian basis set|𝒊 | = &#

3、120783;𝒊 , 𝒋 , 𝐚𝐧𝐝 𝒌 , 相互垂直，為矢量unit vector投影大小、模shi.xqChap. 4: Vector Calculus3Vector Multiplication4.1 Basic Vector AlgebraMultiplication of VectorsExample:1) Scalar product點(diǎn)乘、標(biāo)量乘積平行乘積為標(biāo)量平行、三對(duì)shi.xqChap. 4: Vector CalculusVector Multiplication4.1 Basi

4、c Vector Algebra面積2) Vector product叉乘、矢量乘積力矩畫(huà)垂直 寫(xiě)乘積為矢量垂直、六對(duì)容易記憶shi.xqChap. 4: Vector Calculus5VectorMultiplication4.1 Basic Vector Algebra3) Scalar triple product體積Rotationalinvariant 旋轉(zhuǎn)不變性旋轉(zhuǎn)對(duì)稱(chēng)性輪換性𝐚 x 𝐛 = 𝐛 x 𝐚= 𝐜 (𝐚 x 𝐛)shi.xqChap. 4: Vector C

5、alculus6Differentiation4.2 Vector Differential4.2 Differentiation & Integration of Vectors譬如：把u理解為時(shí)間tExample:度shi.xqChap. 4: Vector Calculus7Differentiation4.2 Vector DifferentialDifferentiation withPolar coordinatesCartesian & Polar Coordinatesare fixed (constant),𝒊 , 𝒋 , &#

6、119834;𝐧𝐝 𝒌 ,time-independent basis vectors.u Polar coordinates are time- dependent (variable of t).For polar coordinates, the basis vectors themselves must also be differentiated.Polar coordinates:constant magnitude 1shi.xqChap. 4: Vector Calculus8Differentiation4.2 Vector D

7、ifferentialVector a(u) with constant magnitudeIf a vector a(u) has a constant magnitude as u varies, then it is perpendicular to the vector da/du. (compare with polar basis)shi.xqChap. 4: Vector Calculus9Differentiation4.2 Vector DifferentialDifferentiation: Example 1力矩角動(dòng)量證明：𝑑𝐋 =

8、19827;𝑑𝑡F=mashi.xqChap. 4: Vector Calculus10Differentiation4.2 Vector DifferentialExample 2The position vector rof the small mass sweeps outequal areas in equal timesshi.xqChap. 4: Vector Calculus11橢圓軌道Space Curves4.3 Space Curves & Surfaces4.3.1 Space CurvesExample: the velocity

9、 vector of a particle is a tangent to the curve in space along which the particle moves.切向量parametric equationsIfu can be x, t, or the arc length s etc.An infinitesimal vector displacement along thecurvearc lengthr：從原點(diǎn)算s：在運(yùn)動(dòng)軌跡上shi.xqChap. 4: Vector Calculus12Space Curves4.3 Space Curves & Surfac

10、esExampleshi.xqChap. 4: Vector Calculus13Space Curves4.3 Space Curves & Surfaces切向量dr/ds is a unit tangent vector to C andis denoted by 𝐭r：從原點(diǎn)算𝐭 = dr/dss：在運(yùn)動(dòng)軌跡上The rate the unit tangent 𝐭 changes with respect to s is defined as the curvature = 1/: radius of curvatureprinc

11、ipal normal主法向binormalshi.xq次法向Chap. 4: Vector Calculus144.3 SpaceExample 2= 𝑣𝐭𝐭 = dr/dsshi.xq向心度Space Curves4.3 Space Curves & SurfacesVector functions of several argumentsAn example, the infinitesimal change in an electric field E in moving from a position r to a neighb

12、ouring one r + dr is given byshi.xqChap. 4: Vector Calculus16Surfaces4.3 Space Curves & Surfaces4.3.2 Surfaces (空間的曲面)tangent planeIf譬如：把𝝀理解為時(shí)間t坐標(biāo)曲線shi.xqChap. 4: Vector Calculus17Surfaces4.3 Space Curves & Surfacesinfinitesimal vector displacementelement of areawhere R is the region

13、 in the uv-plane corresponding to the range of parameter values that define the surface.R：uv定義的曲面的區(qū)域shi.xqChap. 4: Vector Calculus18Surfaces4.3 Space Curves & SurfacesAn example for Space Surface𝑢, 𝑣 𝜃, 𝜑19Surfaces4.3 Space Curves & Surfaces𝑢, 𝑣

14、; 𝜃, 𝜑20后面的內(nèi)容4.4 Vector Operators(Chapter 10.6-10.9, MMPE)ü Gradient of scalar標(biāo)量; Divergence of vector; Curl of vectorü Vector Operator formulaeü Cylindrical and Spherical polar coordinatesshi.xqChap. 4: Vector Calculus214.4 Vector OperatorsScalar and Vector fieldsWe n

15、ow turn to the case where a particular scalar or vector quantity is defined not just at a point in space but continuously as a field throughout some region of space R (which is often the whole space).: 空氣的溫度、氣壓的大??；靜電勢(shì): 空氣的速度、電場(chǎng)強(qiáng)度Vector differential operatorsshi.xqChap. 4: Vector Calculus224.4 Vector

16、 OperatorsVector differential operatorsVector differential operators: gradient of a scalar field and thedivergence and curl of a vector fieldCentral to the above three differential operations is the vector operator 𝛁, which is called 𝒅𝒆𝒍 (or sometimes 𝒏⻙

17、8;𝒃𝒍𝒂) and in Cartesian coordinates is defined byThe form of this operator in non-Cartesian coordinatesystems will be discussed latershi.xqChap. 4: Vector Calculus23Gradient4.4 Vector OperatorsGradient of a scalar field𝛻 𝛻 =shi.xqChap. 4: Vector Calculus24Grad

18、ient4.4 Vector OperatorsGradient : geometric meaning如：等溫線(面)shi.xqChap. 4: Vector Calculus25Gradient4.4 Vector OperatorsScalar differential operatorgive the rate of change with distance in the direction 𝐚 of the quantity on which it actsthe infinitesimal change in an electric fieldin moving

19、from r to r+drshi.xqChap. 4: Vector Calculus26Gradient4.4 Vector OperatorsChain rulethe gradient operation also obeys the chain rule as in ordinarydifferential calculusd= d𝜙 d𝜓𝜙𝜓𝑑𝑥d𝜓 𝑑𝑥shi.xqChap. 4: Vector Calculus27Divergence4

20、.4 Vector OperatorsDivergence of a vector field無(wú)散場(chǎng)shi.xqChap. 4: Vector Calculus28Divergence4.4 Vector OperatorsDivergence: interpretationThe divergence can be considered as a quantitative measureof how much a vector field diverges (spreads out分散、發(fā)散)or converges （）at any given point.For example, If

21、the vector field v(x, y, z) describing the localvelocity at any point in a fluid, then · v is equal to the net rate of outflow（流出量）of fluid per unit volume, evaluated at a point (by letting a small volume at that point tend to zero).shi.xqChap. 4: Vector Calculus29Laplacian4.4 Vector OperatorsL

22、aplacian 拉斯算子Now if some vector field a is itself derived from a scalar field via a = , then · a has the form · or, as it is usually written, 2 , where 2 (del squared) is the scalar differential operator2is called the Laplacian of .shi.xqChap. 4: Vector Calculus30Curl4.4 Vector OperatorsCu

23、rl of a vector field無(wú)旋場(chǎng)shi.xqChap. 4: Vector Calculus31Curl4.4 Vector OperatorsCurl: interpretationFor a vector field v(x, y, z) describing the local velocity at anypoint in a fluid, × v is a measure of the angular velocity of the fluid in the neighbourhood of that point.If a small paddle wheel

24、 （葉輪）were placed at various points in the fluid then it would tend to rotate in regions where ×v 0, while it would not rotate in regions where×v = 0 .shi.xqChap. 4: Vector Calculus324.4 Vector OperatorsVector operator formulaeshi.xqChap. 4: Vector Calculus33周一11月11日交作業(yè)與作業(yè)3.4一起交作業(yè) 4.1 prove

25、 that提示：𝑟 =𝑥2 + 𝑦2 + 𝑧2𝐫 = 𝑥𝐢 + 𝑦𝐣 + 𝑧𝐤𝐫 = 𝐫/|𝐫| = 𝐫/𝑟shi.xqChap. 4: Vector Calculus344.4 Vector OperatorsCombinations of grad, div and curl標(biāo)量場(chǎng)的梯度的旋度是0矢量場(chǎng)的旋度的散度是0Laplacians

26、hi.xqChap. 4: Vector Calculus354.4 Vector OperatorsPolar coordinatesPolar coordinatesThe vector operators grad, div, curl and 2 have beendiscussed in Cartesian coordinatesFor many physical situations other coordinate systems are more natural.For example, an isolated charge in space, have spherical s

27、ymmetry and spherical polar coordinates would be the obvious choice.For axisymmetric軸對(duì)稱(chēng)systems, such as fluid flow in a pipe,cylindrical polar coordinates are the natural choice.shi.xqChap. 4: Vector Calculus364.4 Vector OperatorsCylindrical coordinatesCylindrical polar coordinates𝐢 = 𝜕𝐫𝜕𝑥Cylindrical polar coordinates , , zshi.xqChap. 4: Vector Calculus374.4 Vector OperatorsCylindrical coordinatesscale factors直角坐標(biāo)柱坐標(biāo)scale factor比例因子The magnitude ds of the displacement dr is given in cylindrical polar coordinates byshi.xqChap. 4: Vector Cal