EML 4930/5061 Analysis in M.E. II 03/17/11
Closed book Van Dommelen 9:40-10:30 am
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 force field is given as
- Find the divergence and curl of (the curl will turn
out to be zero).
- What can you say about the flux integral
over the entire surface of some
volume?
- Why must be the gradient of some potential ?
Now find in a systematic way that leaves no doubt that the
most general possible has been found.
- What can you say about the
work
integral
along any path
going from a given point A to a given point B? What about the
contour integral from a point back
to that same point?
Solution.
- A curvilinear coordinate system satisfies
Show that this coordinate system is orthogonal, then derive the
metric indices , , and . Also derive
without using Cartesian components.
Solution.
- Simplify the PDE
by rotation of the coordinate system. Then reduce it further until
its derivative terms are those of the wave equation for unit wave
speed, with an additional nonderivative term.
Solution.