Vector Algebra

1. IBU
International Burch University
Vector algebra
Assist. Prof. Dr. Srđan Lale
MS Mehrija Hasičić

2. Coordinate systems
[Engineering Electromagnetics Sixth Edition, William H. Hayt, Jr. and John A. Buck, McGraw Hill,
2005]
Cartesian coordinate system
Known also as Rectangular coordinate system
Infinitesimal displacement vector
Differential volume element

3. Coordinate systems
Cartesian coordinate system - Vectors
Vector in Cartesian coordinate system
Unit vector
Projection of the vector on x, y, z axes
[Engineering Electromagnetics Sixth Edition,
William H. Hayt, Jr. and John A. Buck, McGraw Hill,
2005]

4. Coordinate systems
Cartesian coordinate system - Vectors
Magnitude of vectors
Unit vector
[Engineering Electromagnetics Sixth Edition, William H. Hayt, Jr. and John A. Buck, McGraw Hill,
2005]

5. Coordinate systems
Cartesian coordinate system - Vectors
Multiplying vectors
Scalar (dot) product
Commutative
[Engineering Electromagnetics Sixth Edition,
William H. Hayt, Jr. and John A. Buck, McGraw Hill, 2005]

6. Coordinate systems
Cartesian coordinate system - Vectors
Multiplying vectors
Vector (cross) product
Not commutative
[Engineering Electromagnetics Sixth Edition, William H. Hayt, Jr. and John A. Buck, McGraw Hill,
2005]

7. Coordinate systems
Circular cylindrical coordinate system
Three coordinate axes (, , z)
Increase by a differential amount per axis = infinitesimal
displacement vector
[Engineering Electromagnetics Sixth Edition, William H. Hayt, Jr. and John A. Buck, McGraw Hill,
2005]

8. Coordinate systems
Circular cylindrical coordinate system
Three coordinate axes (, , z)
Correlation to Cartesian coordinate system
[Engineering Electromagnetics Sixth Edition, William H. Hayt, Jr. and John A. Buck, McGraw Hill,
2005]

9. Coordinate systems
Circular cylindrical coordinate system - Vectors
Multiplication of the vector and a unit vector
Scalar (dot) product
[Engineering Electromagnetics Sixth Edition, William H. Hayt, Jr. and John A. Buck, McGraw Hill,
2005]

10. Coordinate systems
Spherical coordinate system
Three coordinate axes (r, , )
Increase by a differential amount per axis = infinitesimal
displacement vector
[Engineering Electromagnetics Sixth Edition, William H. Hayt, Jr. and John A. Buck, McGraw Hill,
2005]

11. Coordinate systems
[Engineering Electromagnetics Sixth Edition, William H. Hayt, Jr. and John A. Buck, McGraw Hill,
2005]
Spherical coordinate system
Three coordinate axes (r, , )
Increase by a differential amount per axis = infinitesimal
displacement vector

12. Coordinate systems
Spherical coordinate system
[Engineering Electromagnetics Sixth Edition, William H. Hayt, Jr. and John A. Buck, McGraw Hill,
2005]

13. Vector field
Separation vector
A vector from the source point
(where electric charge is located) and field point
Unit vectors of r and r` defined
Cartesian coordinates
[ElectroIntroduction to Electrodynamics Third Edition, David J. Griffiths, Prentice Hall, 1999]

14. Differentials and vector fields
Gradient
Changes in an intensity of electric/magnetic field at particular
points
The change not the same per all directions
Derivative as an estimate of rate change
[Essential Calculus: Early Transcendentals 2nd Edition, James Stewart, Cengage
Learning, 2013]

15. Differentials and vector fields
Gradient
Function of three variables
Partial derivatives and its application
[ElectroIntroduction to Electrodynamics Third Edition, David J. Griffiths, Prentice Hall, 1999]

16. Differentials and vector fields
Gradient
The gradient as vector quantity
Infinitesimal displacements per axes
Scalar product of vectors
Maximum value obtained for
 The gradient points in the direction of maximum increase of a function.
 The magnitude del T gives the slope along the direction of maximum
increase of function.

17. Differentials and vector fields
Gradient
The  operator understood as vector operator acting upon T
Not seen as simple multiplication
Instruction to differentiate
 Application of del on a scalar function T defines the gradient.
 Application of del on a vector function v via scalar product defines the divergence.
 Application of del on a vector function v via vector product defines the curl.

18. Differentials and vector fields
Gradient
 Application of del on a scalar function T defines the gradient.
 Application of del on a vector function v via scalar product defines the divergence.
 Application of del on a vector function v via vector product defines the curl.

19. Differentials and vector fields
Gradient
 Application of del on a scalar function T defines the gradient.
 Application of del on a vector function v via scalar product defines the divergence.
 Application of del on a vector function v via vector product defines the curl.

20. Differentials and vector fields
Divergence
Scalar product of del and vector v
 The divergence is a measure of how much the vector v spreads out from
particular point.

21. Differentials and vector fields
Divergence
Scalar product of del and vector v
 The divergence is a measure of how much the vector v spreads out from
particular point.
[ElectroIntroduction to Electrodynamics Third
Edition,
David J. Griffiths, Prentice Hall, 1999]

22. Differentials and vector fields
Curl
Vector product of del and vector v
 The curl is a measure of how much
the vector v curls around particular point (spinning effect).

23. Differentials and vector fields
Curl
Vector product of del and vector v
 The curl is a measure of how much
the vector v curls around particular point (spinning effect).
[ElectroIntroduction to Electrodynamics Third
Edition,
David J. Griffiths, Prentice Hall, 1999]

24. Differentials and vector fields
Second derivatives
Divergence of gradient
Curl of gradient
Gradient of divergence
Divergence of curl
Curl of curl

25. Vector operations
1. Addition/substraction
 𝐴 + 𝐵 = 𝐵 + 𝐴
 𝐴 – 𝐵 = 𝐴 + (- 𝐵 )
2. Multiplication by scalar
 a(𝐴 + 𝐵) = a 𝐴 + a 𝐵

26. Vector operations (contd.)
3. Dot product of two vectors
𝐴 ∙ 𝐵 = 𝐴𝐵𝑐𝑜𝑠𝜃
(If 𝐴 𝑎𝑛𝑑 𝐵 are perpendicular then 𝐴 ∙ 𝐵 = 0)
4. Cross product of two vectors
𝐴 × 𝐵 = 𝐴𝐵𝑠𝑖𝑛𝜃𝑛
(If 𝐴 𝑎𝑛𝑑 𝐵 are parallel then 𝐴 × 𝐵 = 0)

27. Component form
𝐴 = 𝐴𝑥𝑥 + 𝐴𝑦 𝑦 + 𝐴𝑧 𝑧
1) To add two vectors add like components
2) To multiply by scalar multiply each
component
3) To calculate the dot product, multiply like
components and add
4) To calculate the cross product, form the
determinant whose first row is 𝑥, 𝑦 𝑎𝑛𝑑 𝑧
whose second row is 𝐴 (in component form),
and whose third row is 𝐵.

28. Position, displacement and separation vectors
𝑟 = 𝑥𝑥 + y𝑦 + z𝑧
𝑟 = 𝑥2 + 𝑦2 + 𝑧2
𝑟 =
𝑟
𝑟
=
𝑥𝑥 + y𝑦 + z𝑧
𝑥2 + 𝑦2 + 𝑧2
𝑑𝑙 = 𝑑𝑥𝑥 + 𝑑𝑦 𝑦 + 𝑑𝑧 𝑧

29. Gradient
The gradient is a vector operation which
operates on a scalar function to produce a
vector whose magnitude is the maximum
rate of change of the function at the point
of the gradient and which is pointed in the
direction of that maximum rate of change.

30. Gradient (contd.)

31. Example
Find the gradient of the function
𝑓 𝑥, 𝑦, 𝑧 = 𝑥2 + 𝑦3 + 𝑧4

32. The operator ∇
Nabla or del is not a vector, in the usual sense.
Indeed, it is without specific meaning until we
provide it with a function to act upon.
Furthermore, it does not "multiply" T; rather, it is
an instruction to differentiate what follow

33. The operator ∇ (contd.)
There are three ways ∇ can act:
1. On a scalar function T
∇𝑇 GRADIENT
1. On a vector function 𝑣; via dot product
∇ ∙ 𝑣 DIVERGENCE
1. On a vector function 𝑣; via cross product
∇ × 𝑣 CURL

34. Divergence
Divergence is a vector operator that produces a
scalar field, giving the quantity of a vector
field's source at each point.
More technically, the divergence represents the
volume density of the outward flux of a vector
field from an infinitesimal volume around a
given point.

35. Example
Suppose the functions in Fig. 1.18 are 𝑣𝑎 = 𝑟 = 𝑥𝑥 + 𝑦𝑦 + 𝑧𝑧 , 𝑣𝑏 = 𝑧 and 𝑣𝑧 = 𝑧𝑧. Calculate their
divergences.

36. Curl
Curl is a vector operator that describes
the infinitesimal rotation of a vector
field in three-dimensional Euclidean
space.
At every point in the field, the curl of that
point is represented by a vector. The
attributes of this vector (length and
direction) characterize the rotation at that
point.
A vector field whose curl is zero is called
irrotational

37. Example
Suppose the function sketched in Fig. 1.19a is 𝑣𝑎 = 𝑟 = −𝑦𝑥 + 𝑥𝑦, and that in Fig. 1.19b is 𝑣𝑏 =
𝑟 = 𝑥𝑦. Calculate their curls

38. Second derivatives
Divergence of gradient
Curl of gradient
Gradient of divergence
Divergence of curl
Curl of curl
Notice that is not the same as Laplacian
vector

39. Integral calculus
Line integral
𝑎
𝑏
𝑣 d𝑙 or 𝑣d𝑙 for closed loop
Surface Integral
𝑠
𝑣𝑑𝑎 = 𝑣𝑑𝑎 or 𝑣d𝑎 for closed surface
Volume integral
𝑠
𝑣𝑑𝑉 = 𝑠
𝑣𝑑𝜏 = 𝑣 𝑑𝑥𝑑𝑦 𝑑𝑧

40. Examples

41. Fundamental theorems
Of calculus
For gradient
For divergence
For curl
Divergence theorem
Stoke‘s theorem

42. Examples

43. Questions?
Thank You!