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Fundamental Concepts of Vector Calculus

Overview of vector calculus focusing on gradient, divergence, curl, and Laplacian operators. Includes vector identities and theorems like Stokes's and Divergence Theorem. Helpful for understanding electromagnetic wave equations.

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Fundamental Concepts of Vector Calculus

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  1. ECE 3317 Applied Electromagnetic Waves Prof. David R. Jackson Fall 2018 Notes 3 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: http://courses.egr.uh.edu/ECE/ECE3318 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 (Vector) Laplacian (Scalar)

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

  5. “Del” Operator (cont.) A few more details about calculating the divergence and the curl:

  6. Vector Identities Two fundamental “zero” identities:

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

  8. 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).

  9. 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.

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

  11. Gradient (cont.) Rectangular Cylindrical Spherical

  12. Divergence The divergence measures the rate at which the “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.

  13. Divergence (cont.) Water flow Many small pipes injecting water Tub of water A(x,y,z) = velocity vector of water inside tub

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

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

  16. Curl (cont.) “Exploded view” 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 in the exploded view for clarity).

  17. Curl (cont.) Rectangular Cylindrical Spherical

  18. Divergence Theorem A = arbitrary vector function

  19. 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.)

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