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Introduction

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Differential calculus , as introduced by Sir Isaac Newton and Gottfried Wilhelm
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Leibniz in the late 17th century, opened up a wealth of new possibilities for
mathematical modeling in the natural and – later on – in the life sciences and in
technology. Partial Differential Equations (PDEs), entirely based on the concepts
of differential and integral calculus, relate one or more state variables to their
variations (differentials) with respect to certain independent variables like time,
space, velocity etc.
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Just to name a few examples, PDEs were used by James Clerk Maxwell to
model electromagnetic fields interacting with electrical charges and currents,
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by Ludwig Boltzmann to describe the non-equilibrium dynamics of rarified
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gases, by Albert Einstein to phrase the laws of gravitation in the general theory
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of relativity and by Erwin Schrödinger and Werner Heisenberg to formulate
quantum mechanics in mathematical-analytical terms.
The purpose of this book is to illustrate the fact that PDEs govern, or put in
more modest terms, model many aspects of the nature surrounding us, of the
technology we use on a daily basis and of our socio-economic interactions: PDEs
have a significant importance for the scientific and technological progress of our
society. Two entirely different descriptive levels are used in this book: firstly,
in the subsequent eleven Chapters different scientific and technological prob-
lems are presented, modeled and analyzed by PDE methodology and secondly,
photographic images are shown to illustrate these problems and some of their
specific features. Every Chapter contains comments on the photographs which
relate them directly to the presented mathematical models. Almost all images
have a significant direct impact on and connection to PDE modeling issues, a few
exceptions to this rule have ‘only’ an allegoric meaning. The attentive reader will
easily find out which images belong to the latter class.
It is important to understand that the main purpose of the photographs is
NOT to depict particular solutions of the partial differential equations under
considerations – although some photographs do precisely that, but only as a
by-product. Much more importantly, the photographs show concrete modeling

1 http://en.wikipedia.org/wiki/Calculus
2 http://en.wikipedia.org/wiki/Sir_Isaac_Newton
3 http://en.wikipedia.org/wiki/Gottfried_Leibniz
4 http://de.wikipedia.org/wiki/James_Clerk_Maxwell
5 http://de.wikipedia.org/wiki/Ludwig_Boltzmann
6 http://de.wikipedia.org/wiki/Albert_Einstein
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http://de.wikipedia.org/wiki/Erwin_Schroedinger
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http://de.wikipedia.org/wiki/Werner_Heisenberg