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Unformatted text preview: al inquiry, whether at the high school or university level.)
Moreover, the objects modeled make it clear that there should be a constant
tension between the discrete and continuous. For example, a vibrating string
can be regarded proﬁtably as a continuous object, yet if one looks at a ﬁne
enough scale, the string is made up of molecules, suggesting a discrete model
with a large number of variables. Moreover, we will see that although a partial
diﬀerential equation provides an elegant continuous model for a vibrating membrane, the numerical method used to do actual calculations may approximate
this continuous model with a discrete mechanical system with a large number of
degrees of freedom. The eigenvalue problem for a diﬀerential equation thereby
becomes approximated by an eigenvalue problem for an n × n matrix where n
is large, thereby providing a link between the techniques studied in linear algebra and those of partial diﬀerential equations. The reader should be aware
that there are many cases in which a discrete model may actually provi...
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- Winter '14