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The use of digital elevation models, which represent the surface of the earth by a matrix of heights, has proved an ideal introductory topic for developing both the matrix-handling and computational skills of undergraduate mathematicians. This article looks at the solving of an easily understandable, but not necessarily simple problem: given a particular date, time and terrain, and within the software skills available to first-year students via a computer algebra system, produce a map which can show those parts of the terrain which lie in shadow. (Contains 7 figures.)

This article illustrates how a computer algebra system, such as Maple[R], can assist in the study of theoretical fluid mechanics, for both Newtonian and non-Newtonian fluids. The continuity equation, the stress equations of motion, the Navier-Stokes equations, and various constitutive equations are treated, using a full, but straightforward, notation. Programs are presented that can produce these equations in various coordinate systems, as is the algebra system used to obtain solutions for specific examples in parallel and rotational flow.

If the point of suspension of a multiple pendulum is suitably oscillated then the pendulum can remain in motion in an upside-down position. Since such pendulums can model flexible materials, this inverted motion is sometimes referred to as an 'Indian rope trick'. Despite the complexity of the governing differential equations, this rope trick can be successfully demonstrated using a popular computer algebra system to formulate and solve the equations, and display an animation of the motion.

The way in which computer algebra systems, such as Maple, have made the study of physical problems of some considerable complexity accessible to mathematicians and scientists with modest computational skills is illustrated by solving the multiple pendulum problem. A solution is obtained for four pendulums with no restriction on the size of the angles through which they move, and for as many as a hundred pendulums if the angles remain sufficiently small. An attractive and informative graphical display of the motion in animated form can be obtained from the solution, and the effect of varying the initial conditions can be investigated visually. (Contains 8 figures.)

The way in which computer algebra systems, such as Maple®, have made the study of complex problems accessible to undergraduate mathematicians with modest computational skills is illustrated by some large matrix calculations, which arise from representing the Earth's surface by digital elevation models. Such problems are often considered to lie in the field of computer mapping and thus addressed by geographical information systems. The problems include simple identification of local maximum points, visualization by cross-sectional profiles, contour maps and three-dimensional views, consideration of the visual impact of the placement of large buildings and issues arising from reservoir creation. Motion through a virtual landscape can be simulated by an animation facility. This approach has been successful with first year students: the ‘real world’ problems considered are more accessible than many alternatives, and the attraction of using large matrices is retained. [ABSTRACT FROM AUTHOR]

If the point of suspension of a multiple pendulum is suitably oscillated then the pendulum can remain in motion in an upside-down position. Since such pendulums can model flexible materials, this inverted motion is sometimes referred to as an 'Indian rope trick'. Despite the complexity of the governing differential equations, this rope trick can be successfully demonstrated using a popular computer algebra system to formulate and solve the equations, and display an animation of the motion. [ABSTRACT FROM AUTHOR]