1. UNIVERSITI MALAYSIA PERLIS
EKT 241/4: ELECTROMAGNETIC THEORY
UNIVERSITI MALAYSIA PERLIS
CHAPTER 2 – VECTOR ANALYSIS
PREPARED BY: NORDIANA MOHAMAD SAAID

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 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

4. Scalar A scalar is a quantity that has only magnitude E.g. of Scalars:
Time, mass, distance, temperature, electrical potential etc

5. Vector A vector is a quantity that has both magnitude and direction.
E.g. of Vectors:
Velocity, force, displacement, electric field intensity etc.

6. Basic Laws of Vector Algebra
Cartesian coordinate systems

7. Vector in Cartesian Coordinates
A vector in Cartesian Coordinates maybe represented as OR

8. Vector in Cartesian Coordinates
Vector A has magnitude A = |A| to the direction of propagation.
Vector A shown may be represented as

9. Component Vectors
The vector A has three component vectors, which are Ax, Ay and Az.

10. Unit Vectors A unit vector along vector A is;
A vector with magnitude = 1 (unity)
Directed along the coordinate axes in the direction of increasing coordinate values

11. Unit Vectors Vector A can be represented as
The magnitude of A is written as |A| and is calculated by

12. Unit Vectors
Unit vector in the direction of vector A is

13. Example 1: Unit Vector
Specify the unit vector extending from the origin towards the point

14. Solution to Example 1
Construct the vector extending from origin to point G
Find the magnitude of

15. Solution to Example 1
So, unit vector is

16. Equality of vectors
A and B are equal when they have equal magnitudes and identical unit vectors.

17. Vector Algebra
For addition and subtraction of A and B,

18. Example 2: Vector AlgebraIf
Find:
(a) The component of along
(b) The magnitude of
(c) A unit vector along

19. Solution to Example 2
(a) The component of along is
(b)

20. Solution to Example 2
Hence, the magnitude of is:
(c) Let

21. Solution to Example 2
So, the unit vector along is:

22. Position and Distance Vectors
A point P in Cartesian coordinate maybe represented as
The position vector of point P is the vector from origin O to point P

23. Position and Distance Vectors

24. Position and Distance Vectors
If we have two position vectors, and , the third vector or distance vector can be defined as :-

25. Example 3: Position Vectors
Point P and Q are located at and Calculate:
The position vector P
The distance vector from P to Q
The distance between P and Q
A vector parallel to with magnitude of 10

26. Solution to Example 3 (a) (b) (c)
Since is a distance vector, the distance between P and Q is the magnitude of this distance vector.

27. Solution to Example 3 Distance, d (d) Let the required vector be then
Where is the magnitude of

28. Solution to Example 3
Since is parallel to , it must have the same unit vector as or
So,

29. Multiplication of Vectors
When two vectors and are multiplied, the result is either a scalar or vector, depending on how they are multiplied.
Two types of multiplication:
Scalar (or dot) product
Vector (or cross) product

30. Scalar or Dot Product
The dot product of two vectors, and is defined as the product of the magnitude of , the magnitude of and the cosine of the smaller angle between them.

31. Dot Product in Cartesian
The dot product of two vectors of Cartesian coordinate below yields the sum of nine scalar terms, each involving the dot product of two unit vectors.

32. Dot Product in Cartesian
Since the angle between two unit vectors of the Cartesian coordinate system is , we then have:
And thus, only three terms remain, giving finally:

33. Dot Product in Cartesian
The two vectors, and are said to be perpendicular or orthogonal (90°) with each other if;

34. Laws of Dot Product Dot product obeys the following: Commutative Law
Distributive Law

35. Properties of dot product
Properties of dot product of unit vectors:

36. Vector or Cross Product
The cross product of two vectors, and is a vector, which is equal to the product of the magnitudes of and and the sine of smaller angle between them

37. Vector or Cross Product
Direction of is perpendicular (90°) to the plane containing A and B

38. Vector or Cross Product
It is also along one of the two possible perpendiculars which is in direction of advance of right hand screw.

39. Cross product in Cartesian
The cross product of two vectors of Cartesian coordinate:
yields the sum of nine simpler cross products, each involving two unit vectors.

40. Cross product in Cartesian
By using the properties of cross product, it gives
and be written in more easily remembered form:

41. Laws of Vector Product Cross product obeys the following:
It is not commutative
It is not associative
It is distributive

42. Properties of Vector Product
Properties of cross product of unit vectors:
Or by using cyclic permutation:

43. Example 4:Dot & Cross Product
Determine the dot product and cross product of the following vectors:

44. Solution to Example 4
The dot product is:

45. Solution to Example 4
The cross product is:

46. Scalar & Vector Triple Product
A scalar triple product is
A vector triple product is
known as the “bac-cab” rule.

47. Example 5
Given , and
Find (A×B)×C and compare it with A×(B×C).

48. Solution to Example 5
A similar procedure gives

49. Coordinate Systems Cartesian coordinates
Circular Cylindrical coordinates
Spherical coordinates

50. Cartesian coordinates
Consists of three mutually orthogonal axes
and a point in space is denoted as

51. Cartesian Coordinates
Unit vector of in the direction of increasing coordinate value.

52. Cartesian Coordinates
Differential in Length

53. Cartesian Coordinates
Differential Surface

54. Cartesian Coordinates
Differential Surface

55. Cartesian Coordinates
Differential Volume x y z

56. Circular Cylindrical Coordinatesx y z

57. Circular Cylindrical Coordinates
Form by three surfaces or planes:
Plane of z (constant value of z)
Cylinder centered on the z axis with a radius of Some books use the notation .
Plane perpendicular to x-y plane and rotate about the z axis by angle of
Unit vector of in the direction of increasing coordinate value.

58. Circular Cylindrical Coordinates
Differential in Length

59. Circular Cylindrical Coordinates
Increment in length for direction is:
is not increment in length!

60. Circular Cylindrical Coordinates
Differential Surface

61. Circular Cylindrical Coordinates
Differential volume

62. Example 6 A cylinder with radius of and length of Determine:
(i) The volume enclosed.
(ii) The surface area of that volume.

63. Solution to Example 6
(i) For volume enclosed, we integrate;

64. Solution to Example 6
(ii) For surface area, we add the area of each surfaces;

65. Example 7 The surfaces define a closed surface. Find:
The enclosed volume.
The total area of the enclosing surface.

66. Solution to Example 7 (a) The enclosed volume;
Must convert into radians

67. Solution to Example 7
(b) The total area of the enclosed surface:

68. Spherical Coordinates

69. Spherical Coordinates
Point P in spherical coordinate,
 distance from origin. Some books use the notation
 angle between the z axis and the line from origin to point P
 angle between x axis and projection in z=0 plane

70. Spherical Coordinates
Unit vector of in the direction of increasing coordinate value.

71. Spherical Coordinates
Differential in length

72. Spherical Coordinates
Differential Surface

73. Spherical Coordinates
Differential Surface

74. Spherical Coordinates
Differential Volume

75. Spherical Coordinates
However, the increment of length is different from the differential increment previously, where:
 distance between two radius
 distance between two angles
 distance between two radial planes at angles

76. Example 8 A sphere of radius 2 cm contains a volume charge
density ρv given by
Find the total charge Q contained in the sphere.
Solution

77. Summary

78. Cartesian to Cylindrical Transformations
Relationships between (x, y, z) and (r, φ, z) are shown.

79. Cartesian to Spherical Transformations
Relationships between (x, y, z) and (r, θ, Φ) are shown.

80. Cartesian to Spherical Transformations
Relationships between (x, y, z) and (r, θ, Φ) are shown.

81. Example 9 Solution Express vector in spherical coordinates.
Using the transformation relation,
Using the expressions for x, y, and z,
Solution

82. Example 9: contd
Similarly,
Following the procedure, we have
Hence,

83. Transformations Distance d between two points is
Converting to cylindrical equivalents
Converting to spherical equivalents

84. Gradient of a scalar field
Suppose is the temperature at ,
and is the temperature at
as shown.

85. Gradient of a scalar field
The differential distances are the components of the differential distance vector :
However, from differential calculus, the differential temperature:

86. Gradient of a scalar field
But,
So, previous equation can be rewritten as:

87. Gradient of a scalar field
The vector inside square brackets defines the change of temperature corresponding to a vector change in position
This vector is called Gradient of Scalar T.
For Cartesian coordinate, grad T:
The symbol is called the del or gradient operator.

88. Gradient of a scalar field
Gradient operator needs to be scalar quantity.
Directional derivative of T is given by
Gradient operator in cylindrical coordinates:
Gradient operator in spherical coordinates:

89. Example 10 Find the directional derivative of
along the direction and evaluate it at
(1,−1, 2).

90. Solution to Example 10 GradT : We denote l as the given direction,
Unit vector is
and

91. Example 11
Find the gradient of these scalars:
(a)
(b)
(c)

92. Solution to Example 11
(a) Use gradient for Cartesian coordinate:

93. Solution to Example 11
(b) Use gradient for cylindrical coordinate:

94. Solution to Example 11
(c) Use gradient for Spherical coordinate:

95. Divergence of a vector field
Illustration of the divergence of a vector field at point P:
Positive Divergence
Negative Divergence
Zero Divergence

96. Divergence of a vector field
The divergence of A at a given point P is the net outward flux per unit volume:

97. Divergence of a vector field
Vector field A at closed surface S
What is ??

98. Divergence of a vector field
Where,
And, v is volume enclosed by surface S

99. Divergence of a vector field
For Cartesian coordinate:
For Circular cylindrical coordinate:

100. Divergence of a vector field
For Spherical coordinate:

101. Divergence of a vector field
Example: A point charge q
Total flux of the electric field E due to q is

102. Divergence of a vector field
Net outward flux per unit volume i.e the div of E is

103. Example 12
Find divergence of these vectors:
(a)
(b)
(c)

104. Solution to Example 12
(a) Use divergence for Cartesian coordinate:

105. Solution to Example 12
(b) Use divergence for cylindrical coordinate:

106. Solution to Example 12
(c) Use divergence for Spherical coordinate:

107. Divergence Theorem
It states that the total outward flux of a vector field E at the closed surface S is the same as volume integral of divergence of E.
stands for the divergence of vector E

108. Example 13
A vector field exists in the region between two concentric cylindrical surfaces defined by r = 1 and r = 2, with both cylinders extending between z = 0 and z = 5. Verify the divergence theorem by evaluating:
(a)
(b)

109. Solution to Example 13 (a) For two concentric cylinder, the left side:
Where,

110. Solution to Example 13

111. Solution to Example 13
Therefore:

112. Solution to Example 13
(b) For the right side of Divergence Theorem, evaluate divergence of D
So,

113. Curl of a vector field
The curl of vector A is an axial (rotational) vector whose magnitude is the maximum circulation of A per unit area
Curl direction is the normal direction of the area when the area is oriented so as to make the circulation maximum.

114. Curl of a vector field
The circulation of B around closed contour C:

115. Curl of a vector field
Curl of a vector field B is defined as:

116. Curl of a vector field
Curl is used to measure the uniformity of a field
Uniform field, circulation is zero
Non-uniform field, e.g azimuthal field, circulation is not zero

117. Vector identities involving curl
For any two vectors A and B:

118. Curl in Cartesian coordinates
For Cartesian coordinates:

119. Curl in cylindrical coordinates
For cylindrical coordinates:

120. Curl in spherical coordinates
For spherical coordinates:

121. Example 14
Find curl of these vectors:
(a)
(b)
(c)

122. Solution to Example 14
(a) Use curl for Cartesian coordinate:

123. Solution to Example 14
(b) Use curl for cylindrical coordinate

124. Solution to Example 14
(c) Use curl for Spherical coordinate:

125. Solution to Example 14
(c) continued…

126. Stokes’s Theorem
Converts surface integral of the curl of a vector over an open surface S into a line integral of the vector along the contour C bounding the surface S

127. Example 15
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, 0 ≤ z ≤ 3 as shown.

128. Solution to Example 15 Stokes’s theorem states that:
Left-hand side: First, use curl in cylindrical coordinates

129. Solution to Example 15
The integral of over the specified surface S with r = 2 is:

130. Solution to Example 15 Right-hand side:
Definition of field B on segments ab, bc, cd, and da is

131. Solution to Example 15 At different segments, Thus,
which is the same as the left hand side (proved!)

132. Laplacian of a Scalar Laplacian of a scalar V is denoted by .
The result is a scalar.

133. Example 16
Find the Laplacian of these scalars:
(a)
(b)
(c)

134. Solution to Example 16
(a)
(b)
(c)

135. Laplacian of a vector For vector E given in Cartesian coordinates as:
the Laplacian of vector E is defined as:

136. Laplacian of a vector
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