New Tips & Techniques

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

Maplesoft — 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.

Classroom Tips and Techniques: Diffusion with a Generalized Robin Condition

Robert Lopez — Fri, 17 Sep 2010 04:00:00 Z

The one-dimensonal heat equation with a generalized Robin condition is solved on [0, 1] by a finite-difference scheme and by the Laplace transform, with the inversion implemented numerically. The left end is insulated and the initial temperature is zero. The Robin condition at the right end is driven by a function governed by an ODE, that is in turn, driven by the endpoint temperature.

Classroom Tips and Techniques: Real Distinct Roots of a Cubic

Robert Lopez — Tue, 10 Aug 2010 04:00:00 Z

The real distinct roots of the cubic equation z³ + a z² + b z + c = 0 can be expressed compactly in terms of trig functions. However, Maple's solve command does not return this compact form, so we explore how we can interpret and compact Maple's solution of this equation.

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.