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.
- A surface is given by
- What sort of surface is this?
- Show that the point P with position vector is
on the surface.
- Find the equation for the plane that is tangential to the
surface at P.
- Find the vectorial description for the line through P normal
to the surface at P, in terms of a parameter.
- A ping-pong ball with velocity hits the surface
at P and bounces off according to an elastic (Snell) reflection.
What is the velocity of the bounced ball?
Vector calculus only, absolutely no trig.
Solution.
- For the same surface as the previous question
- Write in terms of Cartesian coordinates.
- If a velocity field is given as
write the volumetric flow rate through
the part of the surface with using your obtained expression.
- Convert to polar and integrate.
Solution.
- Consider the PDE
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 and are positive. You should get a very simple
equation for the transformed PDE. Find a particular solution
this way. In terms of the original coordinates, of course. What
equation does the remainder of satisfy? What is the name
of that simple equation?
Solution.