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Q&A #3886 |
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I asked a friend of mine for some help with your question and received the following response. Her name is Dr. Evelyn Sylvia. She is a mathematics professor at the University of California, Davis. Her field of specialization is complex variables. Here's her response: "...I will offer a general description of the principle on which many of the applications of complex variables are built. They build on the power of mapping (graphing with complex functions) and the basic idea behind math modeling in general. I don't know of any web sites offhand. You might have your students do a Google search for the topic "Joukowski's curves" or "Joukowski's functions" or maybe "conformal mapping applications." There is a nice little book that I recently acquired (but have not yet read) that might offer some direction to the students. It is An Imaginary Tale The Story of "the square root of -1" by Paul J. Nahin (Princeton University Press, Princeton, NJ, 1998. It appears to an "i version" analog to the books The Story of e and The Story of pi. Chapter 5 of Nahin's book is on the Uses of Complex numbers and section 5.5 in titled "Complex Numbers in Electrical Engineering." The section after that describes "A Famous Electronic Circuit that Works Because of the square root of -1." A Brief General Description of the Idea Whether it be in Physics, Aerodynamical Engineering, or the host of other areas where complex variables is applied, the desire to model a real world phenomenon so that it can be studied is a common base on which the work is built. The word model here refers to describing or approximately describing the behavior with equations or sets of equations that have meaning. The reason why complex variables is so useful is that when we "graph" complex functions, because we have two dimensions (x,y) =x+iy going into two dimensions f(z)=u(z)+iv(z), we concentrate on graphing shapes into shapes. The key here is that we stick to what are "nice shapes" for the given functions. If you have been doing some playing with complex functions, try to figure out why I claim that the sector {(x,y): x^2 + y^2 <2 and x>0 and y < 0} is a nice shape for the function f(z)=z^2. Knowing what nice shapes go with certain functions is an important part of what is known as "conformal mapping applications." You don't have to figure out the answer to my question to understand the following. If we can map shapes into shapes that we want to work with then there are laws of physics (engineering, etc.) that allow us to use the formulas to calculate desired estimates for things being studied. Here is a nice "concrete" example. A completed airplane is really big and very heavy. When engineers and physicists design such things they build a model that is tested in a windtunnel to make sure that the "loaded" plane will take off before they build the real one. But before they build the model, they have to have a way of estimating the lift that will come from various parts of the plane. Well, imagine yourself cutting across the body of the plane; when you open it, you see something that is about a circle. It you slice the wing, the cross section looks something like a "squashed ellipse." It turns out that these are the nice curves for functions known as Joukowski's curves. The formulas can be used in other formulas to calculate the estimated lift. Then they have ways todetermine the collected effect. The information used helps them to design the model that is then built and tested in a wind tunnel. I was recently at Black Bird Park in Palmdale CA where they had the amazing Black Bird reconnaissance plane on display along with the 1/12 scale models that had been tested in wind tunnels. You might be able to find information about this park and the design of the plane on the Web." I hope this helps. -Jeanne, for the Teacher2Teacher service
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