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EM & Vector calculus #4 Physical Systems, Tuesday 13 Feb. 2007, EJZ

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

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EM & Vector calculus #4 Physical Systems, Tuesday 13 Feb. 2007, EJZ

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  1. 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

  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) →

  11. Solving Laplace w/ Separation of Variables

  12. 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.

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