Mathematics our cars accelerate smoothly and brake effectively,
whether our mobile phones stay connected as we travel,
whether the pills we take are absorbed by our bodies at the correct
rates. Mathematics determines whether the lift we travel in stops
exactly in line with the floor, whether an aeroplane stays in the
air, whether the Mars Lander does in fact land, or gets lost in space
because it thought a metre was a foot.
So applied mathematics is
pretty vital for design engineers. Precision is the key to most
of our work - the consequences of a misplaced decimal point or
a misunderstood unit are too dreadful to contemplate, and we’ve
talked before about the pros and cons of spreadsheets, programming
languages and maths software tools in trying to avoid that sort
of thing.
Whatever
tool you use, there’s always a limit to the number of digits
you can work with to the right of the decimal point. While a rounding
error may not matter too much after a single calculation, the cumulative
effect following a whole string of computational procedures may
have unfortunate effects.
The design of tools like
spreadsheets encourages you to numerically evaluate your equations at the first possible instance, and in doing so you lose control of your parameters and lose sight of the physical assumptions and information that your equations contain. In essence they only address the final part of mathematical problem solving - that of number crunching.
That’s why many design engineers prefer symbolic maths programs. Algebra was never my favourite subject at school, but I can’t deny that a parameter represented by x or y remains the same however many computations you put it through.
Symbolic maths software has significant advantages over purely
numeric solvers. By keeping your equations in a symbolic form and
only evaluating them when you need to you retain that physical
information and control of your parameters. Optimization techniques
are easier to manage and ‘what-if’ explorations stay
meaningful.
Although there are other products around, the symbolic maths software market is dominated by two products: Maple (maple.adeptscience.co.uk) and Mathematica (www.wolfram.com). Maple has traditionally been favoured in universities as a maths teaching tool, while Mathematica has enjoyed a higher profile in industry - there’s no fundamental reason for this dichotomy, and I’d recommend that design engineers in industry consider both products.
They both make justifiable claims
to power, speed and scalability; they both claim to be easy to use but that, often, is down to personal preference. Maple offers a very sophisticated system of unit management, and its code generation capabilities are particularly noteworthy as it handles a very wide range of languages and programming structures. Mathematica offers features such as automatic selection of algorithms, and has a wide range of add-on packages available.
One thing both packages - at least in their more recent versions - share: neither is solely a symbolic maths program. They both integrate symbolic, numeric and graphical computations to provide, as their marketing people would say, a comprehensive technical computing environment.
Because programs like these incorporate powerful mathematical functions, we ordinary engineers tend to think of them as too complex for everyday use. They are, in fact, less frightening and a good deal easier to use than many of us give them credit for. The advantages they can bring to the design engineering process make them well worth a look.
Dr Know's recommended download is the Maple 9.5 Demo - download yours today.