1. ENTC 3331
RF Fundamentals
Dr. Hugh Blanton
ENTC 3331

2. Gradient, Divergence and Curl: the Basics

3. We first consider the position vector, l:
where x, y, and z are rectangular unit vectors.
Dr. Blanton - ENTC Gradient, Divergence, & Curl

4. Since the unit vectors for rectangular coordinates are constants, we have for dl:
Dr. Blanton - ENTC Gradient, Divergence, & Curl

5. The operator, del: Ñ is defined to be (in rectangular coordinates) as:
This operator operates as a vector.
Dr. Blanton - ENTC Gradient, Divergence, & Curl

6. Gradient
If the del operator, Ñ operates on a scalar function, f(x,y,z), we get the gradient: 
Dr. Blanton - ENTC Gradient, Divergence, & Curl

7. We can interpret this gradient as a vector with the magnitude and direction of the maximum change of the function in space.
We can relate the gradient to the differential change in the function: 
Dr. Blanton - ENTC Gradient, Divergence, & Curl

8. a T dl dT ˆ × Ñ = Directional derivatives: l
Dr. Blanton - ENTC Gradient, Divergence, & Curl

9. Since the del operator should be treated as a vector, there are two ways for a vector to multiply another vector:
dot product and
cross product.
Dr. Blanton - ENTC Gradient, Divergence, & Curl

10. Divergence We first consider the dot product:
The divergence of a vector is defined to be:
This will not necessarily be true for other unit vectors in other coordinate systems.
Dr. Blanton - ENTC Gradient, Divergence, & Curl

11. To get some idea of what the divergence of a vector is, we consider Gauss' theorem (sometimes called the divergence theorem).
Dr. Blanton - ENTC Gradient, Divergence, & Curl

12. Gauss' Theorem (Gaub’s Theorem
We start with:
Surface Areas
Dr. Blanton - ENTC Gradient, Divergence, & Curl

13. We can see that each term as written in the last expression gives the value of the change in vector A that cuts perpendicular through the surface.
Dr. Blanton - ENTC Gradient, Divergence, & Curl

14. For instance, consider the first term:
The first part:
gives the change in the x-component of A
Dr. Blanton - ENTC Gradient, Divergence, & Curl

15. The second part,
gives the yz surface (or x component of the surface, Sx) where we define the direction of the surface vector as that direction that is perpendicular to its surface.
Dr. Blanton - ENTC Gradient, Divergence, & Curl

16. The other two terms give the change in the component of A that is perpendicular to the xz (Sy) and xy (Sz) surfaces.
Dr. Blanton - ENTC Gradient, Divergence, & Curl

17. We thus can write: where the vector S is the surface area vector.
Dr. Blanton - ENTC Gradient, Divergence, & Curl

18. Thus we see that the volume integral of the divergence of vector A is equal to the net amount of A that cuts through (or diverges from) the closed surface that surrounds the volume over which the volume integral is taken.
Hence the name divergence for
Dr. Blanton - ENTC Gradient, Divergence, & Curl

19. So what?
Divergence literally means to get farther apart from a line of path, or
To turn or branch away from.
Dr. Blanton - ENTC Gradient, Divergence, & Curl

20. Consider the velocity vector of a cyclist not diverted by any thoughts or obstacles:
Goes straight ahead at constant velocity.
 (degree of) divergence  0
Dr. Blanton - ENTC Gradient, Divergence, & Curl

21. Now suppose they turn with a constant velocity
 diverges from original direction
(degree of) divergence  0
Dr. Blanton - ENTC Gradient, Divergence, & Curl

22. Now suppose they turn and speed up.
 diverges from original direction
(degree of) divergence >> 0
Dr. Blanton - ENTC Gradient, Divergence, & Curl

23.  No divergence from original direction (degree of) divergence = 0
Current of water
 No divergence from original direction
(degree of) divergence = 0
Dr. Blanton - ENTC Gradient, Divergence, & Curl

24.  Divergence from original direction (degree of) divergence ≠ 0
Current of water
 Divergence from original direction
(degree of) divergence ≠ 0
Dr. Blanton - ENTC Gradient, Divergence, & Curl

25. E-field between two plates of a capacitor.+ 
E-field between two plates of a capacitor.
Divergenceless
Dr. Blanton - ENTC Gradient, Divergence, & Curl

26. divergenceless  solenoidal
b-field inside a solenoid is homogeneous and divergenceless.
divergenceless  solenoidal
Dr. Blanton - ENTC Gradient, Divergence, & Curl

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28. CURL

29. Two types of vector fields exists:+
Electrostatic Field where the field lines are open and there is circulation of the field flux.
Magnetic Field where the field lines are closed and there is circulation of the field flux.
circulation (rotation) = 0
circulation (rotation)  0
Dr. Blanton - ENTC Gradient, Divergence, & Curl

30. The mathematical concept of circulation involves the curl operator.
The curl acts on a vector and generates a vector.
Dr. Blanton - ENTC Gradient, Divergence, & Curl

31. In Cartesian coordinate system:
Dr. Blanton - ENTC Gradient, Divergence, & Curl

32. Example
Dr. Blanton - ENTC Gradient, Divergence, & Curl

33. Important identities:
for any scalar function V.
Dr. Blanton - ENTC Gradient, Divergence, & Curl

34. Closed boundary of that surface.
Stoke’s Theorem
General mathematical theorem of Vector Analysis:
Closed boundary of that surface.
Any surface
Dr. Blanton - ENTC Gradient, Divergence, & Curl

35. Given a vector field
Verify Stoke’s theorem for a segment of a cylindrical surface defined by:
Dr. Blanton - ENTC Gradient, Divergence, & Curl

36. z y x
Dr. Blanton - ENTC Gradient, Divergence, & Curl

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41. Note that has only one component:
Dr. Blanton - ENTC Gradient, Divergence, & Curl

42. The integral of over the specified surface S is
Dr. Blanton - ENTC Gradient, Divergence, & Curl

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44. z c d b y x a
Dr. Blanton - ENTC Gradient, Divergence, & Curl

45. The surface S is bounded by contour C = abcd.
The direction of C is chosen so that it is compatible with the surface normal by the right hand rule.
Dr. Blanton - ENTC Gradient, Divergence, & Curl

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49. Curl
Dr. Blanton - ENTC Gradient, Divergence, & Curl

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51. curl or rot place paddle wheel in a river no rotation at the center
rotation at the edges
Dr. Blanton - ENTC Gradient, Divergence, & Curl

52. the vector un is out of the screen right hand rule
Ds is surface enclosed within loop
closed line integral
Dr. Blanton - ENTC Gradient, Divergence, & Curl

53. Electric Field Lines Rules for Field Lines
Electric field lines point to negative charges
Electric field lines extend away from positive charges
Equipotential (same voltage) lines are perpendicular to a line tangent of the electric field lines
Dr. Blanton - ENTC Gradient, Divergence, & Curl