1. 1 Vector Calculus

2. Copyright © 2007 Oxford University Press Elements of Electromagnetics Fourth Edition Sadiku2 Figure 3.1 Differential elements in the right-handed Cartesian coordinate system.

3. Copyright © 2007 Oxford University Press Elements of Electromagnetics Fourth Edition Sadiku3 Figure 3.2 Differential normal surface areas in Cartesian coordinates: (a) dS  dy dz a x, (b) dS  dx dz a y, (c) dS  dx dy a z.

4. Copyright © 2007 Oxford University Press Elements of Electromagnetics Fourth Edition Sadiku4 Figure 3.3 Differential elements in cylindrical coordinates.

5. Copyright © 2007 Oxford University Press Elements of Electromagnetics Fourth Edition Sadiku5 Figure 3.4 Differential normal surface areas in cylindrical coordinates: (a) dS   d  dz a , (b) dS  d  dz a , (c) dS   d  d  a z.

6. Copyright © 2007 Oxford University Press Elements of Electromagnetics Fourth Edition Sadiku6 Figure 3.5 Differential elements in the spherical coordinate system.

7. Copyright © 2007 Oxford University Press Elements of Electromagnetics Fourth Edition Sadiku7 Figure 3.6 Differential normal surface areas in spherical coordinates: (a) dS  r 2 sin  d  d  a r, (b) dS  r sin  dr d  a , (c) dS  r dr d  a .

8. Copyright © 2007 Oxford University Press Elements of Electromagnetics Fourth Edition Sadiku8 Figure 3.7 For Example 3.1.

9. Copyright © 2007 Oxford University Press Elements of Electromagnetics Fourth Edition Sadiku9 Figure 3.8 For Practice Exercise 3.1 (and also Review Question 3.3).

10. Copyright © 2007 Oxford University Press Elements of Electromagnetics Fourth Edition Sadiku10 Figure 3.9 Path of integration of vector field A.

11. Copyright © 2007 Oxford University Press Elements of Electromagnetics Fourth Edition Sadiku11 Figure 3.10 The flux of a vector field A through surface S.

12. Copyright © 2007 Oxford University Press Elements of Electromagnetics Fourth Edition Sadiku12 Figure 3.11 For Example 3.2.

13. Copyright © 2007 Oxford University Press Elements of Electromagnetics Fourth Edition Sadiku13 Figure 3.12 For Practice Exercise 3.2, L is a closed path.

14. Copyright © 2007 Oxford University Press Elements of Electromagnetics Fourth Edition Sadiku14 Figure 3.13 Gradient of a scalar.

15. Copyright © 2007 Oxford University Press Elements of Electromagnetics Fourth Edition Sadiku15 Figure 3.14 For Example 3.5; plane of intersection of a line with an ellipsoid.

16. Copyright © 2007 Oxford University Press Elements of Electromagnetics Fourth Edition Sadiku16 Figure 3.15 Illustration of the divergence of a vector field at P: (a) positive divergence, (b) negative divergence, (c) zero divergence.

17. Copyright © 2007 Oxford University Press Elements of Electromagnetics Fourth Edition Sadiku17 Figure 3.16 Evaluation of  A at point P(x 0, y 0, z 0 ).

18. Copyright © 2007 Oxford University Press Elements of Electromagnetics Fourth Edition Sadiku18 Figure 3.17 Volume v enclosed by surface S.

19. Copyright © 2007 Oxford University Press Elements of Electromagnetics Fourth Edition Sadiku19 Figure 3.18 For Example 3.7.

20. Copyright © 2007 Oxford University Press Elements of Electromagnetics Fourth Edition Sadiku20 Figure 3.19 Contour used in evaluating the x-component of   A at point P(x o, y o, z o ).

21. Copyright © 2007 Oxford University Press Elements of Electromagnetics Fourth Edition Sadiku21 Figure 3.20 Illustration of a curl: (a) curl at P points out of the page, (b) curl at P is zero.

22. Copyright © 2007 Oxford University Press Elements of Electromagnetics Fourth Edition Sadiku22 Figure 3.21 Determining the sense of dl and dS involved in Stokes’s theorem.

23. Copyright © 2007 Oxford University Press Elements of Electromagnetics Fourth Edition Sadiku23 Figure 3.22 Illustration of Stokes’s theorem.

24. Copyright © 2007 Oxford University Press Elements of Electromagnetics Fourth Edition Sadiku24 Figure 3.23 For Example 3.9.

25. Copyright © 2007 Oxford University Press Elements of Electromagnetics Fourth Edition Sadiku25 Figure 3.24 Typical fields with vanishing and nonvanishing divergence or curl.

26. Copyright © 2007 Oxford University Press Elements of Electromagnetics Fourth Edition Sadiku26 Figure 3.25 For Review Question 3.1.

27. Copyright © 2007 Oxford University Press Elements of Electromagnetics Fourth Edition Sadiku27 Figure 3.26 For Review Question 3.2.

28. Copyright © 2007 Oxford University Press Elements of Electromagnetics Fourth Edition Sadiku28 Figure 3.27 For Review Question 3.8.

29. Copyright © 2007 Oxford University Press Elements of Electromagnetics Fourth Edition Sadiku29 Figure 3.28 For Problem 3.7.

30. Copyright © 2007 Oxford University Press Elements of Electromagnetics Fourth Edition Sadiku30 Figure 3.29 For Problem 3.31.

31. Copyright © 2007 Oxford University Press Elements of Electromagnetics Fourth Edition Sadiku31 Figure 3.30 For Problem 3.32.

32. Copyright © 2007 Oxford University Press Elements of Electromagnetics Fourth Edition Sadiku32 Figure 3.31 For Problem 3.33.

33. Copyright © 2007 Oxford University Press Elements of Electromagnetics Fourth Edition Sadiku33 Figure 3.32 Volume in form of ice cream cone for Problem 3.38.