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.

(Theme music)

For examples of how engineers create such "potential fields," J. H. Lienhard IV and J. H. Lienhard V, *A Heat Transfer Textbook*, 5th ed., Dover Pubs. Inc., Mineola, NY, 2019. You can easily download the entire book, free of charge, at https://ahtt.mit.edu/ . We deal with heat flow fields graphically in Section 5.7 and we describe fluid flow fields analytically in Chapter 6.

Heat flow field in a quarter circle, heated on upper right side, cooled on curved periphery. Lines of constant temperature curve upward. Heat flow paths curve downward.

(From the cover of *A Heat Transfer Textbook*.)