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

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

3. Vector and scalar quantities

4.
What is Vector?

5. Vector Addition and Subtraction
Vector Addition and Subtraction

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

7. Vector Component Breakdown
Vector Component Breakdown

8.
Ex 1 Find

10. Manipulation of vectors
To find a vector from point 1 to 2

11.
Ex2 Find
Solution

12. Rectangular Coordinate System

13. Point Locations in Rectangular Coordinates

14. Differential Volume Element

15. Summary

16. Orthogonal Vector Components

17. Orthogonal Unit Vectors

18. Vector Representation in Terms of Orthogonal Rectangular Components

19. Summary

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

21. Ex3

22. 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)

23. The Dot Product
Commutative Law:

24. 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

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

26. Cross Product

27. 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.

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

29. Orthogonal Unit Vectors in Cylindrical Coordinates

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

31. Summary

32. Point Transformations in Cylindrical Coordinates

33. Dot Products of Unit Vectors in Cylindrical and Rectangular Coordinate Systems

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

35. Then:

36. Finally:

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

38. Constant Coordinate Surfaces in Spherical Coordinates

39. Unit Vector Components in Spherical Coordinates

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

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

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

43. Summary Illustrations

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