Coordinate System

VECTOR REPRESENTATION 3 PRIMARY COORDINATE SYSTEMS: RECTANGULAR CYLINDRICAL SPHERICAL Choice is based on symmetry of problem Examples: Sheets - RECTANGULAR Wires/Cables - CYLINDRICAL Spheres - SPHERICAL

Orthogonal Coordinate Systems: (coordinates mutually perpendicular) Spherical Coordinates Cylindrical Coordinates Cartesian Coordinates P (x,y,z) P (r, Θ, Φ) P (r, Θ, z) x y z P(x,y,z) θ z r x y z P(r, θ, z) θ Φ r z y x P(r, θ, Φ) Page 108 Rectangular Coordinates

Cartesian Coordinates P(x,y,z) Spherical Coordinates P(r, θ, Φ) Cylindrical Coordinates P(r, θ, z) x y z P(x,y,z) θ z r x y z P(r, θ, z) θ Φ r z y x P(r, θ, Φ)

Coordinate Transformation Cartesian to Cylindrical (x, y, z) to (r,θ,Φ) (r,θ,Φ) to (x, y, z)

Cartesian to Cylindrical Vectoral Transformation Coordinate Transformation

Cartesian to Spherical (x, y, z) to (r,θ,Φ) (r,θ,Φ) to (x, y, z)

Cartesian to Spherical Vectoral Transformation Coordinate Transformation

Page 109 x y z Z plane y plane x plane x1x1 y1y1 z1z1 AxAx AyAy Unit vector properties Vector Representation Unit (Base) vectors A unit vector a A along A is a vector whose magnitude is unity

Page 109 x y z Z plane y plane x plane x1x1 y1y1 z1z1 AxAx AyAy AzAz Vector representation Magnitude of A Position vector A Vector Representation

x y z AxAx AyAy AzAz Dot product: Cross product: Back Cartesian Coordinates Page 108

x z y VECTOR REPRESENTATION: UNIT VECTORS Unit Vector Representation for Rectangular Coordinate System The Unit Vectors imply : Points in the direction of increasing x Points in the direction of increasing y Points in the direction of increasing z Rectangular Coordinate System

r  z P x z y VECTOR REPRESENTATION: UNIT VECTORS Cylindrical Coordinate System The Unit Vectors imply : Points in the direction of increasing r Points in the direction of increasing  Points in the direction of increasing z

Base Vectors A1A1 ρ radial distance in x-y plane Φ azimuth angle measured from the positive x-axis Z Cylindrical Coordinates Pages Back ( ρ, Φ, z) Vector representation Magnitude of A Position vector A Base vector properties

Dot product: Cross product: B A Back Cylindrical Coordinates Pages

VECTOR REPRESENTATION: UNIT VECTORS Spherical Coordinate System r  P x z y  The Unit Vectors imply : Points in the direction of increasing r Points in the direction of increasing  Points in the direction of increasing 

Spherical Coordinates Pages Back (R, θ, Φ) Vector representation Magnitude of A Position vector A Base vector properties

Dot product: Cross product: Back B A Spherical Coordinates Pages

RECTANGULAR Coordinate Systems CYLINDRICAL Coordinate Systems SPHERICAL Coordinate Systems NOTE THE ORDER! r, , zr, ,  Note: We do not emphasize transformations between coordinate systems VECTOR REPRESENTATION: UNIT VECTORS Summary

METRIC COEFFICIENTS 1. Rectangular Coordinates: When you move a small amount in x-direction, the distance is dx In a similar fashion, you generate dy and dz Unit is in “meters”

Back Cartesian to Cylindrical Transformation Page 115