1. 1-1 Engineering Electromagnetics Chapter 1: Vector Analysis

2. 1-2 Vector Addition Associative Law: Distributive Law:

3. 1-3 Rectangular Coordinate System

4. 1-4 Point Locations in Rectangular Coordinates

5. 1-5 Differential Volume Element

6. 1-6 Summary

7. 1-7 Orthogonal Vector Components

8. 1-8 Orthogonal Unit Vectors

9. 1-9 Vector Representation in Terms of Orthogonal Rectangular Components

10. 1-10 Summary

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

12. 1-12 Example

13. 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)

14. 1-14 The Dot Product Commutative Law:

15. 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

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

17. 1-17 Cross Product

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

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

20. 1-20 Orthogonal Unit Vectors in Cylindrical Coordinates

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

22. 1-22 Summary

23. 1-23 Point Transformations in Cylindrical Coordinates

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

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

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

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

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

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

30. 1-30 Constant Coordinate Surfaces in Spherical Coordinates

31. 1-31 Unit Vector Components in Spherical Coordinates

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

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

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

35. 1-35 Summary Illustrations