1. Applied Electricity and Magnetism
ECE 3318
Applied Electricity and Magnetism
Spring 2017
Prof. David R. Jackson
ECE Dept.
Notes 17

2. Note the circulation about the z axis in this stream of water.
Curl of a Vector
The curl of a vector function measures the tendency of the vector function to circulate or rotate (or “curl”) about an axis.
Note the circulation about the z axis in this stream of water. x y
The curl of the water velocity vector has a z component.

3. Here the water also has a circulation about the z axis.
Curl of a Vector (cont.)
Here the water also has a circulation about the z axis. x y
This is more obvious if we subtract a constant velocity vector from the water,
as seen on the next slide.

4. Curl of a Vector (cont.) x y = x y + x y

5. Curl of a Vector (cont.) Cz Cy Cx z Sz Sy y Sx
Curl is calculated here
It turns out that the results are independent of the shape of the paths, but rectangular paths are chosen for simplicity.
Note: The paths are defined according to the “right-hand rule.”
The paths are all centered at the point of interest
(a separation between them is shown for clarity).

6. Assume that V represents the velocity of a fluid.
Curl of a Vector (cont.)
“Curl meter”
Assume that V represents the velocity of a fluid.
The term V  dr measures the force on the paddles at each point on the paddle wheel.
Hence,
(If this component is positive, the paddle wheel will spin counterclockwise.)

7. Curl Calculation Path Cx : The x component of the curl zy z
Each edge is numbered.
Path Cx :
(1)
(2)
(3)
(4)
Pair
Pair

8. Curl Calculation (cont.)
We have multiplied and divided by y.
We have multiplied and divided by z. or

9. Curl Calculation (cont.)
From the last slide,
From the curl definition:
Hence,

10. Curl Calculation (cont.)
Similarly,
Hence, we have:

11. Del Operator
Recall:

12. Del Operator (cont.) Hence, in rectangular coordinates, we have
Note: The del operator is only defined in rectangular coordinates.
For example:
See Appendix A.2 in the Hayt & Buck book for a
general derivation of curl that holds in any coordinate system.

13. Summary of Curl Formulas
Rectangular
Cylindrical
Spherical

14. Example
Calculate the curl of the following vector function:

15. Example Calculate the curl: Hence,
Velocity of water flowing in a river

16. Example (cont.)
Note: The paddle wheel will not spin if the axis is pointed in the x or y directions.
Hence
Point your thumb in the z direction:
The paddle wheel spins opposite to the fingers of the right hand.

17. Stokes’s Theorem S (open) C (closed)
The unit normal is chosen from a “right-hand rule”
according to the direction along C.
(An outward normal corresponds to a counter clockwise path.)
“The surface integral of circulation per unit area equals the total circulation.”

18. Proof
Divide S into rectangular patches that are normal to x, y, or z axes
(all with the same area S for simplicity).

19. Proof (cont.)
Substitute so

20. leaving only exterior edges.
Proof (cont.)
Cancelation
Interior edges cancel,
leaving only exterior edges.
Proof complete

21. Example
Verify Stokes’s Theorem x y S

22. Example (cont.)

23. Component of Curl Vector
Consider taking the component of the curl vector in an arbitrary direction.
We have the following property:
(The proof is on the next slide.)
Note: This property is obviously true for the x, y, and z directions, due to the definition of the curl vector. This theorem now says that the property is true for any direction in space.

24. Component of Curl Vector (cont.)
Proof:
Stokes’ Theorem:
For the LHS:
Hence,
Taking the limit:

25. Component of Curl Vector (cont.)
Physical interpretation of theorem (water flow)
The component of the curl vector in any direction tells us what the torque on the paddle wheel will be when we point the axis of the paddle wheel in that direction.
We maximize torque by pointing the paddle wheel in the direction of the curl vector.
The direction of the curl vector is thus the “axis of the whirlpooling” of the water.

26. Rotation Property of Curl (cont.)
Example:
From calculations:
Hence, the paddle wheel spins the fastest when the axis is along the z axis:
This is the axis of the “whirlpool” in the water. x y

27. Vector Identity
Proof: