## 2565 Reputation

15 years, 246 days

Dr. Robert J. Lopez, Emeritus Professor of Mathematics at the Rose-Hulman Institute of Technology in Terre Haute, Indiana, USA, is an award winning educator in mathematics and is the author of several books including Advanced Engineering Mathematics (Addison-Wesley 2001). For over two decades, Dr. Lopez has also been a visionary figure in the introduction of Maplesoft technology into undergraduate education. Dr. Lopez earned his Ph.D. in mathematics from Purdue University, his MS from the University of Missouri - Rolla, and his BA from Marist College. He has held academic appointments at Rose-Hulman (1985-2003), Memorial University of Newfoundland (1973-1985), and the University of Nebraska - Lincoln (1970-1973). His publication and research history includes manuscripts and papers in a variety of pure and applied mathematics topics. He has received numerous awards for outstanding scholarship and teaching.

## Look at the definition on the help page...

The help page for SphericalY provides the definition used in Maple for this function. There's a term of the form (lambda-mu)! which evaluates to (-2)! for the choice of parameters lambda=-1, mu=1. Factorials of negative integers are not defined. Moreover, the definition of SphericalY includes the condition that lambda-mu should nto be a negative integer.

I'm not sure if there is some other definition of SphericalY that would allow lambda-mu to be a negative integer. I have yet to see what the conversion to "elementary" does in this case.

## Giant Pysanka (Ukranian Decorated Easter...

A giant (31 ft tall) decorated Ukranian Easter egg was erected in Vegreville, Alberta, Canada back in 1975. Check the link http://www.cuug.ab.ca/VT/vegreville.html

While I was on the faculty at Memorial University of Newfoundland, St. John's, Newfoundland, Canada, I had a colleague in the CS department who worked on the design of the egg, a great part of which was determining just what is the shape of an egg.

I'll spoil the fun: The shape of the Vegreville egg isn't the same as the shape in this post to Primes.

## Tricking Maple...

Carl, I'm headed out the door and won't be back till next Wednesday. The following seems to work. I should test it by doing a numeric derivative on U, and comparing it to the comparable value of W. Perhaps you can confirm that this trick succeeded before I get back. If not, I'll have to go back to the drawing board next week.

PDE:={diff(u(x,t),t)=w(x,t), diff(u(x,t),x)=-w(x,t)};

IBC:= {u(x,0)=sin(2*Pi*x),u(0,t)=-sin(2*Pi*t)}:
pds:= pdsolve(PDE, IBC, numeric, time=t, range=0..1);

W:=rhs(pds:-value(output=listprocedure)[4]):

## Tricking Maple...

Carl, I'm headed out the door and won't be back till next Wednesday. The following seems to work. I should test it by doing a numeric derivative on U, and comparing it to the comparable value of W. Perhaps you can confirm that this trick succeeded before I get back. If not, I'll have to go back to the drawing board next week.

PDE:={diff(u(x,t),t)=w(x,t), diff(u(x,t),x)=-w(x,t)};

IBC:= {u(x,0)=sin(2*Pi*x),u(0,t)=-sin(2*Pi*t)}:
pds:= pdsolve(PDE, IBC, numeric, time=t, range=0..1);

W:=rhs(pds:-value(output=listprocedure)[4]):

## What is the real problem here?...

The relaxation method of solving a system of linear equations goes back to Southwell in the 1930s when computers had yet to be invented. It is a pencil-and-paper method that's at best linearly convergent.

There are better ways to use a computer to solve such systems. But what is the system? What's the real problem here? The title "Finding the Catenary" seems to suggest that what needs to be solved is either the differential equation for the catenary, or the algebraic equation that singles out a specific catenary that fits initial data.

Short of knowing what the real problem, it's hard to suggest alternates to the implementation of relaxation.

RJL Maplesoft

## VC vs. DG...

The vector calculus packages work with orthogonal coordinate systems. The Student package is limited to the five systems I mentioned in my earlier comments, but the "parent" VC package admits all the orthogonal systems that Maple knows.

In differential geometry, one can define non-orthogonal systems, and in these systems compute, for example, divergence. There's a lot of machinery that has to be setup to use the DG package, so  you made the right choice by sticking with VC. There's another option for the kind of calculations that the Student VC package handles, namely, the Physics:-Vectors package. I've looked at this, but am not as facile with it as I am with the VC packages. From what I've observed, I think the Physics:-Vectors approach is also a viable way to do vector calculus with as little overhead as necessary. On my to-do list is a thorough comparison of Physics:-Vectors and Student VectorCalculus. Unfortunately, days have only 24 hours.

## VC vs. DG...

The vector calculus packages work with orthogonal coordinate systems. The Student package is limited to the five systems I mentioned in my earlier comments, but the "parent" VC package admits all the orthogonal systems that Maple knows.

In differential geometry, one can define non-orthogonal systems, and in these systems compute, for example, divergence. There's a lot of machinery that has to be setup to use the DG package, so  you made the right choice by sticking with VC. There's another option for the kind of calculations that the Student VC package handles, namely, the Physics:-Vectors package. I've looked at this, but am not as facile with it as I am with the VC packages. From what I've observed, I think the Physics:-Vectors approach is also a viable way to do vector calculus with as little overhead as necessary. On my to-do list is a thorough comparison of Physics:-Vectors and Student VectorCalculus. Unfortunately, days have only 24 hours.

## Student VectorCalculus Package...

Note that the Student VectorCalculus package has commands for returning the Curvature and the RadiusOfCurvature of the osculating circle.

The Space Curves tutor draws and/or animates the osculating circle for a space curve. (You can trick the tutor into drawing it for a plane curve by giving zero as the third component of the vector defining the curve.)

## Identify q with f...

Sorry about that typo - I tend to use q when naming expressions. As I experimented with the expression f, I named it q in my worksheet and simply copied and pasted without looking carefully enough.

## Identify q with f...

Sorry about that typo - I tend to use q when naming expressions. As I experimented with the expression f, I named it q in my worksheet and simply copied and pasted without looking carefully enough.

## Conditional approach...

If f = sqrt(2*x+2)/(x+1), then 1/rationalize(1/f) returns 2/sqrt(2*x+2) and no obvious manipulation in Maple "cancels" the 2s.

For this f, the following gets the desired sqrt(2)/sqrt(x+1)

simplify(sqrt(simplify(1/rationalize(1/q^2)))) assuming x>-1

Hence, the title "Conditional approach" where squaring and then taking the square root will generally present sign issues for most expressions.

## Conditional approach...

If f = sqrt(2*x+2)/(x+1), then 1/rationalize(1/f) returns 2/sqrt(2*x+2) and no obvious manipulation in Maple "cancels" the 2s.

For this f, the following gets the desired sqrt(2)/sqrt(x+1)

simplify(sqrt(simplify(1/rationalize(1/q^2)))) assuming x>-1

Hence, the title "Conditional approach" where squaring and then taking the square root will generally present sign issues for most expressions.

## Automatic Simplification...

@Markiyan Hirnyk Typing sqrt((x+1)/2) and pressing the Enter key results in the "somewhat different expression" that the rationalize command produces. It's a result of Maple's automatic "simplification" rules. There's no simple way to get the expression in the form of the square root of the fraction.

## Automatic Simplification...

@Markiyan Hirnyk Typing sqrt((x+1)/2) and pressing the Enter key results in the "somewhat different expression" that the rationalize command produces. It's a result of Maple's automatic "simplification" rules. There's no simple way to get the expression in the form of the square root of the fraction.

## Automatic Simplifications...

If you type in sqrt((x+1)/2), and press the Enter key, Maple will automatically "simplify" the expression to what you get when rationalize is applied to (x+1)/sqrt(2*x+2).

There's nothing simple you can do to get Maple to output the expression in a different form.

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