This text presents models of transport in continuous media and a corre-
sponding body of mathematical techniques. Within this text, I have em-
bedded a subtext of problems. Topics and problems are listed together in
the table of contents. Each problem is followed by a detailed solution em-
phasizing process and craftsmanship. These problems and solutions express
the practice of applied mathematics as the examination and re-examination
of essential ideas in many interrelated examples.
Since the science that falls under the headings “transport” or “fluids”
is so broad, this introductory text for a one-term advanced undergraduate
or beginning graduate course must select a highly specific path. The main
requirement is that topics and exercises be logically interconnected and form
a self-contained whole.
Briefly, the physical topics are: convection and diffusion as the sim-
plest models of transport, local conservation laws with sources as a general
“frame” of continuum mechanics, ideal fluid as the simplest example of an
actual physical medium with mass, momentum and energy transport, and
finally, free surface waves and shallow water theory. The idea behind this
lineup is the progression from purely geometric and kinematic to genuinely
The mathematical prerequisites for engaging the practice of this text
are: fluency in advanced calculus and vector analysis, and acquaintance
with PDEs from an introductory undergraduate course.
The mathematical skills developed in this text have two tracks: First,
classical constructions of solutions to linear PDEs and related tools, such as
the Dirac δ-function, are presented with a relentless sense of connection to
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