rlopez

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14 years, 257 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 answers submitted by rlopez

In Maple, stepwise calculations are available for a limited number of operations. They include

 

Differentiation - use the Differentiation Methods tutor in the Student Calculus1 package

Limit              - use the Limit Methods tutor in the Student Calculus1 package

Integration     - use the Limit Methods tutor in the Student Calculus1 package

Solving a single linear equation - Use the LinearSolveSteps command in the Student Basics package

Expand a product of polynomials - Use the ExpandSteps command in the Student Basics package

Solving a linear or quadratic equation - Use the Equation Manipulator (Assistant) launched from the Context Menu

Finding eigenvalues - use the Eigenvalues tutor in the Student LinearAlgebra package

Finding eigenvectors - use the Eigenvectors tutor in the Student Linear Algebra package

Gauss reduction of a matrix to an upper triangular form - use the Gaussian Elimination tutor in Student LinearAlgebra, or load this package and use the Context Menu options Elementary Operations

Gauss reduction of a matrix to reduced row-echelon form - use the Gauss-Jordan Elimination tutor in Student LinearAlgebra, or load this package and use the Context Menu options Elementary Operations

Invert a matrix - use the Matrix Inverse tutor in Student LinearAlgebra, or load this package and use the Context Menu options Elementary Operations

Obtain an implicit derivative - Use one of three Task Templates in Calculus-Differential/Derivatives/Implicit Differentiation

For most calculations in mathematics, what Maple does "under the hood" is not, in general, useful for the student to experience. Maple commands are coded to produce results, not steps. On the other hand, Maple is robust enough that the individual steps in most calculations can be requested of Maple and assembled to represent a typical "by hands" solution of most problems.

Look at the examples in the "Teaching Concepts" section on the Maple web site. Here, you will find some 150+ examples of how to implement standard calculations in math, from precalculus, through calculus, linear algebra, differential equations, and vector calculus.

In addition, look at the ebook study guides for Calculus and Multivariate Calculus where hundreds of examples in these subjects are presented in a format that begins with a high-level Maple solution, followed by a stepwise solution in which Maple is induced to implement each of the requisite steps.

This two-step approach to using Maple in engineering mathematics can be seen in the Advanced Engineering Mathematics with Maple ebook. For a glimpse of how Maple can be used to augment insight and understanding in some engineering math examples, look at the recorded AEM webinar where six (out of 273) sections in the ebook are demonstrated.

Feel free to contact me if you want details on any of these options.

 

RJL Maplesoft

At first, the word "contour" had me pretty confused because I associate it with a curve on a surface. No surface was given, so what "contour" could the OP have meant. Eventually, I realized that the OP probably wanted to see the field arrows emanating from the plane z=0.

Well, early in 2009 I was asked the question "How can a slice of a vector field be graphed?" I answered this in the Maple Reporter's Tips&Techniques article Plotting a Slice of a Vector Field, available in the Maple Application Center here:

http://www.maplesoft.com/applications/search.aspx?term=plotting+a+slice&rcid=1337&sa.x=25&sa.y=13

In essence, do the following.

with(Student:-VectorCalculus):
F:=VectorField(<sin(x),cos(y),sqrt(sin(x)^2+sin(y)^2)>);

R:=PositionVector([x,y,0]);

PlotPositionVector(R,x=-1..1,y=-1..1,vectorfield=F);

There are many options available in the PlotPositionVector command for adjusting the look of the field arrows.

I hope this is what the OP wants.

RJL Maplesoft

Prior to Maple 17, the multivariate limit was an iterated limit taken along the coordinate axes, not a true multivariate limit wherein deleted neighborhoods are used as per the definition of a multivariate limit.

In Maple 17, a new funtionality was added to Maple, the bivariate limit for rational functions. By means of a new algorithm, the bivariate limit of a rational function is returned as if neighborhoods were used, and the limit is no longer just an iterated limit taken along the coordinate axes.

So, if the goal is to take the true bivariate limit as (x,y)->(a,b), then use limit(x-a+y-2,{x=a,y=b}); it will return the true bivariate limit in the plane, provided it is executed in Maple 17 or beyond.

(Recently, the restriction to rational function has been removed, and this version of the bivariate limit will appear in an upcoming version of Maple.)

RJL Maplesoft

From the help page for the ExpandSteps command:

The ExpandSteps command accepts a product of polynomials and displays the steps required to expand the expression.

Sorry, but there is no command in this package, or in Maple, that will show the steps for factoring.

RJL Maplesoft

Here's how I would do it.

In a Document block using typeset math, write the notation you want. For example, form x-hat and hit the Enter key.

Then execute the lprint command with argument a reference to the echoed symbol. It can either be an equation label or the % symbol.

The return of the lprint command shows the underlying Maple code that typeset math uses to create the notation x-hat. Copy and paste that into the code-edit region. Executing the contents of the code-edit region will re-create the x-hat symbol.

There may be other ways to do this, but this is a trick I learned from the Maple developers long ago.

RJL Maplesoft

The output shown for equation b is not what you would get if the input were processed by Maple. In the denominator of the output it appears that ycos has been typed without either a space or an explicit multiplication. It looks like the display assigned to your letter b might have been modified elsewhere and used in the solve command.

Compare your output display with the display obtained by gkolovidis. Similar inputs, but very different outputs.

RJL Maplesoft

To escape the tedium of using textplot, I've moved to the use of the drawing tools (click on Drawing in the plotting tools toolbar). Insert a text region, and in it use typeset math to write math notation. The text region can be moved with the mouse to exactly the right place.

Because a re-execution of the worksheet will not preserve what was written on the graph with the drawing tools, I export the marked-up graph, generally as a PNG file, and then import it back into the worksheet.

This is not a perfect solution, but it has allowed me to write content more quickly than otherwise.

RJL Maplesoft

Check the task template "Tangent Line" in Tools/Tasks/Browse : Calculus-Differential/Applications

RJL Maplesoft

In a Worksheet (red prompts, etc.) use Control T or the T icon in the toolbar to enter a text region. In such a region, text and typeset math can be entered. However, the typeset math cannot be evaluated (it is not "live").

In a Document (Document Blocks, etc.) simply insert a Document Block where needed, and change the input mode from Math to Text using either function key F5 or the buttons in the toolbar. Text and typeset math can be mixed in such a region but here, all typeset math is "live" so it can be evaluated, say, with Control =.

I have to disagree with Carl Love on this issue. In my experience and opinion, it was not easier to write math in the Classic interface. I wrote the 273 worksheets that became the print copy of Advanced Engineering Mathematics using Maple's Classic interface, and its limited ability to incorporate mathematical notation was a constant headache. When Maplesoft proposed writing AEM as an ebook in Maple 10, the enhanced ability to write mathematical notation was a significant improvement in exposition. In the years since, I've written a lot of content for Maplesoft in the various versions since Maple 10, and I continue to find that mathematical exposition in the "Standard" interface, especially in the Document mode, is far superior to writing exposition in the old Classic interface.

Insofar as this is an expression of preference, let's not have this ignite a new round of warfare between advocates of the old and new Maple interfaces, or between advocates of Worksheets vs. Documents. We all know that there are opinions on this and I don't want to throw gasoline on that fire.

RJL Maplesoft

Apply combine(expand... to the right-hand side of the solution given by Maple's dsolve command. The result will be exactly the same as the "hand" solution.

My attention was drawn to this example because of past experience with Maple's use of variation-of-parameters to find a particular solution to a linear, constant-coefficient ODE. In the past, Maple did not check that the particular solution might contain constant multiples of members of the fundamental set. This caused a great deal of grief in my Advanced Engineering Mathematics with Maple ebook, and I was interested in discovering if that was the issue here. Apparently, it's not.

RJL Maplesoft

collect(expand(f2),diff,distributed)

The result will be similar to what Acer's code gives.

RJL Maplesoft

The task template at Tools/Tasks/Browse-Differential Equations/ODEs/Phase Portrait - Autonomous Systems will create a phase portrait interactively. Enter the ODEs, the critical points, ranges for the window and the parameter along the orbits. Then press the Enter Data button. On the resulting direction field, click on a point through which an orbit is desired. A full phase portrait can be built up without the need to tediously enter each initial point into the phaseportrait command.

RJL Maplesoft

interface(imaginaryunit = j)

This frees the default I=sqrt(-1) and sets j = sqrt(-1). I can now be assigned to, and used as an ordinary name.

RJL Maplesoft

Mathematically, the problem presented is this: Given the function psi(x,eta) where eta is itself a function of x and y, obtain the partial derivatives of psi with respect to x and y. The notation for this is psi(x,eta(x,y)). Hence, the derivatives are diff(psi,x)+diff(psi,eta)*diff(eta,x) and diff(psi,eta)*diff(eta,y).

Unfortunately, implementing those commands in Maple won't work once eta is defined because psi will inherit that definition for the "eta" used in defining psi. One way around that is to define psi with x and eta, and define Eta(x,y) instead of eta(x,y). Then, change diff(eta...) to diff(Eta...). One can always replace eta by Eta in the resulting answers to see the form of the solution obtained by Preben.

Preben's method takes less coding in this case, but could become more cumbersome in more complex cases where an explicit display of the functional dependencies might just be too messy.

RJL Maplesoft

Not all the Large Operators are connected to the Maple language. In particular, none of the integral symbols are anything more than notation.

RJL Maplesoft

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