1. VECTOR CALCULUS 1.10 GRADIENT OF A SCALAR 1.11 DIVERGENCE OF A VECTOR
DIVERGENCE THEOREM
CURL OF A VECTOR
STOKES’S THEOREM
LAPLACIAN OF A SCALAR

2. 1.10 GRADIENT OF A SCALAR Suppose is the temperature at ,
and is the temperature at
as shown.

3. GRADIENT OF A SCALAR (Cont’d)
The differential distances are the components of the differential distance vector :
However, from differential calculus, the differential temperature:

4. GRADIENT OF A SCALAR (Cont’d)
But,
So, previous equation can be rewritten as:

5. GRADIENT OF A SCALAR (Cont’d)
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:

6. GRADIENT OF A SCALAR (Cont’d)
For Circular cylindrical coordinate:
For Spherical coordinate:

7. EXAMPLE 10
Find gradient of these scalars:
(a)
(b)
(c)

8. SOLUTION TO EXAMPLE 10
(a) Use gradient for Cartesian coordinate:

9. SOLUTION TO EXAMPLE 10 (Cont’d)
(b) Use gradient for Circular cylindrical
coordinate:

10. SOLUTION TO EXAMPLE 10 (Cont’d)
(c) Use gradient for Spherical coordinate:

11. DIVERGENCE OF A VECTOR
Illustration of the divergence of a vector field at point P:
Positive Divergence
Negative Divergence
Zero Divergence

12. DIVERGENCE OF A VECTOR (Cont’d)
The divergence of A at a given point P is the outward flux per unit volume:

13. DIVERGENCE OF A VECTOR (Cont’d)
Vector field A at closed surface S
What is ??

14. DIVERGENCE OF A VECTOR (Cont’d)
Where,
And, v is volume enclosed by surface S

15. DIVERGENCE OF A VECTOR (Cont’d)
For Cartesian coordinate:
For Circular cylindrical coordinate:

16. DIVERGENCE OF A VECTOR (Cont’d)
For Spherical coordinate:

17. EXAMPLE 11
Find divergence of these vectors:
(a)
(b)
(c)

18. SOLUTION TO EXAMPLE 11
(a) Use divergence for Cartesian coordinate:

19. SOLUTION TO EXAMPLE 11 (Cont’d)
(b) Use divergence for Circular cylindrical
coordinate:

20. SOLUTION TO EXAMPLE 11 (Cont’d)
(c) Use divergence for Spherical coordinate:

21. DIVERGENCE THEOREM
It states that the total outward flux of a vector field A at the closed surface S is the same as volume integral of divergence of A.

22. EXAMPLE 12
A vector field exists in the region between two concentric cylindrical surfaces defined by ρ = 1 and ρ = 2, with both cylinders extending between z = 0 and z = 5. Verify the divergence theorem by evaluating:
(a)
(b)

23. SOLUTION TO EXAMPLE 12 (a) For two concentric cylinder, the left side:
Where,

24. SOLUTION TO EXAMPLE 12 Cont’d)

25. SOLUTION TO EXAMPLE 12 Cont’d)
Therefore

26. SOLUTION TO EXAMPLE 12 Cont’d)
(b) For the right side of Divergence Theorem,
evaluate divergence of D
So,

27. CURL OF A VECTOR
The curl of vector A is an axial (rotational) vector whose magnitude is the maximum circulation of A per unit area tends to zero and whose direction is the normal direction of the area when the area is oriented so as to make the circulation maximum.

28. CURL OF A VECTOR (Cont’d)
Where,

29. CURL OF A VECTOR (Cont’d)
The curl of the vector field is concerned with rotation of the vector field. Rotation can be used to measure the uniformity of the field, the more non uniform the field, the larger value of curl.

30. CURL OF A VECTOR (Cont’d)
For Cartesian coordinate:

31. CURL OF A VECTOR (Cont’d)
For Circular cylindrical coordinate:

32. CURL OF A VECTOR (Cont’d)
For Spherical coordinate:

33. EXAMPLE 13
Find curl of these vectors:
(a)
(b)
(c)

34. SOLUTION TO EXAMPLE 13
(a) Use curl for Cartesian coordinate:

35. SOLUTION TO EXAMPLE 13 (Cont’d)
(b) Use curl for Circular cylindrical coordinate

36. SOLUTION TO EXAMPLE 13 (Cont’d)
(c) Use curl for Spherical coordinate:

37. SOLUTION TO EXAMPLE 13 (Cont’d)

38. STOKE’S THEOREM
The circulation of a vector field A around a closed path L is equal to the surface integral of the curl of A over the open surface S bounded by L that A and curl of A are continuous on S.

39. STOKE’S THEOREM (Cont’d)

40. EXAMPLE 14
By using Stoke’s Theorem, evaluate for

41. EXAMPLE 14 (Cont’d)

42. SOLUTION TO EXAMPLE 14 Stoke’s Theorem,
Evaluate right side to get left side,
where, and

43. SOLUTION TO EXAMPLE 14 (Cont’d)

44. EXAMPLE 15 Verify Stoke’s theorem for the vector field
for given figure by evaluating:
(a) over the semicircular contour.
(b) over the surface of semicircular contour.

45. SOLUTION TO EXAMPLE 15
(a) To find
Where,

46. SOLUTION TO EXAMPLE 15 (Cont’d)

47. SOLUTION TO EXAMPLE 15 (Cont’d)
Therefore the closed integral,

48. SOLUTION TO EXAMPLE 15 (Cont’d)
(b) To find

49. SOLUTION TO EXAMPLE 15 (Cont’d)
Therefore

50. LAPLACIAN OF A SCALAR
The Laplacian of a scalar field, V written as:
Where, Laplacian V is:

51. LAPLACIAN OF A SCALAR (Cont’d)
For Cartesian coordinate:
For Circular cylindrical coordinate:

52. LAPLACIAN OF A SCALAR (Cont’d)
For Spherical coordinate:

53. EXAMPLE 16 You should try this!! Find Laplacian of these scalars: (a)
(b)
(c)
You should try this!!

54. SOLUTION TO EXAMPLE 16
(a)
(b)
(c)

55. END
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