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Prof. Ji Chen

ECE 3317. Prof. Ji Chen. Spring 2014. Notes 3 Review of Vector Calculus. Adapted from notes by Prof. Stuart A. Long. “Del” Operator. This is an “operator.”. Gradient. (Vector). Laplacian. (Scalar). “Del” Operator (cont.). Vector A : . Divergence . (Scalar). Curl .

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Prof. Ji Chen

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  1. ECE 3317 Prof. Ji Chen Spring 2014 Notes 3 Review of Vector Calculus Adapted from notes by Prof. Stuart A. Long

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

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

  4. Vector Identities Two fundamental “zero” identities:

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

  6. 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).

  7. 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 scalarHelmholtz equation, which the components of the fields satisfy.

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

  9. Divergence Operator z y x The divergence measures the rate at which flux emanates from a region of space.

  10. Curl Operator z C z y y "right-hand rule" z A = velocity vector River Paddle wheel y A component of the curl tells us the rotation about that axis.

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