New Tips & Techniques

Classroom Tips and Techniques: Visualizing Regions of Integration

Dr. Robert Lopez — Tue, 06 Jul 2010 04:00:00 Z

Five of the new task templates in Maple 14 are designed to help visualize regions of integration for iterated integrals. In particular, there are task templates for double integrals in Cartesian and polar coordinates, and for triple integrals in Cartesian, cylindrical, and spherical coordinates. These task templates can be found at the end of the path

Tools ≻ Tasks ≻ Browse: Calculus - Multivariate ≻ Integration ≻ Visualizing Regions of Integration

Each of these task templates provides for iterating the relevant multiple integral in any of its possible orders. An example for each task template is provided.

The Astroid and Its Tangent Lines

Dr. Robert Lopez — Thu, 10 Jun 2010 04:00:00 Z

Tangents to the astroid have fixed length if restricted to a single quadrant. In fact, the envelope of a line of fixed length that has its endpoints on the coordinate axes in the first quadrant is just the astroid. In this month's article, we explore these two aspects of the astroid, a curve also known as the four-cusped hypocycloid, as well as the tetracuspid, cubocycloid, and paracycle.

Classroom Tips and Techniques: Fitting Circles in Space to 3-D Data

Dr. Robert Lopez — Mon, 17 May 2010 04:00:00 Z

In "A Project on Circles in Space," Carl Cowen provided an algebraic solution for the problem of fitting a circle to a set of points in space. His technique used the singular value decomposition from linear algebra, and was recast as a project in the volume ATLAST: Computer Exercises for Linear Algebra. Both versions of the problem used MATLAB® for the calculations. In this worksheet, we implement the algebraic calculations in Maple, then add noise to the data to test the robustness of the algebraic method. Next, we solve the problem with an analytic approach that incorporates least squares, and appears to be more robust in the face of noisy data. Finally, the analytic approach leads to explicit formulas for the fitting circle, so we end with graphs of the data, fitting circle, and plane lying closest to the data in the least-squares sense.

_{Simulink is a registered trademark of The MathWorks, Inc.}

Classroom Tips and Techniques: A Note on Parametric Plotting

Dr. Robert Lopez — Wed, 05 May 2010 04:00:00 Z

Under suitable conditions, the equations f(x, y, t) = g(x, y, t) and g(x, y, t) = 0 define the curve x = x(t), y = y(t) parametrically. If the algebra permits an explicit solution, the resulting curve can be drawn with Maple's basic parametric plotting functionality. However, when the algebra is recalcitrant, the equations must be solved numerically for a sequence of points, or converted to differential equations that are solved numerically. All three of these alternatives are explored in this month's article.

Classroom Tips and Techniques: Stepwise Solutions in Maple - Part 3 - Vector Calculus

Dr. Robert Lopez — Fri, 09 Apr 2010 04:00:00 Z

In our previous two articles we have discussed the commands, Assistants, Tutors, and Task Templates that implement stepwise calculations in algebra, calculus (both of one and several variables) and linear algebra. In this month's article, we illustrate the stepwise tools available in vector calculus.

Classroom Tips and Techniques: Stepwise Solutions in Maple - Part 2 - Linear Algebra

Dr. Robert Lopez — Mon, 08 Mar 2010 05:00:00 Z

In our previous article we have discussed the commands, Assistants, Tutors, and Task Templates that implement stepwise calculations in algebra and calculus (both of one and several variables). In this month's article, we illustrate the stepwise tools available in linear algebra.

Classroom Tips and Techniques: Stepwise Solutions in Maple - Part 1

Dr. Robert Lopez — Wed, 10 Feb 2010 05:00:00 Z

In Maple, there are commands, Assistants, Tutors, and Task Templates that show stepwise calculations in algebra, calculus (single-variable, multivariable, vector), and linear algebra. In this article we discuss Maple's functionality for providing these stepwise solutions to mathematical problems in algebra and calculus (both of one and several variables).

Classroom Tips and Techniques: Geodesics on a Surface

Dr. Robert Lopez — Tue, 08 Dec 2009 05:00:00 Z

Several months ago we provided the article Tensor Calculus with the Differential Geometry Package in which we found geodesics in the plane when the plane was referred to polar coordinates. In this month's article we find geodesics on a surface embedded in R³. We illustrate three approaches: numeric approximation, the calculus of variations, and differential geometry.

Classroom Tips and Techniques: Series Expansions

Dr. Robert Lopez — Thu, 05 Nov 2009 05:00:00 Z

Maple has the ability to provide various series expansions and their truncations, as well as complete formal series for a variety of elementary and special functions. In this month's article, we examine the relevant commands and interface devices that access these functionalities.

Classroom Tips and Techniques: Visualizing Regions of Integration

Dr. Robert Lopez — Wed, 21 Oct 2009 04:00:00 Z

In this month's article, the synergy between the visual and the analytic is demonstrated with a learning tool built with Maple's embedded components.

Cooling a Solid with a Cold Air Stream with MapleSim and Maple

Maplesoft — Fri, 10 Jul 2009 04:00:00 Z

This article looks at each of the components of cooling a solid with a cold air stream by convection.

Point-and-Click Tools for Control Systems Design

Maplesoft — Wed, 08 Jul 2009 04:00:00 Z

Maple 13 offers point-and-click access to powerful tools for control systems design. The DynamicSystems package, first introduced in Maple 12, offers a large selection of analytic and graphing tools for linear time-invariant systems, which are essential in control systems development. These powerful control systems analysis tools can now be accessed at the click of a button through context-sensitive menus, with no knowledge of commands or syntax required. In this Tips & Techniques document, you will learn how to use Maple's context-sensitive menus to access control systems design tools, without needing prior knowledge of specific Maple commands and syntaxes.

Classroom Tips and Techniques: Jordan Normal Form of a Matrix

Dr. Robert Lopez — Mon, 06 Jul 2009 04:00:00 Z

A square matrix is similar either to a diagonal matrix or to a Jordan matrix, that is, a diagonal matrix with a mix of 1s and 0s on the superdiagonal. This month's article explores computations by which the matrix of transition to Jordan form can be constructed, and in so doing, illustrates some of the underlying theory.

Classroom Tips and Techniques: Point-and-Click Access to the Differential Operators of Vector Calculus

Dr. Robert Lopez — Thu, 04 Jun 2009 04:00:00 Z

This month's article describes new tools that allow a syntax-free setting of coordinate systems so that the differential operators of vector calculus can be implemented via VectorCalculus[Nabla], the "del" or "nabla" symbol.

Using the New Fly-through Feature in Maple 13

Maplesoft — Wed, 03 Jun 2009 04:00:00 Z

As part of its new 3-D plotting facilities, Maple 13 provides fly-through animations. These animations help you to gain additional insight from your 3-D plots as a virtual camera flies through, over, under, into, and around your surface, letting you focus on points of interest from any angle to understand the trends and behavior of surfaces. In this Tips & Techniques, you will learn how to create fly-through animations using built-in and user-defined camera paths, how to select effective camera paths, and how to a variety of options to further customize the animation.

Classroom Tips and Techniques: An Implicit Analytic Solution of the Lotka-Volterra Model

Dr. Robert Lopez — Tue, 05 May 2009 04:00:00 Z

Maple can provide both an implicit and an explicit general solution to the Lotka-Volterra predator-prey model. Moreover, Maple has a variety of tools for obtaining these solutions and a phase portrait for the model.

Setting Initial Conditions on Integrator Blocks

Maplesoft — Mon, 04 May 2009 04:00:00 Z

This article discusses two ways of setting initial conditions on integrator blocks.

Notational Devices for ODEs

Maplesoft — Tue, 07 Apr 2009 04:00:00 Z

One of the earliest notational devices relevant to differential equations first appeared in Maple V Release 6. This is the declare command found in the PDEtools package. In this month's article, we will recollect the functionality provided by declare, and show how it compares to typesetting in Maple 12. In particular, we are interested in how to write and work with ordinary differential equations using the most natural and standard notations as provided by the Typesetting package.

Classroom Tips and Techniques: Trigonometric Parametrization of an Ellipse

Dr. Robert Lopez — Tue, 07 Apr 2009 04:00:00 Z

Computing a line integral around an ellipse is more easily done if the ellipse is described parametrically. In this month's article, we delineate how to obtain an exact trigonometric parametrization of an ellipse, even one whose axes are rotated with respect to the coordinate axes.

Classroom Tips and Techniques: Plotting a Slice of a Vector Field

Dr. Robert Lopez — Fri, 06 Mar 2009 00:00:00 Z

This month's article answers a user's question about plotting a "slice" of a vector field, and superimposing this graph on a density plot of the underlying scalar field. The "slice" of the vector field is a graph of the field arrows emanating from a coordinate plane.