ENE/EIE 325 Electromagnetic Fields and Waves Lecture 2 Vectors and Coordinate Systems

Outline Review of vector operations Orthogonal coordinate systems and change of coordinates

Vector and scalar quantities

http://www.physicstutorials.org What is Vector?

Vector Addition and Subtraction http://www.physicstutorials.org Vector Addition and Subtraction

Vector addition and Subtraction Vector multiplication vector  vector = vector vector  scalar = vector

Vector Component Breakdown http://www.physicstutorials.org Vector Component Breakdown

http://www.physicstutorials.org Ex 1 Find

http://www.physicstutorials.org

Manipulation of vectors To find a vector from point 1 to 2 http://www.colorado.edu/geography/gcraft/notes/coordsys/coordsys_f.html

http://www.physicstutorials.org Ex2 Find Solution

Rectangular Coordinate System

Point Locations in Rectangular Coordinates

Differential Volume Element

Summary

Orthogonal Vector Components

Orthogonal Unit Vectors

Vector Representation in Terms of Orthogonal Rectangular Components

Summary

Vector Expressions in Rectangular Coordinates General Vector, B: Magnitude of B: Unit Vector in the Direction of B:

Ex3

Vector Field We are accustomed to thinking of a specific vector: A vector field is a function defined in space that has magnitude and direction at all points: where r = (x,y,z)

The Dot Product Commutative Law:

Vector Projections Using the Dot Product B • a gives the component of B in the horizontal direction (B • a)a gives the vector component of B in the horizontal direction

Operational Use of the Dot Product Given Find where we have used: Note also:

Cross Product

Operational Definition of the Cross Product in Rectangular Coordinates Therefore: Or… Begin with: where Cross product can be expressed in a matrix form as formal determinant. It can be computed via a cofactor expansion along the first row.

Circular Cylindrical Coordinates Point P has coordinates Specified by P(z)

Orthogonal Unit Vectors in Cylindrical Coordinates

Differential Volume in Cylindrical Coordinates dV = dddz

Summary

Point Transformations in Cylindrical Coordinates

Dot Products of Unit Vectors in Cylindrical and Rectangular Coordinate Systems

Ex4 Transform the vector, into cylindrical coordinates: Start with:

Then:

Finally:

Spherical Coordinates Point P has coordinates Specified by P(r)

Constant Coordinate Surfaces in Spherical Coordinates

Unit Vector Components in Spherical Coordinates

Differential Volume in Spherical Coordinates dV = r2sindrdd

Dot Products of Unit Vectors in the Spherical and Rectangular Coordinate Systems

Ex5 Vector Component Transformation Transform the field, , into spherical coordinates and components

Summary Illustrations

Ex6 Convert the Cartesian coordinate point P(3, 5, 9) to its equivalent point in cylindrical coordinates.