7 02/27 F

  1. 2.28c. Show that the equation may be simplified to

    \begin{displaymath}
u_{\xi\xi} = 0
\end{displaymath}

    Solve this equation and write the solution in terms of $x$ and $y$.
  2. 2.28b. Reduce to canonical form II. In 2.23 you diagonalized essentially the same equation by rotating the coordinate system; and you could then have stretched the coordinates to reduce it to the Laplace equation. Are the coordinates that you find now equivalent to those? In particular, are the lines of constant $\xi$ and $\eta$ orthogonal like in 2.23? If not, how come that more than one linear coordinate transformation can turn the equation into the Laplace equation?
  3. 3.41. This is similar to the Laplace version discussed earlier in class. Describe the reason that there is no solution physically, considering it as a heat conduction problem in a circular plate.