EML 4930/5061 Analysis in M.E. II 03/04/15
Closed book Van Dommelen 12:30-1:20 pm

Show all reasoning and intermediate results leading to your answer, or credit will be lost. One book of mathematical tables, such as Schaum's Mathematical Handbook, may be used, as well as a calculator and one handwritten letter-size single formula sheet.

  1. A surface is given by

    \begin{displaymath}
x^2 + y^2 + 8 z^2 = 4
\end{displaymath}

    Vector calculus only, absolutely no trig.

    Solution.

  2. For the same surface as the previous question

    Solution.

  3. Consider the PDE

    \begin{displaymath}
x^2 u_{xx} - 2xy u_{xy} + 2y^2 u_{yy} + x u_x + 2y u_y = \ln(x)
\end{displaymath}

    What is the equation type? Using class procedures, convert this equation to 2D canonical coordinates. Take the straighforward choice for these coordinates, do not try to be smart. You can assume that $x$ and $y$ are positive. You should get a very simple equation for the transformed PDE. Find a particular solution $u_p$ this way. In terms of the original coordinates, of course. What equation does the remainder $u_h$ of $u$ satisfy? What is the name of that simple equation?

    Solution.