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Unformatted text preview: equation is an equation which contains partial derivatives,
such as the equation
in which u is regarded as a function of x and t. Unlike the theory of ordinary
diﬀerential equations which centers upon one key theorem—the fundamental
existence and uniqueness theorem—there is no real uniﬁed theory of partial differential equations. Instead, each type of partial diﬀerential equations exhibits
its own special features, which usually mirror the physical phenomena which
the equation was ﬁrst used to model.
Many of the foundational theories of physics and engineering are expressed
by means of systems of partial diﬀerential equations. The reader may have
heard some of these equations mentioned in previous courses in physics. Fluid
mechanics is often formulated by the Euler equations of motion or the so-called
Navier-Stokes equations, electricity and magnetism by Maxwell’s equations, general relativity by Einstein’s ﬁeld equations. It is therefore important to dev...
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- Winter '14