Today, a look at a secret abstraction. The
University of Houston's College of Engineering
presents this series about the machines that make
our civilization run, and the people whose
ingenuity created them.

Who among us doesn't speak
easily of magnetic fields, gravitational fields,
electric fields? The word *field* is shot
through our language. The fields I've just
mentioned are force fields that act upon matter
within them in various ways. Gravity fields move
objects; electric fields build up the charge on
objects; magnetic fields reveal themselves by
making patterns in iron filings.

We engineers like to solve problems by
*creating* fields: Many common quantities are
distributed through materials -- temperatures,
stresses, voltages. The velocity of water or air is
smoothly distributed through the fluid. So, in each
case, we invent fields from which we can calculate
quantities. And we do it without telling the
public. When we speak outside our circle, we go
directly to our results and pass over the path that
got them.

We tell of lasers or neutrinos, microchips or the
speed of light. But we say almost nothing about the
mathematical brick and mortar of science. Nothing
is so elemental yet so invisible to the public as
the way we use these fields in our everyday
calculations.

The idea is this: Temperatures, stresses, voltages,
and fluid flows all spread out in space according
to the same general mathematics. Whether we're
interested in the way stress is distributed in a
flying buttress or the exquisite pattern of flow in
water coming out of a hole in a tank, the same
equations apply.

Try an example: We have a square copper plate with
a tiny heater in the center. We hold the outside of
the plate at room temperature while heat flows out
from the heater through the plate. It turns out
that we can recreate the mathematical field that
describes temperature and heat flow with an
analogy.

We stretch a rubber membrane over a square wire
frame. Then we press a pencil into the center from
below. A mountain, a little like Fujiyama, but
steeper, rises up from the center. The height of
that mountain reflects the temperature at any point
in the plate. The steepness reflects the heat flow.

The wonderful thing about that mountain in the
rubber sheet is that it also represents other
situations. It can be used to describe the flow
when water is released in the center of a shallow
square container and allowed to flow out over the
four sides. That's exactly the same kind of
mathematical field we saw in the heated bar or even
in the flying buttress.

But we don't build rubber-sheet models to get
answers. It's much quicker to write equations and
let a computer build the mountain for us. Every
engineer learns to describe these fields and use
them to predict mass diffusion, heat flow, stress
distributions, fluid flows, and more. But, because
all this is abstract, we don't even try to tell the
public about it. That's too bad, because this is
some of the neatest stuff we do. And a student,
looking for a field of study, never hears about it.

I'm John Lienhard, at the University of Houston,
where we're interested in the way inventive minds
work.

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