Dr. Hugh Blanton ENTC 3331

Dr. Blanton - ENTC Orthogonal Coordinate Systems 2 Fields and Waves VECTORS and VECTOR CALCULUS

Dr. Blanton - ENTC Orthogonal Coordinate Systems 3 VECTORS Today’s Class will focus on: vectors - description in 3 coordinate systems vector operations - DOT & CROSS PRODUCT vector calculus - AREA and VOLUME INTEGRALS

Dr. Blanton - ENTC Orthogonal Coordinate Systems 4 VECTOR REPRESENTATION 3 PRIMARY COORDINATE SYSTEMS: RECTANGULAR CYLINDRICAL SPHERICAL Choice is based on symmetry of problem Examples: Sheets - RECTANGULAR Wires/Cables - CYLINDRICAL Spheres - SPHERICAL

Dr. Blanton - ENTC Orthogonal Coordinate Systems 5 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

Dr. Blanton - ENTC Orthogonal Coordinate Systems 6 -Parabolic Cylindrical Coordinates (u,v,z) -Paraboloidal Coordinates (u, v, Φ) -Elliptic Cylindrical Coordinates (u, v, z) -Prolate Spheroidal Coordinates (ξ, η, φ) -Oblate Spheroidal Coordinates (ξ, η, φ) -Bipolar Coordinates (u,v,z) -Toroidal Coordinates (u, v, Φ) -Conical Coordinates (λ, μ, ν) -Confocal Ellipsoidal Coordinate (λ, μ, ν) -Confocal Paraboloidal Coordinate (λ, μ, ν)

Dr. Blanton - ENTC Orthogonal Coordinate Systems 7 Parabolic Cylindrical Coordinates

Dr. Blanton - ENTC Orthogonal Coordinate Systems 8 Paraboloidal Coordinates

Dr. Blanton - ENTC Orthogonal Coordinate Systems 9 Elliptic Cylindrical Coordinates

Dr. Blanton - ENTC Orthogonal Coordinate Systems 10 Prolate Spheroidal Coordinates

Dr. Blanton - ENTC Orthogonal Coordinate Systems 11 Oblate Spheroidal Coordinates

Dr. Blanton - ENTC Orthogonal Coordinate Systems 12 Bipolar Coordinates

Dr. Blanton - ENTC Orthogonal Coordinate Systems 13 Toroidal Coordinates

Dr. Blanton - ENTC Orthogonal Coordinate Systems 14 Conical Coordinates

Dr. Blanton - ENTC Orthogonal Coordinate Systems 15 Confocal Ellipsoidal Coordinate

Dr. Blanton - ENTC Orthogonal Coordinate Systems 16 Confocal Paraboloidal Coordinate

Dr. Blanton - ENTC Orthogonal Coordinate Systems 17 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)

Dr. Blanton - ENTC Orthogonal Coordinate Systems 18 VECTOR NOTATION VECTOR NOTATION: Rectangular or Cartesian Coordinate System x z y Dot Product Cross Product Magnitude of vector (SCALAR) (VECTOR)

Dr. Blanton - ENTC Orthogonal Coordinate Systems 19 Cartesian Coordinates Page 109 x y z Z plane y plane x plane x1x1 y1y1 z1z1 AxAx AyAy AzAz ( x, y, z) Vector representation Magnitude of A Position vector A Base vector properties

Dr. Blanton - ENTC Orthogonal Coordinate Systems 20 x y z AxAx AyAy AzAz Dot product: Cross product: Back Cartesian Coordinates Page 108

Dr. Blanton - ENTC Orthogonal Coordinate Systems 21 VECTOR REPRESENTATION: CYLINDRICAL COORDINATES Cylindrical representation uses: r, , z UNIT VECTORS: Dot Product (SCALAR) r  z P x z y

Dr. Blanton - ENTC Orthogonal Coordinate Systems 22 VECTOR REPRESENTATION: SPHERICAL COORDINATES r  P x z y  Spherical representation uses: r, ,  UNIT VECTORS: Dot Product (SCALAR)

Dr. Blanton - ENTC Orthogonal Coordinate Systems 23 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

Dr. Blanton - ENTC Orthogonal Coordinate Systems 24 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

Dr. Blanton - ENTC Orthogonal Coordinate Systems 25 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 

Dr. Blanton - ENTC Orthogonal Coordinate Systems 26 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

Dr. Blanton - ENTC Orthogonal Coordinate Systems 27 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”

Dr. Blanton - ENTC Orthogonal Coordinate Systems 28 Cartesian Coordinates Differential quantities: Length: Area: Volume: Page 109

Dr. Blanton - ENTC Orthogonal Coordinate Systems 29 METRIC COEFFICIENTS 2. Cylindrical Coordinates: Distance = r d  x y dd r Differential Distances: ( dr, rd , dz )

Dr. Blanton - ENTC Orthogonal Coordinate Systems Spherical Coordinates: Distance = r sin  d  x y dd r sin  Differential Distances: ( dr, rd , r sin  d  ) r  P x z y  METRIC COEFFICIENTS

Dr. Blanton - ENTC Orthogonal Coordinate Systems 31 Representation of differential length dl in coordinate systems: rectangular cylindrical spherical METRIC COEFFICIENTS

Dr. Blanton - ENTC Orthogonal Coordinate Systems 32 AREA INTEGRALS integration over 2 “delta” distances dx dy Example: x y AREA == 16 Note that: z = constant In this course, area & surface integrals will be on similar types of surfaces e.g. r =constant or  = constant or  = constant et c….

Dr. Blanton - ENTC Orthogonal Coordinate Systems 33 Representation of differential surface element: Vector is NORMAL to surface SURFACE NORMAL

Dr. Blanton - ENTC Orthogonal Coordinate Systems 34 DIFFERENTIALS FOR INTEGRALS Example of Line differentials or Example of Surface differentials or Example of Volume differentials

Dr. Blanton - ENTC Orthogonal Coordinate Systems 35 Base Vectors A1A1 r radial distance in x-y plane Φ azimuth angle measured from the positive x-axis Z Cylindrical Coordinates Pages Back ( r, θ, z) Vector representation Magnitude of A Position vector A Base vector properties

Dr. Blanton - ENTC Orthogonal Coordinate Systems 36 Dot product: Cross product: B A Back Cylindrical Coordinates Pages

Dr. Blanton - ENTC Orthogonal Coordinate Systems 37 Cylindrical Coordinates Differential quantities: Length: Area: Volume: Pages

Dr. Blanton - ENTC Orthogonal Coordinate Systems 38 Spherical Coordinates Pages Back (R, θ, Φ) Vector representation Magnitude of A Position vector A Base vector properties

Dr. Blanton - ENTC Orthogonal Coordinate Systems 39 Dot product: Cross product: Back B A Spherical Coordinates Pages

Dr. Blanton - ENTC Orthogonal Coordinate Systems 40 Spherical Coordinates Differential quantities: Length: Area: Volume: Pages Back

Dr. Blanton - ENTC Orthogonal Coordinate Systems 41 Back Cartesian to Cylindrical Transformation Page 115