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Help with Implicit Differentiation, etc...

(This message was originally intended for a local University Question &
Answer ( session, but if anyone else knows
how to answer my questions, kindly let me know via a post. (MathCentral
couldn't provide me with too many good resources on this subject. Also,
websites are preferred to books because of the size of our mathematics



I have the following inplicitly defined curve that I found in the
American Mathematical Monthly (1975, I think). Apparently,
Gauss had studied a few properties of the curve. First of all, let me
call funct[x,y] constant. For graphing purposes later on, I
chose funct = 7. I then equated funct as follows: funct[x_,y_]:=
DefiniteIntegral ( (x Sin[ at ])**2 + (y Cos[ at ])**2)**(-1/2)
D(@)  with <<[0<@<pi/2]>> as the integration limits (where @ means
"theta"). This integral, upon evaluation for certain x and
y, gives funct as a function of x and y (I later equate this to the
constant, 7, but I could probably equate to almost any function
of x and y in general). Amongst other interesting properties of this
integral, replacing x by (x+y)/2 and y by sqrt(x y) gives the
same indefinite integral as before (assuming that limits of integration
change correspondingly).

So, I gave the integral few x,y values to chew on. ImplicitPlot
originally tried convincing me that 0,0 is a bad first spot to
choose, so I tried a table of possible interesting regions of x,y values
and came up with the following Mathematica command to
make life easy:

(For[j = 1/4, j < 6,
    For[i = 1/3, i < 6,
      Print["x=", i, "  y=", j, "  funct[x,y]=", funct[i, j],
        " ~ ", N[funct[i, j], 5]]; i = i + 1/3]; j = j + 1/4])  //
Notice that I ignored the 0 values. They seem to want to make the
integral diverge.

We can then call ImplicitPlot into Mathematica and use the contour plot
version of ImplicitPlot (regular ImplicitPlot apparently
hates evaluating integrals in the middle of a plot command). I had to
experiment with a few constants to equate funct to to
detemine which choice gave the best looking plot near 0. Zero didn't
appear to be a good choice and 1, 1/2, 1/3, 2/3, 3/2, 2, 3
weren't making the graph look any prettier. I typed in seven, and I
thought, wow, maybe this function is dresed up as a
hyperbola! (I wonder how the graph behaves as the constant -->oo). My
Plot range was 1/7<x,y<7 if you are wondering and
my AspectRatio=1. You might also wish to try [-7, -1/7]. Here's what
another of my commands looks like:

(For[j = 1, j < 7,
    ImplicitPlot[funct[x, y] == j/7, {x, 1/7, 7}, {y, 1/7, 7}]; j = j +

Now that I think that I've got something that behaves somewhat like an
hyperbola, I want to differentiate this seemingly
complex function just to be sure. My questions: By differentiating
implicitly with respect to x (or y, I don't think that it matters
because of the variety of symmetry that this particular function
exhibits), how can I (if possible) explicitly solve for dy/dx (I want
to explicitly find dy/dx and then integrate my result in terms of x to
see if I really am getting an hyperbola), d2y/dx2, the
curvature function, the arclength function, etc or integate this curve
funct[x,y] with respect to y. Can y be explicitly
rewritten as a function of x only (ie no extra parameters such as @ or t
(for time) hanging around). Can funct[x,y] be rewritten
as a doubly infinite series/product of x and y = const? Or better yet:
can the x and y terms be seperated from each other in such
a way that double series would distribute into two seperate series with
one solely as x; one solely as y (ie if absolute (or
conditional) convergence holds, is (sum(sum(g[x,y]))=sum(g1[x])
sum(g2[y]) for some trio of functions (g,g1,g2))? This part
would be useful for doing some partial approximations and to see whether
the partials converge uniformily as either x, y, or the
constant approach oo or 0 (or other singularities). What is else is
known about this function, funct[x,y]?

There also exist functions such as those found in Stewart's calculus
which are called Fresnel functions in which the function
depend on the value of limits of integration (see p. 287, p. 292 #11,
etc). How could we write these functions as power series
in terms of x and not only differentiate, but integrate, with respect to
x? Can we also classify any types of functions as functions
whose nth derivative is h[n_,x_] where n in this case would require to
be an integer until we extended the idea of differentiation
to include non-integer differentiation? What about functions where as
n->oo, does h[n,x] have a limiting function, h[x]? Or, by
summing all of up the nth derivatives of the function, are there some
non-trivial functions that become limiting? Polynomials,
exponential and trig functions might be called trivial. I am thinking
more along the lines of Bessel and Hypergeometric functions.
If you know of any accesible resources to start at (particularily the
internet) could you please post in the virtual
resource room?

This could make for an interesting laboratory investigation for curve
sketching lovers of Math 110 (although, it might seem as if
computer software would also be beneficial, if not necessary). Disguise
a few partial approximations of funct that do not
converge uniformly and tell them to try and approximate funct by funct1,
funct2, etc.. They'd have no idea what funct1, funct2
would look like because they'd first have to evaluate the implicitly
defined partail approximation to funct1, funct2 (which would
also be definite integral equations that are not necessarily reducible
to functions such as y=1/x, etc until we'd let n->oo as funct
n ->funct[x,y]). Of course, the intelligent mathematicians in the class
would realize that curve sketching not only depends on
differentiation, but on the importance of evaluating limits in their
proper order. I am surprised that we don't cover more
sketching techniques such as how one family of curves becomes asymptotic
to any curve (not just the straight lines y=0, x=0 or
y=mx+b). I think that a whole elementary math class could be centred
around curve sketching techniques, etc. In particular, I
don't really see why we don't define curves in calculus (as opposed to
functions of one variable in particular); with the power of
the implicit function theorem, it seems that many of our questions for
multiple variables obey the same rules as they do for
y=f[x]. To me, being introduced to something like const= g[x,y,z] seems
to be more easily applicable to a special case of
const= g[x,y,0].

As you probably have guessed by now,

Kai G. Gauer

(oh yeah, and have all the fun that you in

your investigations of this curve)  ;-)

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