# Maplesoft Blogger Profile: Robert Lopez

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## Maple Fellow

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.

## Volumes by Slicing

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Some calculus texts compute volumes of solids by the method of "slices" before they discuss the methods of disks and shells. On the other hand, there are texts that start with disks and shells, then throw in a few examples of slices. In any event, these calculations are supposed to be illustrations of how definite integration is an additive process. Unfortunately, students often get lost in the details of the individual examples, and fail to see that all these calculations are just demonstrations that definite integration is a process of addition.

## The Challenge of the Challenger Puzzle

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Do an internet search on "Challenger Puzzle" and you will find descriptions and solvers for a puzzle that involves sums of integers from one to nine. Indeed, on a 4 × 4 grid where sixteen integers would fit, four are given, along with the row, column, and diagonal sums of the numbers not shown. The object of the puzzle is to discover the missing twelve numbers.

Unlike Sudoku, the digits can repeat. And unlike Sudoku, the puzzle can have multiple solutions. In fact, "There may be more than one solution" is explicitly stated below the directions, copyrighted by King Features Syndicate, Inc., that appear in my local newspaper, the Waterloo Region Record.

## Using Sliders to Vary Parameters in an Initial...

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Recently, I received an email from a physics instructor asking for help in building a tool that would display the solution of the initial value problem

with the four parameters under the control of sliders. (Of course, we recognize that this equation governs the damped, driven linear oscillator, and that the request to endow its solution with sliders is in service of visualization of the change in the nature of the solution as the parameters vary.)

## Why Slopes of Perpendicular Lines Are Negative...

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It was years since I "derived" the result that slopes of perpendicular lines were negative reciprocals of each other. So I thought it would be easy to show that when , where, in Figure 1, is the slope of line (black) and is the slope of line (red). Clearly, lines and are perpendicular when .

## It's Better With Maple

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Recently, I had to write a brief introduction to the precalculus topic "Vertical Translation of Graphs." Figure 1 ( in black, in red) says just about everything.

 Figure 1   The red curve () is the black curve () vertically translated upward by one unit.

But is the issue all that trivial? Although the curves are vertically separated by one unit, they don't look uniformly spaced. The animation in Figure 2 helps overcome the optical illusion that makes it seem like the black curve bends towards the red curve, even though the curves are congruent.

## Visualizing a Parallel Field in a Curved Manifold

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It was 1992 when Mel Maron and I had just published the third edition of Numerical Analysis: A Practical Approach.  One of our editors made the suggestion that a Maple version of an advanced engineering math book should be written. For the next five years I steadfastly resisted the challenge.  Finally, in 1997 I signed a contract with Addison Wesley for a 1000-page AEM text, the manuscript due in two years.

Rose-Hulman Institute of Technology where I was teaching in the math department is on the quarter system, and math faculty normally teach twelve contact hours.  Calculus classes are five hours per week, so for each calculus course taught, a faculty member picks up an extra hour.  To minimize prep time, I wrangled three courses all the same, but they had to be calculus courses, so I was teaching fifteen contact hours and writing what turned out to be a 1200-page text.

After the first two quarters of academic year 1997, I needed to come up for air, so I set aside the project and spent several months putting together a Maple-based tensor calculus course. Happily, I even got to teach it in the following school year. One of the high points for me was animating a parallel vector field along a latitude on a sphere.

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