User:Egm6322.s09/Report table/Best of report R2

Under construction; not final (but close to being stable after a couple of weeks). The intention here is to document the best features in any report for the readers (including you); if you see any excellent features in any of the reports (including your team's) that I may have missed noticing, don't hesitate to let me know. I don't have time to read all reports in detail. Due to time constraint, I would only selectively looked at a few important features. I also added some annotations (e.g., related to errors, improvements, etc.) where I looked through for the benefit of everyone. Egm6322.s09 12:18, 7 February 2009 (UTC)

Latest archived version (possibly with comments): Team Bit Team Mafia Team Three

• classnotes (including simple HW problems), graded over 100%

• 2nd-order quasilinear PDEs
  • examples
  • verification of linearity

• 2nd-order linear PDEs
  • matrix-operator form
  • constant coefficients
  • varying coefficients
  • equivalence between two forms: regular form and matrix form
    • HW: Condition for equality of mixed derivatives $\displaystyle u_{xy} = u_{yx}$. Best solution: Team Bit simpler to understand; Team Mafia the proof is a bit more complicated to read, but essentially the same. Note: Team Three good starting point, which coincides with the proofs in the other two reports, but need to use mean-value-theorem to conclude, and can use different step size along $\displaystyle x$ and $\displaystyle y$ to be more general as in the proofs in the other two reports; appears to do this proof on their own; good.
    • linearity of diffusion operator
      • linearity of derivative. Best solution: Team Mafia. Note: Team Bit Team Three gave a statement, but not a proof.
      • linearity of gradient operator
      • linearity of divergence operator
  • nonlinear transformation of coordinates
    • matrix-operator form in new coordinates

• linearity, 2 definitions
  • definition 1 in class
  • definition 2 in Kolmogorov & Fomin, Selvadurai
    • additivity
    • homogeneity
  • examples of linear maps in higher dimensions, matrices, homogeneity
    • transformation of coordinates, rotation of coordinate system. Best solution: Team Bit see better explanation in class. Note: Team Mafia only display the rotation matrix, and referred to a book, but not derived it. Team Three did not do this problem.
    • affine mappings, not linear; example, rotation and translation
  • equivalence between two definitions, proof
    • sufficient condition
    • necessary condition