rlopez

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14 years, 343 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.

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These are replies submitted by rlopez

RJL Maplesoft

Given a surface defined implicitly by an equation of the form f(x,y,z)=0, you can pick anything "convenient" for x=x(u,v), y=y(u,v) and solve f(x(u,v),y(u,v),z)=0 for z=z(u,v). Determining what the "convenient" choices are is an art, based on previously accumulated knowledge. There is no recipe by which you can determine what is going to be most "convenient."

Maple's ability to "do" algebraic manipulations would certainly be helpful, but without the insight into what functions to pick for x(u,v) and y(u,v), Maple is just a servant waiting for instructions.

RJL Maplesoft

Maple makes provision for grouping symbols to form a name. Such groupings are called atomic variables (earlier called atomic identifiers). These objects are simplest to construct using Typeset math. Form the grouping of symbols (in Typeset math), select all of it, and convert to atomic variable. There is a keyboard shortcut for this conversion: Control+Shift+A. Otherwise, use the Format menu and select "Convert to".

There is a complicated string of symbols created behind the scenes by this process. If you apply the lprint command to the atomic variable, you will see the ascii code that is associated. You could use that code in text-mode (1D math in a worksheet) if necessary, but it can be pretty ugly stuff, depending on what atomic variable was created.

In a document, if the option "Atomic Variables" in the View menu is selected, atomic variables will appear in color (purple?) in the document. Unfortunately, every time you need to re-use the atomic variable, you have to re-create it, or copy/paste it.

If an assignment is made to an atomic variable, the associated string will show up in the Variables palette. It might be possible to create a "snippets palette" containing the atomic variables, but clicking on an item in such a palette inserts it at the left margin of a new line. Again, you have to copy/paste to put it where you want it. Perhaps the simplest thing to do is to copy/paste to the bottom of the worksheet or to another parallel worksheet.

@leiniu Are you using Maple 2016.1 or just 2016? Have you added something to your Maple library that might be interfering with the calculation? Have you done a restart? Do you have access to Maple 2015 just to see if the problem is in your particular installation of Maple 2016?

RJL Maplesoft

Just copied and pasted your code into Maple 2016.1, and executed it. I obtained a graph of a curve. Not sure what happened in your session, or if Maple 2016 differs from 2016.1.

RJL Maplesoft

Have you tried contacting Maplesoft's Technical Support?

RJL Maplesoft

The vector remails parallel in the world of the 2D being who lives in the tangent plane. That observer sees no change in the vector. That is the meaning of having the derivative of the vector orthogonal to the tangent plane. To the eye of the 2D observer, the vector is not "changing." That is the essence of the struggle I had as a graduate student 50 years ago: what does the parallel field look like to an external observer, and what does it look like to the internal observer, which I can never be. I could only be the external observer, and had to rely on the mathematics to provide a prescription of what the internal observer would see. That observer sees no change in the field because the derivative (change) in the field has no component in the tangent plane.

The change in the vector at the end of the loop is then a measure of the intrinsic curvature of the manifold (sphere).

Joe Riel seems to have an additional insight into what's happening. I hope it doesn't take me another 50 years to fathom same!

The eliminate command provides four solutions, and if each is solved for, say, z, one obtains in each case 0, (y-x)/4, in keeping with what others have reported. Since elimination of the parameters gives the Cartesian representation of a plane, I would assume the manifold so defined is flat.

RJL Maplesoft

LocusOfRoots.mw

@rlopez OK, I just couldn't let this go. I've attached a worksheet where I've carried out the calculations sketched in my answer initially.  

lopez@maplesoft.com@alecjacobson 

The help page for the EulerLagrange command specifically states that the argument for this command is an expression in t, x(t), and x'(t). The Description section suggests that for higher-order functions, use variables to represent derivatives, and gives an example of how this might be done.

Alternatively, using Physics:-diff, you can differentiate with respect to a function such as x'(t). Hence, it is possible to implement the Euler-Lagrange equation from first principles. Decidedly more tedious, but certainly possible.

rlopez@Noor2015 Note that the syntax solve({sin(x)+y=0,y^2-x=0},{x=0..6,y=0..6}); is not valid. The solve command does not take any specification for location of roots. That syntax would work with the fsolve command, the numeric solver. The exact solver, the solve command, does not have that capability.

The solve command will not return "points" in the form (a,b) or [a,b]. Maple just does not do that. Ever.

Executing solve({sin(x)+y=0,y^2-x=0},{x,y}); returns solutions in the form of a RootOf construction. Applying the allvalues command to this expression yields two complicated, but complex, solutions, and a real third solution of the form {x=0,y=0}. This is how Maple indicates that the real solution is the point (0,0).

Assuming this last form of solve has had allvalues applied, and the result is called SOL. One way of picking out the real solution is by executing remove(has,[SOL],I)[]. This will result in the set {x=0,y=0}.

If you have a command that returns a set of equations such as {x=a,y=b} and you needed to have this result in the form [a,b], the simplest way to do that is to execute eval([x,y],{x=a,y=b}), that is, evaluate the template [x,y] using the information in the set of equations.

 

@tazatel Use the options shown in the following form of the PlotPositionVector command.

PlotPositionVector(R,x=-1..1,y=-1..1,vectorfield=F,vectorfieldoptions=[width=.05,color=green],vectorgrid=[3,3],scaling=constrained);

The help page is dense with descriptions of how the graph of curves and surfaces, along with associated vector fields, can be adjusted.

@vv The first appearance of a true bivariate limit functionality in Maple is in Maple 17. Check ?updates,Maple17,BivariateLimits. Initially, the algorithm worked for isolated singularities. Eventually, it was updated to allow for non-isolated singularities. And soon, the restriction to rational functions will disappear.

@pacew Your observation that after the graph has been constructed in the Plot Builder and inserted into the worksheet, the "system exhibits the same behavior." Once the graph is in the worksheet, the connection to the computational engine is lost, and the GUI is able to manipulate just the existing plot data-structure. It can't add new data to the structure, but only manipulate the data that the math engine generated.

This is a shortcoming that our developers have discussed for many releases, and correcting it will require a great change in how graphs are generated and then rendered.

Once a graph has been inserted into the worksheet, connection between the graph and the computational engine of Maple needed for doing more computing is lost. Hence, when the image is moved by the pan operation, it is the GUI that is changing the display, not the math engine. This situation holds for all changes made to the graph via the context menu for the graph.

It would be much better to use the Plot Builder to obtain the graph. In this Assistant, the graph can be manipulated because there is still a connection to the math engine. The Preview button in the Plot Builder lets you see what your graph will look like when you select Plot.

RJL Maplesoft

@Mac Dude In Maple 2016, document block management is available through the Edit menu option "Document Blocks". With this option, document blocks can be created, removed, edited, etc. It also helps to make the "Marker Column" visible by selecting the View menu option "Markers". This column on the left of the workspace shows pairs of opposing triangles that delineate each document block.

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