Darryl Michael/GE CRD Fields and Waves Lesson 2.1 VECTORS and VECTOR CALCULUS

VECTORS Today’s Class will focus on: vectors - description in 3 coordinate systems vector operations - DOT & CROSS PRODUCT vector calculus - AREA and VOLUME INTEGRALS

VECTOR NOTATION VECTOR NOTATION: Rectangular or Cartesian Coordinate System x z y Dot Product Cross Product Magnitude of vector (SCALAR) (VECTOR)

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

VECTOR REPRESENTATION: CYLINDRICAL COORDINATES Cylindrical representation uses: r, , z UNIT VECTORS: Dot Product (SCALAR) r  z P x z y

VECTOR REPRESENTATION: SPHERICAL COORDINATES r  P x z y  Spherical representation uses: r, ,  UNIT VECTORS: Dot Product (SCALAR)

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

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 

VECTOR REPRESENTATION: UNIT VECTORS Summary RECTANGULAR Coordinate Systems CYLINDRICAL Coordinate Systems SPHERICAL Coordinate Systems NOTE THE ORDER! r, , zr, ,  Note: We do not emphasize transformations between coordinate systems Do Problem 1

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 Generate: ( dx, dy, dz ) Unit is in “meters” 2. Cylindrical Coordinates: Distance = r d  x y dd r Differential Distances: ( dr, rd , dz )

3. Spherical Coordinates: Distance = r sin  d  x y dd r sin  Differential Distances: ( dr, rd , r sin  d  ) METRIC COEFFICIENTS r  P x z y 

Representation of differential length dl in coordinate systems: rectangular cylindrical spherical

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….

PROBLEM 2 PROBLEM 2AWhat is constant? How is the integration performed? What are the differentials? Representation of differential surface element: Vector is NORMAL to surface

DIFFERENTIALS FOR INTEGRALS Example of Line differentials or Example of Surface differentials or Example of Volume differentials