1. Applied Electromagnetic Waves
ECE 3317
Applied Electromagnetic Waves
Prof. David R. Jackson
Fall 2018
Notes Review of
Vector Calculus
Adapted from notes by Prof. Stuart A. Long

2. Overview
Here we present a brief overview of vector calculus. A much more thorough discussion of vector calculus may be found in the class notes for ECE 3318:
Notes 13: Divergence
Notes 17: Curl
Notes 19: Gradient and Laplacian
Please also see the textbooks and the following supplementary books
(on reserve in the Library):
H. M. Schey, Div, Grad, Curl, and All That: an Informal Text on Vector Calculus, 2nd Ed., W. W. Norton and Company, 1992.
M. R. Spiegel, Schaum’s Outline on Vector Analysis, McGraw-Hill, 1959.

3. “Del” Operator This is an “operator”. Gradient Laplacian (Vector)
(Scalar)

4. “Del” Operator (cont.) Vector A: Divergence Curl (Scalar) (Vector)
Note:
Results for cylindrical and spherical coordinates are given at the back of your books.

5. Vector Identities
Two fundamental “zero” identities:

6. Vector Identities (cont.)
Another useful identity:
This will be useful in the derivation of the Poynting theorem.

7. Vector Laplacian
The vector Laplacian of a vector function is a vector function.
The vector Laplacian is very useful for deriving the vector Helmholtz equation (the fundamental differential equation that the electric and magnetic fields obey).

8. Vector Laplacian (cont.)
In rectangular coordinates, the vector Laplacian has a very nice property:
This identity is a key property that will help us reduce the vector Helmholtz equation to the scalar Helmholtz equation, which the components of the fields satisfy.

9. Gradient (from calculus)
The gradient vector tells us the direction of maximum change in a function.

10. Gradient (cont.)
Rectangular
Cylindrical
Spherical

11. Divergence
The divergence measures the rate at which flux of the vector function emanates from a region of space.
Divergence > 0: “source of flux”
Divergence < 0: “sink of flux”
Please see the books or the ECE 3318 class notes for a derivation of this property.

12. Divergence (cont.)
Rectangular:
Cylindrical:
Spherical:

13. Curl
"right-hand rule for C"
A component of the curl tells us the rotation of the vector function about that axis.
River
Paddle wheel
Please see the books or the ECE 3318 class notes for a derivation of this property.

14. Curl (cont.)
Curl is calculated here
Note:
The paths are defined according to the “right-hand rule.”
Note:
The paths are all centered at the point of interest
(a separation between them is shown for clarity).

15. Curl (cont.)
Rectangular
Cylindrical
Spherical

16. A = arbitrary vector function
Divergence Theorem
A = arbitrary vector function

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.)