Coordinate System VECTOR REPRESENTATION 3 PRIMARY COORDINATE SYSTEMS

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) Cartesian Coordinates z P(x, y, z) Rectangular Coordinates P (x, y, z) y x z z P(r, θ, z) Cylindrical Coordinates P (r, Θ, z) x r θ y z Spherical Coordinates θ r P (r, Θ, Φ) x Φ P(r, θ, Φ) y Page 108

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

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

Coordinate Transformation • Cartesian to Cylindrical Vectoral Transformation

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

Coordinate Transformation • Cartesian to Spherical Vectoral Transformation

Vector Representation z z 1 Z plane Unit (Base) vectors x plane A unit vector a. A along A is a vector whose magnitude is unity ne la yp y 1 x 1 Ax Ay y x Unit vector properties Page 109

Vector Representation z Vector representation z 1 Z plane Magnitude of A x plane Az Position vector A x 1 Ax ne la yp y 1 Ay y x Page 109

Cartesian Coordinates z Dot product: Az Cross product: Ax Ay y x Back Page 108

VECTOR REPRESENTATION: UNIT VECTORS Rectangular Coordinate System z Unit Vector Representation for Rectangular Coordinate System y x 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

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

Cylindrical Coordinates ( ρ, Φ, z) ρ radial distance in x-y plane Φ azimuth angle measured from the positive x-axis A 1 Z Vector representation Base Vectors Magnitude of A Base vector properties Position vector A Back Pages 109 -112

Cylindrical Coordinates Dot product: B Cross product: Back A Pages 109 -111

VECTOR REPRESENTATION: UNIT VECTORS Spherical Coordinate System z P q r x f y The Unit Vectors imply : Points in the direction of increasing r Points in the direction of increasing q Points in the direction of increasing j

Spherical Coordinates Vector representation (R, θ, Φ) Magnitude of A Position vector A Base vector properties Back Pages 113 -115

Spherical Coordinates Dot product: B A Cross product: Back Pages 113 -114

VECTOR REPRESENTATION: UNIT VECTORS Summary RECTANGULAR Coordinate Systems CYLINDRICAL Coordinate Systems SPHERICAL Coordinate Systems NOTE THE ORDER! r, f, z r, q , f Note: We do not emphasize transformations between coordinate systems

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

Cartesian to Cylindrical Transformation Back Page 115