ECE 305 Electromagnetic Theory Fall 2016 ECE 305 Electromagnetic Theory Lecture 3: Chapter 3 Qiliang Li Dept. of Electrical and Computer Engineering, George Mason University, Fairfax, VA

3 Vector Calculus (Chapter 3 in the book) 3.1 Introduction 3.2 Differential length, area and volume A. Cartesian coordinate system

It is a vector

B.

C.

FIGURE 3.6 Differential normal surface areas in spherical coordinates: (a) dS = r2 sin θ dθ dɸ ar , (b) dS = r sin θ dr dɸ aθ, (c) dS = r dr dθ aɸ.

E.g. 3.1 Solve: Obviously, it is easier to solve it in cylindrical coordinate

(note: for (b), what happen if you choose Cartesian coordinates?)

3.3 Line, surface and volume integrals

Surface integral (or the flux of A through S)

E.g. 3.2

3.4 Del Operator Cylindrical Coordinate Spherical Coordinate

- The gradient of a scalar field V is a vector dl is differential displacement from P1 to P2 (see Fig. 3.13)

In the three coordinate systems

3.6 Divergence of a Vector and Divergence Theorem

Divergence of a Vector in 3 coordinate systems Note:

Divergence Theorem E.g. 3.6

(E.g. 3.6)

3.7 Curl of a Vector and Stokes’s Theorem

Cylindrical coordinate Spherical coordinate

Note: Stokes’s Theorem

3.8 Laplacian of a Scalar