Help with Implicit Differentiation, etc...

*To*: mathgroup at smc.vnet.net*Subject*: [mg18383] Help with Implicit Differentiation, etc...*From*: "Kai G. Gauer" <gauer at sk.sympatico.ca>*Date*: Wed, 30 Jun 1999 14:13:39 -0400*Organization*: SaskTel - Sympatico*Sender*: owner-wri-mathgroup at wolfram.com

(This message was originally intended for a local University Question & Answer (www.Mathcentral.uregina.ca) 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 library.) Thanks) Hi 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 + 1] 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) ;-)