ROCHESTER PHY 217 - Lecture 5B - More Vector Calculus (19 pages)

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Lecture 5B - More Vector Calculus



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Lecture 5B - More Vector Calculus

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Lecture Notes


Pages:
19
School:
University of Rochester
Course:
Phy 217 - E M I

Unformatted text preview:

Today in Physics 217 more vector calculus z Vector derivatives in curvilinear coordinate systems spherical and cylindrical coordinates The Dirac delta function r r y x z z s z s y x 13 September 2002 Physics 217 Fall 2002 1 Curvilinear coordinates Coordinate 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 2 Spherical coordinates The location of a point P can be defined by specifying the following three parameters Radius r distance of P from the origin Polar angle angle between the position vector of P and the z axis Like 90o latitude Azimuthal angle angle between the projection of the position vector P and the x axis Like longitude 13 September 2002 z r y x Physics 217 Fall 2002 3 Spherical coordinates continued The Cartesian coordinates of P are related to the spherical coordinates as follows x r sin cos r x 2 y 2 z2 y r sin sin arctan z r cos arctan y x The unit vectors of spherical coordinate systems are not constant their direction changes when the position of point P changes 13 September 2002 x2 y 2 z x Physics 217 Fall 2002 z r y z y x 4 Spherical coordinates continued In Cartesian coordinates an infinitesimal displacement from point P is equal to z r dl x dx y dy z dz In spherical coordinates an infinitesimal displacement from point P is equal to r sin d dl r dr rd r y x where r is parallel to r is perpendicular to r and lies in the r z plane is perpendicular to this plane and and point in the direction of increasing and 13 September 2002 Physics 217 Fall 2002 5 Example transformation of unit vectors Griffiths problem 1 37 Express the



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