@Carl Love 

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]):

@Carl Love 

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]):

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

@williamov 

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.

@williamov 

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.

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.)

@rlopez 

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.

@rlopez 

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.

@leaky 

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.

@leaky 

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.

@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.

@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.

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.

I think Maplesoft should take a look at the sliders in GeoGebra, that free software that people are using to create apps that don't need some underlying software to run.

In GeoGebra, sliders are one size, and relatively small. All that's there is a line and a disk sliding on the line. No numbers along the line. The only number visible is the value of the slider corresponding to the position of the disk. This permits all kinds of labeling generated by the slider, but removes the problem of providing space for displaying all the value-labels. I've seen this in demos at conferences, and find this approach to sliders most satisfactory.

RJL Maplesoft

Doug pointed this out to me privately by email, and I immediately modified the graph by adding a view option to the plot command. As soon as the original file is replaced with the modified one, downloads will not have this "glitch" in the graph.

I raised Doug's point with our graphics people. Apparently, I've already hit on the best cure by adding the view option.

RJL Maplesoft