1. EM & Vector calculus #4 Physical Systems, Tuesday 13 Feb. 2007, EJZ Vector Calculus 1.4: Curvilinear Coordinates Quick review of quiz and homework Review Cartesian coordinates, unit vectors, and dl Spherical and cylindrical Coordinates, unit vectors, dl, and vector derivatives Ch.3b: Finding potentials using separation of variables Quick review of quiz and homework Example 3.3 Worksheets for Problems 3.12 and 3.23

2. Vector calculus HW & Quiz review Online solutions at http://192.211.16.13/curricular/physys/0607/solns/ HW: VCsoln31.pdf, EMsoln3a.pdf, EMsoln3b.pdf Quiz: VCMidSoln.pdf, EMmodMidSoln.pdf

3. Cartesian Coordinates The infinitesimal displacement vector from (x,y,z) to (x+dx, y+dy, z+dz) is dl:

4. Cylindrical Coordinates

5. Spherical Coordinates

6. Cylindrical Coordinates Derive these (Problem 1.41):

7. Spherical Coordinates Derive these (Problem 1.37):

8. Vector calculus HW due next week: Ch.1.4 Problems: 1.37, 1.38, 1.41, 1.42

9. E&M Ch.3b: Separation of variables Quick review of quiz and homework When to use separation of variables? In charge-free regions With well-specified boundary conditions Without sufficient symmetry to use Gauss’ law How to use separation of variables? Guess form of solutions based on BC Separate variables, insert guessed solutions with constants Apply BC and solve for constants

10. Poisson and Laplace equations Gauss: Potential: combine to get Poisson’s eqn: Laplace equation holds in charge-free regions: Last week we found the general solutions to Laplace’s eqn. in spherical and cylindrical coordinates for the case where V depends only on r (Prob.3.3, p.116) →

12. Solving Laplace w/ Separation of Variables

22. Worksheets for Problems 3.12 (136), 3.23 (145) Homework due next week: work through Ex.3.3, do 3.12 and 3.23. Extra credit: #13, 24.