EM & Vector calculus #4Physical 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

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

Cartesian Coordinates

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

Cylindrical Coordinates

Spherical Coordinates

Cylindrical Coordinates

Derive these (Problem 1.41):

Spherical Coordinates

Derive these (Problem 1.37):

Vector calculus HW due next week:

Ch.1.4 Problems: 1.37, 1.38, 1.41, 1.42

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

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) →

0

E

VE

2

0

V

2 0V

Solving Laplace w/ Separation of Variables

2 0V

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.