I have a couple of short questions on differential forms.  Feel free
to use any standard facts for this subject matter; I can always ask
for clarification later if you use a result I'm not familiar with.

Abbreviations I will use:

R = real numbers                
p = phi    
w = omega
a # b = wedge product of a and b
     *
q = p  [the * is a superscript on the p, i.e., the pull back under p]

1)
                       2
Let p : R x (0,pi) -> R  by 

p(r,t)=(r cos(t), r sin(t)).  Calculate the value of q applied to
either

dx and xy dx # dy

OR

dr and sin(t) dr # dt

whichever makes mathematical sense. 

2) Let w be a differential k-form on a manifold M and let f : M ->
(0,infinity) be smooth.  Show that d(fw) = 0 implies w # dw = 0.

Lastly, I would like an example of a 2-form w s.t. w # w is
non-vanishing.  I am fairly certain this holds for a 2-form 
such as 
dx dx + dx dx , 
  1  2    3  4
but am unsure how to formalize.                            

A solution within 24-48 hours would be greatly appreciated.  Thank you
sincerely for your time.

---

Clarification of Question by dubois-ga on 03 May 2003 15:20 PDT

Hi, if anyone wants to answer, one quick change: remove the
"lastly...formalize" paragraph and replace it with:

"Lastly, please explain why dx would be an example of a closed
one-form on T belonging to a non-zero cohomology class, where T is the
torus regarded as a manifold."

Thanks.