New Tips & Techniques

Classroom Tips and Techniques: Fourier Series and an Orthogonal Expansions Package

Dr. Robert Lopez — Mon, 14 May 2012 04:00:00 Z

The OrthogonalExpansions package contributed to the Maple Application Center by Dr. Sergey Moiseev is considered as a tool for generating a Fourier series and its partial sums. This package provides commands for expansions in 17 other bases of orthogonal functions. In addition to looking at the Fourier series option, this article also considers the Bessel series expansion.

Classroom Tips and Techniques: Custom and Task Palettes

Dr. Robert Lopez — Thu, 12 Apr 2012 04:00:00 Z

New in Maple 16, the Custom palette is a palette added to Maple by the user. It is populated with task templates that are already in the Task Browser Table of Contents. A separate Tasks palette can be populated with task templates created by the "Create Task" option in the Context Menu for any selection in a worksheet. This article sheds light on these new functionalities, and gives an example of a Custom palette developed to capture part of the geom3d package in task templates.

Classroom Tips and Techniques: Caustics for a Plane Curve

Dr. Robert Lopez — Mon, 12 Mar 2012 04:00:00 Z

This article shows how to construct and visualize a caustic, the envelope of lines emanating from a fixed point, and reflecting off a plane curve.

Classroom Tips and Techniques: Sliders for Parameter-Dependent Curves

Dr. Robert Lopez — Tue, 14 Feb 2012 05:00:00 Z

Methods for building slider-controlled graphs are explored, and used to show the variations in the limaçon. Then, the conchoid of a cubic is explored with the same set of tools.

Classroom Tips and Techniques: An Undamped Coupled Oscillator

Dr. Robert Lopez — Tue, 10 Jan 2012 05:00:00 Z

Even for just three degrees of freedom, an undamped coupled oscillator modeled by the ODE system M ü + K u = 0 is difficult to solve analytically because, ultimately, a cubic characteristic equation has to be solve exactly. Instead, we simultaneously diagonalize M and K, the mass and stiffness matrices, thereby uncoupling the equations, and obtaining an explicit solution.

Classroom Tips and Techniques: Simultaneous Diagonalization and the Generalized Eigenvalue Problem

Dr. Robert Lopez — Tue, 06 Dec 2011 05:00:00 Z

This article explores the connections between the generalized eigenvalue problem and the problem of simultaneously diagonalizing a pair of n × n matrices.

Given the n × n matrices A and B, the generalized eigenvalue problem seeks the eigenpairs (lambda_k, x_k), solutions of the equation Ax = lambda Bx, or (A - lambda B) x = 0. If B is nonsingular, the eigenpairs of B^-1 A are solutions. If a matrix S exists for which S^T A S = Lambda, and S^T B S = I, where Lambda is a diagonal matrix and I is the n × n identity, then A and B are said to be diagonalized simultaneously, in which case the diagonal entries of Lambda are the generalized eigenvalues for A and B. Such a matrix S exists if A is symmetric and B is positive definite. (Our definition of positive definite includes symmetry.)

Classroom Tips and Techniques: Gems 21-25 from the Red Book of Maple Magic

Dr. Robert Lopez — Wed, 09 Nov 2011 05:00:00 Z

From the Red Book of Maple Magic, Gems 21-25: Simplifying an absolute value, extracting coefficients from a complete quadratic, "dot and stick" graphs of discrete data, restoring the order of terms in an expression, and finding the smallest positive zero of a non-polynomial function.

Classroom Tips and Techniques: Directional Derivatives in Maple

Dr. Robert Lopez — Fri, 14 Oct 2011 04:00:00 Z

Several identities in vector calculus involve the operator A . (VectorCalculus[Nabla]) acting on a vector B. The resulting expression (A . (VectorCalculus[Nabla]))B is interpreted as the directional derivative of the vector B in the direction of the vector A. This is not easy to implement in Maple's VectorCalculus packages. However, this functionality exists in the Physics:-Vectors package, and in the DifferentialGeometry package where it is properly called the DirectionalCovariantDerivative. This article examines how to obtain (A . (VectorCalculus[Nabla]))B in Maple.

Classroom Tips and Techniques: Gems 16-20 from the Red Book of Maple Magic

Dr. Robert Lopez — Fri, 23 Sep 2011 04:00:00 Z

From the Red Book of Maple Magic, Gems 16-20: Vectors with assumptions in VectorCalculus, aliasing commands to symbols, setting iterated integrals from the Expression palette, writing a slider value to a label, and writing text to a math container.

Classroom Tips and Techniques: Circle Inscribed in a Parabola

Dr. Robert Lopez — Wed, 17 Aug 2011 04:00:00 Z

The center of a fixed-radius circle inscribed in a parabola is found. This is generalized to members of the family y = x^{2 n}, where n is an integer greater than 1. For what values of the radius are there only circles tangent at the origin? How many circles can be inscribed in one of these curves?

Classroom Tips and Techniques: Steepest-Ascent Curves

Dr. Robert Lopez — Tue, 19 Jul 2011 04:00:00 Z

Steepest-ascent curves are obtained for surfaces defined analytically and digitally.

Classroom Tips and Techniques: Nonlinear Fit, Optimization, and the DirectSearch Package

Dr. Robert Lopez — Wed, 15 Jun 2011 04:00:00 Z

In this month's article, I revisit a nonlinear curve-fitting problem that appears in my Advanced Engineering Mathematics ebook, examine the role of Maple's Optimization package in that problem, and then explore the DirectSearch package from Dr. Sergey N. Moiseev.

Classroom Tips and Techniques: Factoring a Quadratic Polynomial

Dr. Robert Lopez — Tue, 24 May 2011 04:00:00 Z

Factoring a quadratic polynomial by inspection - is this a necessary skill, and if it is, how can students be helped to master it?

Classroom Tips and Techniques: "Events" and the Numeric Solution of ODEs

Dr. Robert Lopez — Tue, 19 Apr 2011 04:00:00 Z

Several examples of the use of "events" in solving differential equations numerically are given. The main example is the "skydiver" problem where free-fall changes to descent with an open parachute.

Classroom Tips and Techniques: Yet More Gems from the Little Red Book of Maple Magic

Dr. Robert Lopez — Mon, 21 Mar 2011 04:00:00 Z

Five more bits of accumulated "Maple magic" are shared: the limit of Picard iterates, combining radicals, factoring, yet another trig identity, and sorting strategies.

Classroom Tips and Techniques: More Gems from the Little Red Book of Maple Magic

Dr. Robert Lopez — Tue, 22 Feb 2011 05:00:00 Z

Five more bits of "Maple magic" accumulated in recent months are shared: "if" with certain exact numbers, constant functions, replacing a product with a name, assumptions on subscripted variables, and gradient vectors via the Matrix palette.

Classroom Tips and Techniques: Gems from the Little Red Book of Maple Magic

Dr. Robert Lopez — Fri, 14 Jan 2011 05:00:00 Z

Five bits of "Maple magic" accumulated in recent months are shared: converting the half-angle trig formulas to radicals, tickmarks along a parametric curve, writing unevaluated math on a graph, changing Maple's differentiation formulas, and drawing a decent surface for a function containing a square root.

Classroom Tips and Techniques: Partial Derivatives by Subscripting

Dr. Robert Lopez — Wed, 15 Dec 2010 05:00:00 Z

As output, Maple can display the partial derivative ∂/∂x f(x,y) as f_x; that is, subscript notation can be used to display partial derivatives, and it can be done with two completely different mechanisms. This article describes these two techniques, and then investigates the extent to which partial derivatives can be calculated by subscript notation.

Classroom Tips and Techniques: Maple Meets Marden's Theorem

Robert Lopez — Tue, 16 Nov 2010 05:00:00 Z

Dan Kalman ascribes to Marden, and describes it as the most amazing theorem: If three noncollinear points in the complex plane forming the vertices of a triangle are interpolated by a (monic) cubic polynomial p(z), the zeros of p'(z) are the foci of a unique ellipse (the Steiner in-ellipse) that is tangent to the sides of the triangle at the midpoints of the sides. The charm of this easy-to-state theorem that joins algebra, geometry, and complex analysis is explored with the tools of Maple.

Classroom Tips and Techniques: The One- and Two-Argument Arctangent Functions in Maple

Robert Lopez — Wed, 13 Oct 2010 04:00:00 Z

In general, a "conversion" between the one- and two-argument artangent functions is not mathematically correct. However, we give two examples of calculations that benefit from a formal interchange of these two functions, and show how different programming strategies in Maple can be used to build tools to make these formal interchanges.