rlopez

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15 years, 178 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.

MaplePrimes Activity


These are replies submitted by rlopez

@J4James 

Draw the graph. Right-click on the graph and in the Context Menu that appears, select Probe Info/Nearest datum. This turns the cursor on the graph to cross-hairs that trace the curves. Place the cross-hair on the intersection. Right-click again, select Probe Info/Copy data. The coordinates are now on the clipboard. Paste into the worksheet. Coordinates appear as a column vector.

I wrote a Newton iterator and solved for the intersection coordinates numerically, but that's a challenge for another day.

RJL

 

@J4James 

The following modification of the procedure given earlier computes f(b,S) using the same idea that each fixed pair of values of (b,S) generates a solution satisfying the first three boundary conditions. If the condition f(b,S)=b*S/2 is then imposed, a curve in the bS plane is generated. So, is S an arbitrary parameter whose value you are seeking, or is it a fixed and known number? If it is parameter, then a curve results. If it is a fixed number, then there is a unique value of b that satisfies all the conditions of the problem.

R:=proc(b,s)

local bc,Q,F;

bc:=D(f)(0)=1,f(0)=0,(D@@2)(f)(b)=0;

Q:=dsolve({eval(Eq,S=s),bc},f(y),numeric,output=listprocedure);

F:=rhs(Q[2]);

F(b,S);

I graphed the surface f(b,S) and superimposed the surface defined by b*S/2. These two surfaces intersect in a curve in space, whose projection onto the bS-plane contains an infinite number of (b,S) pairs that satisfy the ODE and the conditions imposed. (I found that the ranges b in [0,1.5] and S in [0,2] generated points for which f was positive.)

Now look at  your specific questions. You ask for a graph of f(b) vs the *parameter* S. If S is a parameter in the ODE, then the solution of the ODE is a function of both b and S, so consider that it is f(b,S), not f(b). What does it mean to graph f(b,S) against S? The graph of f(b,S) over the bS-plane is a surface. It's not clear to me what it is you want here.

Similarly, what does f"(0) mean in the context where S is a parameter? As soon as S becomes a free parameter in the ODE, the "solution" f becomes a function of both b and S, not just b alone. A clarification in the question would help here.

The odeplot command allows you to graph functions of the solution. This is stated on the help page for odeplot, the second bullet in the Description section.

However, it seems Preben has given you a much more detailed solution, one in which you need look up no command or its syntax.

Maple exports to rtf, but when imported into Word, 2D math becomes an image. The export/import process does not result in 2D math changed into the Word equation-editor format. I think one of the products found in the links provided by Markiyan Hirnyk would have to be used, and I think the route has to be via LaTeX to Word. Since the Word equation editor is a cut-down version of the editor in MathType, I suspect that product might be the best tool to employ. Unfortunately, a quick scan of the list of tools Markiyan found seems to indicate that at least one intermediate product would have to be purchased in order to take Maple's 2D math over into Word in a form that the Word equation editor could operate on.

I faced problems like this when I wrote the original version of my Advanced Engineering Math text (Addison Wesley, 2001). At that time, the classic worksheet was inadequate for expressing the full range of mathematical notation, and I used MathType to format some displayed equations, but this worked only in Windows, and not on other platforms. It was a nightmare to embed proper math notation in the classic worksheet, and I welcomed most heartily the extended functionalities in the newer standard interface that Maple provided in 2004.

In the not-too-distant past, I collaborated with an engineering prof who submitted our work to engineering journals that accepted only Word documents. I worked in Maple, but fortunately, my colleague was willing to rewrite everything in Word, a chore that I would not have enjoyed.

This interoperability of math formats is a knotty problem, and I sympathize with waseem, who asked the original question.

plot3d(abs(SphericalY(1,-1,u,v))^2,v=0..2*Pi,v=0..Pi,coords=spherical)

Last time this gave the proper graph was in Maple 13.

A bug report has now been filed.

Converting to elementary functions does help. I would use simplify(convert(...,elementary)) and square the absolute value, avoiding the need for conjugation.

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.

 

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.

 

@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.

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