13 September 2002 Physics 217, Fall 2002 1Today in Physics 217: more vector calculus Vector derivatives in curvilinear coordinate systems: spherical and cylindrical coordinates. The Dirac delta function.xyzrθφˆrˆθˆφxyzsφzˆzˆsˆφ13 September 2002 Physics 217, Fall 2002 2Curvilinear coordinatesCoordinate systems: Cartesian coordinates: used to describe systems without any apparent symmetry. Curvilinear coordinates: used to describe systems with symmetry. We will often find spherical symmetry or axial symmetry in the problems we will do this semester, and will thus use• Spherical coordinates• Cylindrical coordinates There are other curvilinear coordinate systems (e.g. ellipsoidal) that have special virtues, but we won’t get to use them this semester.13 September 2002 Physics 217, Fall 2002 3Spherical coordinatesThe location of a point Pcan be defined by specifying the following three parameters: Radius r: distance of Pfrom the origin. Polar angle θ: angle between the position vector of Pand the zaxis. (Like 90o–latitude.) Azimuthal angle φ: angle between the projection of the position vector Pand the xaxis. (Like longitude.)xyzrθφ13 September 2002 Physics 217, Fall 2002 4Spherical coordinates (continued)The Cartesian coordinates of Pare related to the spherical coordinates as follows:The unit vectors of spherical coordinate systems are not constant: their direction changes when the position of point Pchanges.sin cosxr θφ=sin sinyrθφ=coszr θ=222.rxyz=++()arctanyxφ=()22arctanxyzθ= +xyzxyzrθφ13 September 2002 Physics 217, Fall 2002 5Spherical coordinates (continued)In Cartesian coordinates, an infinitesimal displacement from point Pis equal toIn spherical coordinates, an infinitesimal displacement from point Pis equal towhere is parallel to r, is perpendicular to and lies in the r-zplane, is perpendicular to this plane, and and point in the direction of increasing θand φ.ˆˆˆddxdydz=++lx y zˆˆˆsinddrrd r dθθφ=+ +lr θφˆrˆθˆrˆφˆφˆθxyzrθφˆrˆθˆφ13 September 2002 Physics 217, Fall 2002 6Example: transformation of unit vectorsGriffiths problem 1.37: Express the spherical-coordinate unit vectors in terms of the Cartesian ones.(Worked out on blackboard.)We getxyzrθφˆrˆθˆφˆˆˆˆsin cos sin sin cosˆˆˆˆcos cos cos sin sinˆˆˆsin cosθφ θφ θθφ θφ θφφ=++=+−=− +rxyzθ xyzφ xy13 September 2002 Physics 217, Fall 2002 7Spherical coordinates (continued)In a Cartesian coordinates, an infinitesimal volume element around point Pis equal toIn a spherical coordinates, an infinitesimal volume element around point Pis equal toddxdydzτ=2sinrddldldlrdrddθφτθθφ==xyzrθφdτ13 September 2002 Physics 217, Fall 2002 8Spherical coordinates (continued)In Cartesian coordinates, an infinitesimal area element on a plane containing point PisIn spherical coordinates, the infinitesimal area element on a spherethrough point Pisxyzrθφdaˆ, orˆ, orˆ.d dxdydydzdzdx===azxy2ˆˆsinddldlr ddθφθθφ==ar r13 September 2002 Physics 217, Fall 2002 9Vector derivatives in spherical coordinatesWhat if you want to express a vector derivative in spherical coordinates? (Or someone asks you to, on a test…)Start from the Cartesian-coordinate version, and first use the chain rule to transform the derivatives, e.g.Use the coordinate definitions to reduce the remaining derivatives and eliminate all Cartesian coordinates, e.g.Transform the unit vectors, as we did earlier today.Then multiply the whole mess out and simplify. This is tedious, and takes hours, but is instructive and highly recommended, even though it’s not on the homework explicitly. See also Appendix A in Griffiths.()xTTrT TTxrx x xθφθφ∂∂∂∂∂∂∂== + +∂∂∂∂∂∂∂∇222222sin cosrxxxyzxx rxyzθφ∂∂=++= ==∂∂++13 September 2002 Physics 217, Fall 2002 10Vector derivatives in spherical coordinates (continued)The following operations will be encountered frequently enough that they’re even written on the inside front cover of Griffiths:()()()()() ()()() ()22222211ˆˆˆsin11 1sinsin sin111ˆˆsinsin sin1ˆ11sinsinrrrTT TTrr rrv v vrr rrvv vrvrrrrv vrrTTTrrrrrθφφθ φθθθφθθθ θφθθθ φ θφθθθθθ∂∂ ∂=+ +∂∂ ∂∂∂ ∂⋅= + +∂∂ ∂ ∂∂ ∂∂×= − + − ∂∂ ∂∂ ∂∂+−∂∂∂∂ ∂ ∂ ∇= +∂∂ ∂ ∂ r θφvvrθφ∇∇∇222 21sinTr θφ∂+∂13 September 2002 Physics 217, Fall 2002 11Cylindrical coordinatesSpherical coordinates are useful mostly for spherically symmetric situations. In problems involving symmetry about just one axis, cylindrical coordinates are used:The radius s: distance of P from the z axis.The azimuthal angle φ: angle between the projection of the position vector P and the x axis. (Same as the spherical coordinate of the same name.)The z coordinate: component of the position vector P along the zaxis. (Same as the Cartesian z).xyzPsφz13 September 2002 Physics 217, Fall 2002 12Cylindrical coordinates (continued)The Cartesian coordinates of Pare related to the cylindrical coordinates byAgain, the unit vectors of cylindrical coordinate systems are not constant; their direction changes when the position of point Pchanges.cossinxsyszzφφ===()22arctansxyyxzzφ=+==xyzPsφzxyz13 September 2002 Physics 217, Fall 2002 13Cylindrical coordinates (continued)In cylindrical coordinates: unit vectors infinitesimal displacement infinitesimal volume element infinitesimal area elementˆˆ ˆddsrddzφ=+ +ls φ zd rdrd dzτφ=xyzsφzˆzˆsˆφˆˆ ˆcos sinˆˆ ˆsin cosˆˆφφφφ=+=− +=sx yφ xyzzˆˆ (top of cylinder), (cylinder wall).d sdsd sd dzφφ==az s13 September 2002 Physics 217, Fall 2002 14Cylindrical coordinates (continued)The more common vector derivatives, in cylindrical coordinates:()()()()()() ()()()22222 21ˆˆˆ111ˆˆ1ˆ11szzszsTTTTsss zsv v vss s zvvsvvsz zssv vssTTTTsss sszφφφφφφφφ∂∂∂=+ +∂∂∂∂∂∂⋅= + +∂∂∂∂∂ ∂∂×= − + −∂∂ ∂∂∂∂+−∂∂∂∂ ∂ ∂∇= + +∂∂∂∂φ zvv φz∇∇∇13 September 2002 Physics 217, Fall 2002 15The Dirac delta functionIn problem 1.16, in this week’s homework, you will show that thefollowing vector function:has divergence