EXTRA CREDIT PROBLEMS FOR WD SMTIH, Math127(3).
HAND IN ANY PROBLEM-ANSWER AT ANY TIME, TO GET EXTRA CREDIT.
PLEASE CLEARLY LABEL AND PUT ON OTHER SHEETS FROM REGULAR HW.
(these are *harder* than regular problems and you may have to 
write a little essay to show me how you are doing it...
in some cases you may have to get data from books or the internet
to solve it...)
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YOU GET TO PICK WHICH OF THESE (IF ANY) YOU WANT TO TRY.
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MUST SHOW ALL WORK.  IF GOT HELP, MUST GIVE
EXPLICIT NAMED CREDIT TO ALL HELPERS, BOOKS, WEB SITES, OR
WHATEVER.
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1. Let the vector-valued-function 
   F(x,y) = (ax + by , px + qy) 
where a,b,p,q are constants.
(this is a "linear" function in 2 dimensions.)

Prove the 2D divergence theorem for such a function, assuming the region
we integrate over, is a triangle.  (I.e. explicitly show the equality of 
the outward-flux single integral out of the triangle perimeter,
and the double integral of divF in the triangle interior, for this F.)



2. Suppose that F(n) is a real-valued function of an INTEGER n.
Suppose we are interested in the BEST function F(n), among
all possible such functions, where "best" means
the one with the minimum average (over all
subintervals from k to k+1) value of its squared slope
in the interval  A<=n<=B  (where A,B are fixed integers)
given that F(A) and F(B) are unalterable constants.

Write down the thing that is being minimized, and find the function
F that minimizes it.   
Now by considering the limit as A-B goes to infinity... what can we conclude
about real-valued functions of REALS that minimize their average
     2
F'(x)  value?



3.
Sometimes it is useful to work totally in some non-Cartesian
coordinate system, and express all vector components, etc,
in those coordinates.
For example, if we are using sphercial coordinates, we could
have some vector  (a,b,c)  where a is the component in the r,
b in the theta, and c in the phi, directions... not x,y,z...

If you look in book "CRC standard math tables and formulae"
you will see how to re-express lots of things, like curl(F),
in other coordinate systems, like in spherical coordinates
where F is now a function of r, theta, phi,  rather than x,y,z
and the curl now comes out as a vector (a,b,c) where a is in the r,
b in the theta, and c in the phi, directions!  You never
have to use x,y,z,  you can stay in the r,theta,phi world forever!

So... QUESTION:
  (a) what is this formula for curl(F(r,theta,phi) ?
  (b) use it to compute the curl of this F:
          radial component:   F1(r, theta, phi) = 1
          theta component:    F2(r, theta, phi) = r
          phi component:      F3(r, theta, phi) = 3 sin(theta)
      with answer expressed as a spherical-coords vector.


4.
Suppose you have a closed surface S, defined by some
parameterization   (x,y,z) = F(a,b)  for  0<a<1, 0<b<1, say.
But now suppose that S CHANGES WITH TIME.  I.e., the surface moves.
(Think of a balloon being inflated.)
Suppose the velocity each point on the surface is moving is given
by the VECTOR (d/dt)(x,y,z) = v(a,b).
 QUESTION: what is the rate of increase in the VOLUME surrounded by
 this surface?  (Write as a surface integral involving the 
 v and/or F vector-valued functions.)


5. 
In class we discussed how, if you know   gradF,  to reconstruct F.
But now suppose you know  curl F.  
QUESTION:
Can you devise a procedure for
reconstructing the vector-valued function F? Once you've devised your
procedure, try it out on
   curl F = (x, y, 5-2z)
and
   curl F = (x, x+y, 5-2z).
[And you might want to realize why you can immediately see that no F exists with
   curl F = (x, x+y, z).]


6. 
The scalar element of surface area  for a parameterized surface
(x,y,z) = F(p,q)   is   dArea = |dr/dp X dr/dq| dp dq    
where r=(x,y,z).   Fine.   But this only works for a 2-dimensional
surface in 3-dimensional (x,y,z)-space.  
QUESTION:
Suppose we have a 2-dimensional
surface in 4-dimensional (x,y,z,t)-space!  It is parameterized
in the same way by 2 parameters p,q:
   (x,y,z,t) = F(p,q).
Can you work out what  dArea  should be 
(as some function of p and q times dp dp)
now?


7.
The scalar element of surface area  for a parameterized surface
(x,y,z) = F(p,q)   is   dArea = |dr/dp X dr/dq| dp dq    
where r=(x,y,z).   Fine.   But this only works for a 2-dimensional
surface in 3-dimensional (x,y,z)-space.  
QUESTION:
Suppose we have a 3-dimensional
surface in 4-dimensional (x,y,z,t)-space!  It is parameterized
in the same way by 3 parameters m,p,q:
   (x,y,z,t) = F(m,p,q).
Can you work out what  dArea   (or really, dVolume, since this surface
"area" now is 3-dimensional) should be 
(as some function of p and q and m  times dm dp dp)
now?


8. Find a parameterization of the surface of the torus
got by rotating about the z-axis, a radius-a circle in the xy-plane,
with  center (b,0,0)    for  0<a<b.
[Hint, use as parameters, the angle of rotation about the z axis
and the angle of going around the original circle.]
Use this to find the surface area of the torus.

9.
show
integral from 0 to 1                                             2
   integral from 0 to 1                                  1     pi
          1/(1-xy) dx dy  = sum from n=1 to infinity of ---- = ---
                                                          2    6    .
                                                         n