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Why Do We Derive?

Every time I teach a class I learn something. In this year’s class the following questions came up often.

  • Why should we learn how to derive solutions starting from Maxwell’s equations?
  • Why not just look up the solutions to use for a particular problem?
  • Why should we use u substitution to solve an integral when we can look up the answer?
  • Why derive the skin depth when the result is on Wiki?

All good questions. It took me a while to internalize an answer. As a practicing electrical engineer, you probably won’t use u substitution to solve an integral. You’ll use simulators to design products to specifications. You will know the general relationships that loss increases with frequency, and that is probably good enough. You’ll look things up on Wiki. One of the most important skills of a good engineer is figuring out what short cuts and simplifications can be taken to save money, while still meeting the product specifications, and doing it quickly to stay on schedule. Much of engineering is about understanding finance, specifically the time value of money and the life cycle of a product. The product life cycle is design -> engineering -> validation -> production -> costs of sustaining. Everything costs and needs money to complete. The survival of your project, or even company, can depend on it. Products are measured by return on investment. Money drives pretty much everything in engineering. So why do we derive equations at all?

This is what I figured out this year that I would like to share with you, but to appreciate it you have to remember how we got here. Look at the time line of evolution. Nucleic acids began to self-assemble about 4 billion years ago. Bacteria about 3.5 billion years ago. After 1 and a half billion years cells with a DNA in a nucleus show up on the scene. They reproduce by splitting. These cells floated around for another billion years before multi-celled organisms started to evolve. 600 million years ago simple animals showed up. 2.5 million years ago the genus Homo shows up. 200,000 years ago we have evolved into modern humans.

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Now consider the timeline of our mathematics. 300 years ago we got calculus and Newton’s Laws. That took about 199,700 years. 140 years ago we got Maxwell’s equations. 107 years ago we learned about relativity, and that space and time can bend, and nothing is what it seems. 86 years ago we got Schrodinger’s equation, and solved the mysteries of the atom. We learned that not only are things strange, they are strange in a way that we still do not understand. In 1950, using xray crystallography, we figured out what DNA was. This started the medical revolution, where we can finally begin to understand cancer and the workings of the cell at a chemical level.

These are the equations of our industrial revolution. These are the equations that enabled us to create fertilizer needed to feed the planet, make computers that let us numerically solve more complicated problems, design drugs that have a particular molecular shape, and land robots on Mars. Without these equations, there would be no knowledge about the workings of DNA, medicine would not exist in the form that we have today. We would not have cars, planes, or electricity to power our homes. Life would be drastically different. In terms of evolution, this all happened in an instant.

Here we sit on the cusp of evolution. For the majority, we no longer have the knowledge needed to survive if we were transported back just a few thousand years ago. I know how to calculate energy levels of atoms, but not how to farm a field, or what roots can be eaten, or how to raise chickens. I know how to go to Costco. I know how to drive a car but am totally incapable of building a car myself. In fact, I cannot build a toaster (although I can calculate the inductance of its coils). I would not be able to mine copper. When it comes to survival, I am totally dependent on this technology that we have created. This is where evolution has led us, the result of our equations. As an engineer, you have an obligation to understand these equations and where the solutions come from. You are obligated to understand the last 300 years of mathematics.

Being adept with the use of mathematics, I can make some very simple and very certain predictions. Let’s look at the near future, say 50 to 100 years. This is the future of my children. Very simply, we will not be able to feed the world population with the current oil based technology. We will experience either a population die off, or our technology will evolve. Continued global warming will change our planet to a point that it cannot sustain society in the current form. If we switch from oil to coal we may well damage our planet so that it cannot sustain human life. That really sums it up. As a matter of survival, the future engineers should be able to solve equations like (oil supply – d(oil) / dt)= 0. Solve for t. You, the future engineer, will face some of the most difficult, multi-disciplinary engineering problems ever. You are at a cusp of evolution, the result of 4 billion years of life processes, and the work you do in your lifetime determines the future. We are either going to continue to evolve as a technology based society or we will set the evolutionary clock back thousands, millions or even billions of years. It won’t get done by guessing, or by feel, or by popular vote. There are no formulas, there is no book. You need to be able to write down the equations that describe complex and terrible problems. Then you need to figure out how to derive the solutions. I wish you the best of luck.  favicon

 

Andy Martwick
has been an Electrical Engineer at Intel for the past 17 years. For the last 5 years he has been an adjunct professor at Portland State University and Oregon Tech where he teaches math, electricity and magnetism. He can be reached at martwick@pdx.edu.

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