1. Differential Calculus (revisited):
Derivative of any function f(x,y,z):
Gradient of function f

2. Change in a scalar function f corresponding to a change in position dr
Gradient of a function
Change in a scalar function f corresponding to a change in position dr
 f is a VECTOR

3. Geometrical interpretation of GradientZ P Q dr Y
change in f : X =0
=> f  dr

4. Z Q dr P Y X

5. Magnitude: slope along this maximal direction
For a given |dr|, the change in scalar function f(x,y,z) is maximum when:
=> f is a vector along the direction of maximum rate of change of the function
Magnitude: slope along this maximal direction

6. => df = 0 for small displacements about the point (x0,y0,z0)
If  f = 0 at some point (x0,y0,z0)
=> df = 0 for small displacements about the point (x0,y0,z0)
(x0,y0,z0) is a stationary point of f(x,y,z)

7. The Operator   is NOT a vector, but a VECTOR OPERATOR Satisfies:
Vector rules
Partial differentiation rules

8.  can act: On a scalar function f : f GRADIENT
On a vector function F as: . F
DIVERGENCE
On a vector function F as:  × F
CURL

9. Divergence of a vector is a scalar.
.F is a measure of how much the vector F spreads out (diverges) from the point in question.

10. Physical interpretation of Divergence
Flow of a compressible fluid:
(x,y,z) -> density of the fluid at a point (x,y,z)
v(x,y,z) -> velocity of the fluid at (x,y,z)

11. (rate of flow in)EFGH
(rate of flow out)ABCD Z X Y dy dx dz A D C B E F H G

12. Net rate of flow out (along- x)
Net rate of flow out through all pairs of surfaces (per unit time):

13. Net rate of flow of the fluid per unit volume per unit time:
DIVERGENCE

14. Curl of a vector is a vector
×F is a measure of how much the vector F “curls around” the point in question.

15. Physical significance of Curl
Circulation of a fluid around a loop: Y 3 2 4 1 X
Circulation (1234)

16. Circulation per unit area = ( × V )|z
z-component of CURL

17. Curvilinear coordinates: used to describe systems with symmetry.
Spherical coordinates (r, , Ø)

19. Cartesian coordinates in terms of spherical coordinates:

20. Spherical coordinates in terms of Cartesian coordinates:

21. Unit vectors in spherical coordinatesZ r  Y  X

22. Line element in spherical coordinates:
Volume element in spherical coordinates:

23. Area element in spherical coordinates:
on a surface of a sphere (r const.)
on a surface lying in xy-plane ( const.)

24. Gradient:
Divergence:

25. Curl:

26. Fundamental theorem for gradient
We know df = (f ).dl
The total change in f in going from a(x1,y1,z1)
to b(x2,y2,z2) along any path:
Line integral of gradient of a function is given by the value of the function at the boundaries of the line.

27. Corollary 1:
Corollary 2:

28. E = - V Field from Potential From the definition of potential:
From the fundamental theorem of gradient:
E = - V

29. Electric Dipole
Potential at a point due to dipole: z r  p y  x

30. Electric Dipole
E = - V
Recall:

31. Electric Dipole
Using:

32. Fundamental theorem for Divergence
Gauss’ theorem, Green’s theorem
The integral of divergence of a vector over a volume is equal to the value of the function over the closed surface that bounds the volume.

33. Fundamental theorem for Curl
Stokes’ theorem
Integral of a curl of a vector over a surface is equal to the value of the function over the closed boundary that encloses the surface.

34. THE DIRAC DELTA FUNCTION
Recall:

35. The volume integral of F:

36. Surface integral of F over a sphere of radius R:
From divergence theorem:

37. From calculation of Divergence:
By using the Divergence theorem:

38. Note: as r  0; F  ∞
And integral of F over any volume containing the point r = 0

39. The Dirac Delta Function
(in one dimension)
 Can be pictured as an infinitely high, infinitesimally narrow “spike” with area 1

40. The Dirac Delta Function
(x) NOT a Function
But a Generalized Function OR distribution
Properties:

41. The Dirac Delta Function
(in one dimension)
Shifting the spike from 0 to a;

42. The Dirac Delta Function
(in one dimension)
Properties:

43. The Dirac Delta Function
(in three dimension)

44. The Paradox of Divergence of
From calculation of Divergence:
By using the Divergence theorem:

45. So now we can write:
Such that: