1-1 Engineering Electromagnetics Chapter 1: Vector Analysis

1-2 Vector Addition Associative Law: Distributive Law:

1-3 Rectangular Coordinate System

1-4 Point Locations in Rectangular Coordinates

1-5 Differential Volume Element

1-6 Summary

1-7 Orthogonal Vector Components

1-8 Orthogonal Unit Vectors

1-9 Vector Representation in Terms of Orthogonal Rectangular Components

1-10 Summary

1-11 Vector Expressions in Rectangular Coordinates General Vector, B: Magnitude of B: Unit Vector in the Direction of B:

1-12 Example

1-13 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)

1-14 The Dot Product Commutative Law:

1-15 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

1-16 Operational Use of the Dot Product Given Find where we have used: Note also:

1-17 Cross Product

1-18 Operational Definition of the Cross Product in Rectangular Coordinates Therefore: Or… Begin with: where

1-19 Circular Cylindrical Coordinates Point P has coordinates Specified by P(  z) z x y

1-20 Orthogonal Unit Vectors in Cylindrical Coordinates

1-21 Differential Volume in Cylindrical Coordinates dV =  d  d  dz

1-22 Summary

1-23 Point Transformations in Cylindrical Coordinates

1-24 Dot Products of Unit Vectors in Cylindrical and Rectangular Coordinate Systems

1-25 Transform the vector, into cylindrical coordinates: Example

1-26 Transform the vector, into cylindrical coordinates: Start with:

1-27 Transform the vector, into cylindrical coordinates: Then:

1-28 Transform the vector, into cylindrical coordinates: Finally:

1-29 Spherical Coordinates Point P has coordinates Specified by P(r  )

1-30 Constant Coordinate Surfaces in Spherical Coordinates

1-31 Unit Vector Components in Spherical Coordinates

1-32 Differential Volume in Spherical Coordinates dV = r 2 sin  drd  d 

1-33 Dot Products of Unit Vectors in the Spherical and Rectangular Coordinate Systems

1-34 Example: Vector Component Transformation Transform the field,, into spherical coordinates and components

1-35 Summary Illustrations