Maplesoft Blogger Profile: Robert Lopez

A wealth of knowledge is on display in MaplePrimes as our contributors share their expertise and step up to answer others’ queries. This post picks out one such response and further elucidates the answers to the posted question. I hope these explanations appeal to those of our readers who might not be familiar with the techniques embedded in the original responses.

Before I begin, a quick note that the content below was primarily created by one of our summer interns, Pia, with guidance and advice from me.

The Question: why is 2*cos(x)^2-1 simpler than 1-2*sin(x)^2

The author, nm, asked why 2*cos(x)^2-1  was simpler than 1-2*sin(x)^2 according to Maple. nm wrote:

I looked at help trying to understand why Maple thinks 2*cos(x)^2-1 is simpler than 1-2*sin(x)^2 but did not see it. I was expecting to see cos(2*x) as a result.

Preben Alsholm answered nm’s question by recommending the use of the combine command to obtain the result he was expecting to see, as well as a further explanation on how the simplify command works. Alsholm wrote:

Use combine to obtain what you want:
combine(1-2*sin(x)^2);

simplify has a general preference for cos over sin. That doesn't mean however, that it turns sin into cos at all costs:

simplify(sin(x));
##Try also
simplify(1-2*sin(x)^2,size);

simplify doesn't necessarily get you the simplest result in the common sense of the word 'simplify'. Try as another example

expand((x+y)^3);
simplify(%);
factor(%);

As always, Alsholm provided an accurate, thoughtful response. But for those just learning Maple, I thought some additional explanation could be helpful.

Let’s talk more about the simplify command and combine function

The simplify command applies simplification rules to an expression. Its parameters can be any expression.

The combine function applies transformations which combine terms in sums, products, and powers into a single term. For many functions, the transformations applied by combine are the inverse of the transformations that are applied by expand. For example, consider the well-known identity:

sin(a + b) = sin(a) cos(b) + cos(a) sin(b)

The combine function applies the identity from right to left, whereas the expand function does the reverse.

I hope that you find this useful. If there is a particular question on MaplePrimes that you would like further explained, please feel free to contact me.

You are teaching linear algebra, or perhaps a differential equations course that contains a unit on first-order linear systems. You need to get across the notion of the eigenpair of a matrix, that is, eigenvalues and eigenvectors, what they mean, how are they found, for what are they useful.

Of course, Maple's Context Menu can, with a click or two of the mouse, return both eigenvalues and eigenvectors. But that does not satisfy the needs of the student: an answer has been given but nothing has been learned. So, of what use is Maple in this pedagogical task? How can Maple enhance the lessons devoted to finding and using eigenpairs of a matrix?

In this webinar I am going to demonstrate how Maple can be used to get across the concept of the eigenpair, to show its meaning, to relate this concept to the by-hand algorithms taught in textbooks.

Ah, but it's not enough just to do the calculations - they also have to be easy to implement so that the implementation does not cloud the pedagogic goal. So, an essential element of this webinar will be its Clickable format, free of the encumbrance of commands and their related syntax. I'll use a syntax-free paradigm to keep the technology as simple as possible while achieving the didactic goal.

Notes added on July 7, 2015:

• This live webinar has already been presented and you can view the recording here: http://www.maplesoft.com/webinars/recorded/featured.aspx?id=922
  
  
• For one reaction to a point made in the webinar, see this post.

Fourteen Clickable Calculus examples have been added to the Teaching Concepts with Maple area of the Maplesoft web site. Four are sequence and series explorations taken from algebra/precalculus, four are applications of differentiation, four are applications of integration, and two are problems from the lines-and-planes section of multivariate calculus. By my count, this means some 111 Clickable Calculus examples have now been posted to the section.

Thirteen Clickable Calculus examples have been added to the Teaching Concepts with Maple section of the Maplesoft web site. The additions include examples in algebra, differential and integral calculus, lines-and-planes in multivariate calculus, and linear algebra. By my count, this means some 97 Clickable Calculus examples are now available.

In the Algebra/Precalculus section, examples of an

In a webinar on July 10, 2013, I solved the related rate problem:

Helium is pumped into a spherical balloon at the constant rate of 25 cu ft per min.
At what rate is the surface area of the balloon increasing at the moment when its radius is 8 ft?

A question in the Q&A at the end of the Webinar asked if it were possible to have an animation illustrate the expanding sphere and the rate of change in the surface area thereof. 

In Maple 17, the Student MultivariateCalculus package has been augmented with fifteen new commands relevant for defining and manipulating lines and planes. There already exists a functionality for this in the geom3d package whose structures differ from those in the new Student packages. Students...

Ten more Clickable Calculus solutions have been added to the Teaching Concepts with Maple section of the Maplesoft web site. Solutions to problems include examples in algebra, differential and integral calculus, lines-and-planes in multivariate calculus, linear algebra, and vector calculus.

The algebra additions include an example illustrating how a

Revision Note:
I have updated the graph in the attached Maple document based on Doug Meade's comment below.
CarTalkPuzzler_9-22-.mw 

Car Talk, a humorous phone-in program in which Tom and Ray Magliozzi (Click and Clack, the Tappet Brothers) diagnose and offer solutions for mysterious auto-related maladies, is carried by National Public Radio...

Eleven new Clickable-Calculus examples have been added to the Teaching Concepts with Maple section of the Maplesoft website. That means some 74 of the 154 solved problems in my data-base of syntax-free calculations are now available. Once again, these examples and associated videos illustrate point-and-click computations in support of the pedagogic message of resequencing skills and concepts.

This message has been articulated in ...

This post is a further exploration of the optimization problem of finding a point on f(x) = sinh(x) - xe^-3x closest to the point (1,7).  The problem is part of our Teaching Concepts with Maple web site, a collection of video examples and Maple worksheets designed to illustrate how Maple can be used to generate...

With the addition of ten new Clickable-Calculus examples to the Teaching Concepts with Maple section of the Maplesoft website, we've now posted 63 of the 154 solved problems in my data-base of syntax-free calculations. Once again, these examples and associated videos illustrate point-and-click computations, but more important, they embody the

My list of problems solved with Clickable-Calculus syntax-free techniques now numbers 154, spread over eight subject areas. Recently, Maplesoft posted to its website 44 of these problems, along with videos of their point-and-click solutions. Not only do these solutions demonstrate Maple functionalities, but they also have a pedagogic message, that is resequencing skills and concepts. They show how Maple can be used to obtain a solution, then show how Maple can be used to implement...

Being easy to use is nice, but being easy to learn with is better. Maple’s ease-of-use paradigm, captured in the phrases “Clickable Calculus” and “Clickable Math” provides a syntax-free way to use Maple. The learning curve is flattened. But making Maple easy to use to use badly in the classroom helps neither student nor instructor.

In the mid to late ‘80s,...

The directional derivative of a scalar function , computed in the direction u in Cartesian coordinates, is defined by

A recent Tips and Techniques article in the Maple Reporter contained the following five "gems" from my Red Book of Maple Magic. These 'gems' are tricks and techniques for Maple that I've discovered in my years here at Maplesoft. The previous 15 gems have appeared in three other issues of the Reporter, as...