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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 answers submitted by rlopez

The Gibbs phenomenon: near a jump discontinuity in f(x), every partial sum of the Fourier series for f(x) exhibits the Gibbs spike, which tends to approximately 9% of the jump in f(x).

Maple does not even have built-in tools for the Fourier series, let alone for the Gibbs phenomenon. See other posts on this forum for add-on packages for Fourier series. To my knowledge, none of these packages address the Gibbs phenomenon, but it's been a number of years since I wrote a series of articles on these packages for the Tips&Techniques column in the Maple Reporter. These articles can be found in the Application Center.

You must have kept the default setting for the behavior of the underscore. In this default mode subscripted quantities become Atomic Variables. (Select "Atomic Variables" in the View menu to see all such in magenta.) The mess you are getting for the derivative of theta_sub_1 is because Maple is corrupting an Atomic Variable upon differentiation. If you made theta_sub_1 a table entry rather than an Atomic Variable, its derivative would be displayed with an overdot, just as it is on the x(t).

Given the setting you have for the underscore, when you subscript theta, press Control+Shift+minus. This will make theta_sub_1 an entry in a table whose name is theta, and the differentiation will be displayed properly. Of course, I'm assuming that there is no intrinsic reason why theta_sub_1 has to be Atomic, and that you are free to render it as a table entry.

Yes, this has happened to me, and more than once. I believe the cure I apply is to resize the window, making it slightly larger or smaller.

If the input that creates equation (2) has square brackets as in [...], change them to round parentheses as in (...). Next time, upload a worksheet via the green up-arrow in the toolbar. The embedded image is much too small to see clearly what you did.

You really don't want to calculate all the determinants needed for Cramer's rule. Instead, obtain the LU decomposition with the LUDecomposition command in the LinearAlgebra package.

It would help to make the Marker column visible. Go to the View menu and select the option "Markers". A thin gray column will open at the left edge of the Maple workspace and yoiu will see in it pairs of opposing triangles. Each such pair delineates a document block in which you can enter typeset math if you are in math mode, or text, if you are in text mode. In math mode, 2D math will appear in a blue rectangle. If you right-click on this math, you will have the option of changing this executable math to non-executable math that will then be in a gray rectangle.

Somewhere in here is the reason why your 2D math is not executing.

There are integrals for which a general result is obtained, but  which are not valid for particular values of a parameter such as n. These coefficients have to be evaluated with a separate integral for the particular values of n at stake.

At least one of the packages that users have created for generating Fourier series gets this right - the code identifies the special cases and determines the appropriate integral to obtain those coefficients. In fact, a recent thread on this forum dealt with at least two of those add-on packages.

A valid link in a Maple worksheet exported to pdf does not seem to be preserved in the pdf. I did find that I could set the link in the pdf itself. Is this a pdf-thing, or is there a way to export links from Maple to pdf?

This would be a useful feature for Maple if the shortcoming is in Maple. If the shortcoming is in pdf itself, shame.


FlowLine( VectorField( <-y -z, x + y/5, 1/5 + (x-5/2)*z>),[<1,1,1>], fieldoptions=[transparency=.3,fieldstrength=fixed,grid=[5,5,5],arrows=SLIM], flowlineoptions=[color=black,thickness=2], axes=frame,caption="");

The number of initial points can be more than one: Just add them to the list containing the initial point (1,1,1).

The arrows of the field and integral curves (flow lines) emanating from the initial points are drawn.

The ApplyLinearTransformPlot command in the Student LinearAlgebra package may give you what you want, provided you can express the linear transformation as a matrix. The command then applies this matrix to a plot data structure.

Additional discussion and examples can be found in the November 2013 Tips & Techniques article "Locus of Eigenvalues," available in the Application Center.


To obtain a locus of continuity class C1 (continuously turning tangent), one often needs to change branches. This can be seen for characteristic equations that can be solved in closed form. Once the degree gets high enough, only a numeric solution for the eigenvalues is feasible. Maple's numeric solvers return a sorted list of eigenvalues. The article concludes with the observation "There is no user-control of this sort, but even if there were, what sorting rule could be invoked across an eigenvalue with algebraic multiplicity greater than 1? It would seem that the only way to define a unique locus of eigenvalues is to require that it be of class C1, that is, that it have a continuously turning tangent."

(Not sure if the links included here are effective.)

I assume the question means "how do I include a link to a URL in a Maple worksheet?"

If so, then here's howto Hyperlink a word to the URL. To do that, select the "hot" word (it has to be text, not typeset math), right-click and select Convert To/Hyperlink. In the Hyperlink dialog that opens, select URL in the "Type" dropdown box. Paste the complete URL in the "Target" box. Click OK.

An alternate path to the Hyperlink dialog is via the Format menu option Convert To/Hyperlink.

If this wasn't the question, please clarify.

The Euler-Lagrange equation can be displayed using typeset (2D) math and items from appropriate palettes. To write a semblance of the equation that actually executes, you will have to accept a compromise.

For display purposes, you only need the differentiation templates available in the Calculus palette, or for displaying the partial derivatives use the partial derivative "squiggle" (PartialD) found in the Operators palette. The overdot can be set with one of the templates in the Layout palette, or can be set from the keyboard by pressing Control Shift Quote to jump on top of q and pressing the period key.

The compromise you have to accept to make the equation executable is that the L cannot be in the numerator of the partial derivative operator. It must be to the right of the operator that is obtained from the Calculus palette.

The following two steps worked for me.



The first command produces a list of real numbers; the second, returns the "location" of the minimum.

You could combine both into one step: min[inded](abs~(b))

The tilde maps the abs command onto each element of the list b. (The tilde is a newer construct than the older map command.)

In addition to the LinearSolveSteps command, there are other tools in Maple to help with solving such linear equations.

For example, bring up the Context Menu (right-click in Windows, etc., elsewhere) and select "Manipulate Equation." An interactive tool pops up. In this tool, you can apply the various transformations you see in the stepwise solution provided by the LinearSolveSteps command.

You can also select terms in the equation and wait for Maple to provide suggestions for transformations (i.e., a next step).

If you know what steps you want to apply, you can also implement them stepwise from the keyboard. For example, to add 3 to both sides of the equation x-3=7, write (x-3=7)+3 and Maple will return x=10.

If you want to start out knowing what the solution is (so that you can recognize it when you find it stepwise) bring up the Context Menu and select the Solve option.

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