The exam will have a take home part. This part is due at the beginning of class
on Monday. 

The exam consists of simple calculational problems (or lengthier ones to ba
taken home) and conceptual (theoretical) problems. 
The conceptual ones should require little time to answer and will make up 
some 20 or 25% of the total point score.

Some useful trig formulas will be provided. No technology, closed
notes and books. 

Contents is 
Harmonic functions, 
Separation of variables and 
Fourier series.


I expect you to know the formula for the Laplace operator
in cartesian and polar coordiantes, but you do not need to memorize it  
in other coordinate systems. 
The formula how to get the Laplacian in a general coordinate system, given the
length element (p.6 of notes) was given for your general education, but will 
not be required from you in exams in this class. 



You should know the results about existence and uniqueness for the Dirichlet
boundary value problem of the Laplacian, the maximum principle and the mean
value theorem for for harmonic functions. 

You should know the basic separation solutions in cartesian and polar
coordinates. Also remember that the separation constant becomes discrete rather than
continuous when boundary or periodicity conditions enter. 

You should know the Poisson kernel and the Poisson formula. (The idea is not
that you need to have it memorized for all the semester, but with the studies 
we did about it recently, I'll assume you know the formula for the time being.) 
Also have a look at the qualitative properties of the Poisson kernel (the hwk
problems with level lines and graphs). 





You should know that the smoother the function the faster the Fourier
coefficients become small. And you should know the formulas how to obtain 
the Fourier, sine Fourier, and cosine Fourier series of a function. 

You should be aware of the overshoot (Gibbs) phenomenon (graphically) and that
a Fourier series converges for a piecewise C^1 function in every point of 
continuity. 
Finer issues mentioned in class like uniform convergence (how it is
defined etc) are NOT required for the exam. 




Have a look at all pertinent hwk problems. 
Read them both for calculational skills and `lesson learned' messages. 
Most solutions are posted already, and all will be posted Wednesday 3pm
They are written to give a `second digest' after you have solved them. 

Good success!



WED 5PM, WE'LL HAVE A QUESTION AND ANSWER SESSION FOR THE EXAM.
MEET IN USUAL CLASSROOM. IF AVAILABLE, WE'LL TAKE IT; ELSE WE'LL MIGRATE 
WHERE AVAILABLE (AND LEAVE A NOTE WHERE WE WENT; FOR LATECOMERS)