Dr Hugh Blanton ENTC 3331 Fields and Waves

Dr. Hugh Blanton ENTC 3331

Fields and Waves VECTORS and VECTOR CALCULUS Darryl Michael/GE CRD

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 3331 - Orthogonal Coordinate Systems 3

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 3331 - Orthogonal Coordinate Systems 4

Orthogonal Coordinate Systems: (coordinates mutually perpendicular) z P(x, y, z) Cartesian Coordinates P (x, y, z) y x Rectangular Coordinates z z P(r, θ, z) Cylindrical Coordinates P (r, Θ, z) x r θ y z Spherical Coordinates P (r, Θ, Φ) θ r x Φ Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems P(r, θ, Φ) y 5 Page 108

-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 3331 - Orthogonal Coordinate Systems 6

Parabolic Cylindrical Coordinates Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 7

Paraboloidal Coordinates Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 8

Elliptic Cylindrical Coordinates Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 9

Prolate Spheroidal Coordinates Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 10

Oblate Spheroidal Coordinates Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 11

Bipolar Coordinates Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 12

Toroidal Coordinates Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 13

Conical Coordinates Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 14

Confocal Ellipsoidal Coordinate Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 15

Confocal Paraboloidal Coordinate Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 16

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) z Cylindrical Coordinates P(r, θ, z) z P(r, θ, z) x θ r Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems y 17

VECTOR NOTATION: Rectangular or Cartesian Coordinate System z Dot Product (SCALAR) Cross Product y (VECTOR) x Magnitude of vector Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 18

Cartesian Coordinates ( x, y, z) Vector representation z z 1 Z plane Magnitude of A x plane Az ne la yp y 1 Position vector A x 1 Ax Ay y x Base vector properties Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 19 Page 109

Cartesian Coordinates z Dot product: Az Cross product: Ax y Ay x Dr. Blanton - ENTC 3331 - Orthogonal Back Coordinate Systems 20 Page 108

VECTOR REPRESENTATION: CYLINDRICAL COORDINATES Cylindrical representation uses: r , f , z UNIT VECTORS: Dot Product (SCALAR) z r P z x f y Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 21

VECTOR REPRESENTATION: SPHERICAL COORDINATES Spherical representation uses: r , q , f UNIT VECTORS: Dot Product (SCALAR) z P q r x f y Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 22

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 Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 23

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 Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 24

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 Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 25

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 Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 26

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 Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 27

Cartesian Coordinates Differential quantities: Length: Area: Volume: Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 28 Page 109

METRIC COEFFICIENTS 2. Cylindrical Coordinates: y Distance = r df Differential Distances: ( dr, rdf, dz ) df r x Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 29

METRIC COEFFICIENTS 3. Spherical Coordinates: y Distance = r sinq df Differential Distances: ( dr, rdq, r sinq df ) z df P q r sinq x r x Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems f y 30

METRIC COEFFICIENTS Representation of differential length dl in coordinate systems: rectangular cylindrical spherical Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 31

AREA INTEGRALS • integration over 2 “delta” distances dy dx Example: y AREA = 6 2 = 16 Note that: z = constant 3 7 x In this course, area & surface integrals will be on similar types of surfaces e. g. r =constant or f = constant or q = constant et c…. Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 32

SURFACE NORMAL Representation of differential surface element: Vector is NORMAL to surface Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 33

DIFFERENTIALS FOR INTEGRALS Example of Line differentials or or Example of Surface differentials or Example of Volume differentials Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 34

Cylindrical Coordinates ( r, θ, z) r 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 Dr. Blanton - ENTC 3331 - Orthogonal. Back Coordinate Systems 35 Pages 109 -112

Cylindrical Coordinates Dot product: B A Cross product: Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems Back 36 Pages 109 -111

Cylindrical Coordinates Differential quantities: Length: Area: Volume: Pages 109 -112 Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems 37

Spherical Coordinates Vector representation (R, θ, Φ) Magnitude of A Position vector A Base vector properties Dr. Blanton - ENTC 3331 - Orthogonal Coordinate Systems Back 38 Pages 113 -115