Prelude to Mathematics by W. W. Sawyer

Author:W. W. Sawyer [Sawyer, W. W.]
Language: eng
Format: epub, pdf
Tags: Applied, Mathematics, Applied Sciences, History & Philosophy, Science
ISBN: 9780486152776
Google: YuqUrTzxGbQC
Publisher: Courier Corporation
Published: 2012-04-19T20:45:47+00:00

But x and y are small. So x2, xy, y2, etc., are very small indeed, and can be neglected. To this order of approximation,

f(x, y) = k + ax + by.

When x = y = 0, f(x, y) is k. So k = f(0, 0). It follows

f(x, y) -f(0, 0) = ax + by.

The expression on the left-hand side represents the change in f in going from (0, 0) to (x, y). On the right-hand side, we have ax + by, exactly the type of expression that led us to matrices earlier. But of course we need two such expressions to give a matrix. So in any problem concerned with the changes in two functions f(x, y) and g(x, y) - and there are many such problems — matrices will arise.

The small vibrations of any structure are an important engineering application of matrices. The matrices come in, as explained above, because the vibrations are small. Serious breakdowns can occur if in some machine the beat of an engine happens to coincide with the natural rate of vibration of some other part of the machine. (You have probably noticed in some motor-cars, at a certain speed some object in the car begins to chatter.) In certain conditions, the perpetual vibration can lead to a serious rupture. It is like the old story of soldiers breaking step when crossing a bridge. (As undergraduates some of us used to tap the Cambridge lamp-posts in the hope of finding their natural rate of vibration and causing them to disintegrate. We never succeeded.)

Quantum theory can be developed by means of matrices (Heisenberg, Dirac). The tensors of relativity are a generalization of matrices.

Matrices occur too in many branches of pure mathematics. Conic sections are an elementary example; projective geometry, groups, differential equations somewhat more advanced instances.

Matrix algebra is in fact one of the most striking examples of a pattern that arises in the most varied circumstances.

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