1. Fundamentals of Applied Electromagnetics
Chapter 2 - Vector Analysis

2. Chapter Objectives Operations of vector algebra
Dot product of two vectors
Differential functions in vector calculus
Divergence of a vector field
Divergence theorem
The curl of a vector field
Stokes’s theorem

3. Chapter Outline 2-1) Basic Laws of Vector Algebra
Orthogonal Coordinate Systems
Transformations between Coordinate Systems
Gradient of a Scalar Field
Divergence of a Vector Field
Curl of a Vector Field
Laplacian Operator
2-2)
2-3)
2-4)
2-5)
2-6)
2-7)

4. 2-1 Basic Laws of Vector Algebra
Vector A has magnitude A = |A| to the direction of propagation.
Vector A shown may be represented as

5. 2-1.1 Equality of Two Vectors
A and B are equal when they have equal magnitudes and identical unit vectors.
For addition and subtraction of A and B,
Vector Addition and Subtraction

6. 2-1.3 Position and Distance Vectors
Position vector is the vector from the origin to point.

7. 2-1.4 Vector Multiplication
3 different types of product in vector calculus:
Simple Product with a scalar
Scalar or Dot Product
where θAB = angle between A and B
+ve
-ve

8. 2-1.4 Vector Multiplication
3. Vector or Cross Product
Cartesian coordinate system relations:
In summary,

9. 2-1.5 Scalar and Vector Triple Products
A scalar triple product is
A vector triple product is
known as the “bac-cab” rule.

10. Example 2.2 Vector Triple Product
Given , , and , find (A× B)× C and compare it with A× (B× C).
A similar procedure gives
Solution

11. 2-2 Orthogonal Coordinate Systems
Orthogonal coordinate system has coordinates that are mutually perpendicular.
Differential length in Cartesian coordinates is a vector defined as
Cartesian Coordinates

12. 2-2.2 Cylindrical Coordinates
Base unit vectors obey right-hand cyclic relations.
Differential areas and
volume in cylindrical coordinates are shown.

13. Example 2.4 Cylindrical Area
Find the area of a cylindrical surface described by
r = 5, 30° ≤ Ф ≤ 60°, and 0 ≤ z ≤ 3
For a surface element with constant r, the surface area is
Solution

14. 2-2.3 Spherical Coordinates
Base unit vectors obey right-hand cyclic relations.
where R = range coordinate sphere radius
Θ = measured from the positive z-axis

15. Example 2.6 Charge in a Sphere
A sphere of radius 2 cm contains a volume charge
density ρv given by
Find the total charge Q contained in the sphere.
Solution

16. 2-3 Transformations between Coordinate Systems
Cartesian to Cylindrical Transformations
Relationships between (x, y, z) and (r, φ, z) are shown.
Relevant vectors are defined as

17. 2-3 Transformations between Coordinate Systems
Cartesian to Spherical Transformations
Relationships between (x, y, z) and (r, θ, Φ) are shown.
Relevant vectors are defined as

18. Example 2.8 Cartesian to Spherical Transformation
Express vector in spherical coordinates.
Using the transformation relation,
Using the expressions for x, y, and z,
Solution

19. Solution 2.8 Cartesian to Spherical Transformation
Similarly,
Following the procedure, we have
Hence,

20. 2-3 Transformations between Coordinate Systems
Distance between Two Points
Distance d between 2 points is
Converting to cylindrical equivalents.
Converting to spherical equivalents.

21. 2-4 Gradient of a Scalar Field
Differential distance vector dl is
Vector that change position dl is gradient of T, or grad.
The symbol ∇ is called the del or gradient operator.

22. 2-4 Gradient of a Scalar Field
Gradient operator needs to be scalar quantity.
Directional derivative of T is given by
Gradient operator in cylindrical and spherical coordinates is defined as
Gradient Operator in Cylindrical and Spherical Coordinates

23. Example 2.9 Directional Derivative
Find the directional derivative of along the direction and evaluate it at (1,−1, 2).
Gradient of T :
We denote l as the given direction,
Unit vector is
and
Solution

24. 2-5 Divergence of a Vector Field
Total flux of the electric field E due to q is
Flux lines of a vector field E is

25. 2-5.1 Divergence Theorem The divergence theorem is defined as
∇ ·E stands for the divergence of vector E.

26. Example 2.11 Calculating the Divergence
Determine the divergence of each of the following vector fields and then evaluate it at the indicated point:
Solution

27. 2-6 Curl of a Vector Field
Circulation is zero for uniform field and not zero for azimuthal field.
The curl of a vector field B is defined as

28. 2-6.1 Vector Identities Involving the Curl
(1) ∇ × (A + B) = ∇× A+∇× B,
(2) ∇ ·(∇ × A) = 0 for any vector A,
(3) ∇ × (∇V ) = 0 for any scalar function V.
Stokes’s theorem converts surface into line integral.
Stokes’s Theorem

29. Example 2.12 Verification of Stokes’s Theorem
A vector field is given by Verify Stokes’s theorem for a segment of a cylindrical surface defined by r = 2, π/3 ≤ φ ≤ π/2, and 0 ≤ z ≤ 3, as shown.
Solution
Stokes’s theorem states that
Left-hand side: Express in cylindrical coordinates

30. Solution 2.12 Verification of Stokes’s Theorem
The integral of ∇ × B over the specified surface S with r = 2 is
Right-hand side:
Definition of field B on segments ab, bc, cd, and da is

31. Solution 2.12 Verification of Stokes’s Theorem
At different segments,
Thus,
which is the same as the left hand side (proved!)

32. 2-7 Laplacian Operator Laplacian of V is denoted by ∇2V.
For vector E given in Cartesian coordinates as
the Laplacian of E is defined as

33. 2-7 Laplacian Operator
In Cartesian coordinates, the Laplacian of a vector is a vector whose components are equal to the Laplacians of the vector components.
Through direct substitution, we can simplify it as