1. EM & Vector calculus #3 Physical Systems, Tuesday 30 Jan 2007, EJZ Vector Calculus 1.3: Integral Calculus Line, surface, volume integrals Fundamental theorems Integration by parts Ch.3a: Special Techniques (Electrostatics) Quick homework review Poisson’s and Laplace’s equations (Prob. 3.3 p.116) Uniqueness Method of images (Prob. 3.9 p.126)

2. 1.2.6 Product rules

3. 1.2.7 Second derivatives

4. 1.3.1 Line, Surface, Volume integrals

9. 1.3.2 Fundamental theorem of calculus

10. 1.3.3 Fundamental theorem for Gradients

12. 1.3.4 Fundamental theorem for Divergences

15. 1.3.5 Fundamental theorem for Curls

17. Consequences Gauss’s law and fundamental theorem for divergences: Ampere’s Law and fundamental theorem for curls:

18. E&M Ch.3: Techniques for finding V Why? Easy to find E from V Scalar V superpose easily How? Poisson’s and Laplace’s equations (Prob. 3.3 p.116) Guess if possible: unique solution for given BC Method of images (Prob. 3.9 p.126) Separation of variables (next week)

19. Poisson’s equation Gauss: Potential: combine to get Poisson’s eqn: Laplace equation holds in charge-free regions: Prob.3.3 (p.116): Find the general solution to Laplace’s eqn. In spherical coordinates, for the case where V depends only on r. Do the same for cylindrical coordinates, assuming V(s). (See Laplacian on p.42 and 44)

23. Method of images A charge distribution  induces  on a nearby conductor. The total field results from combination of  and . + - Guess an image charge that is equivalent to . Satisfy Poisson and BC, and you have THE solution. Prob.3.9 p.126 (cf 2.2 p.82)