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Physics, Math and Mental Gymnastics

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Mental Gymnastics

While we enter physics to study the fascinating world of black holes, quarks and the quantum, the brutal truth is that mathematics is the central tool of the physicist. Gauss called mathematics the "Queen of the Sciences", and with good reason. If you don't have a solid grasp of mathematics, you aren't going to get very far.

One thing I noticed when getting my degrees in physics was that many of the students found math to be a painful "aside". In one case that really stands out in my memory, I was in a mechancs course and one of the homeworks required the calculation of a brutal integral. I worked very hard by myself over the weekend and managed to get the calculation out with a couple of pages of work. When I returned to class, I was surprised to find that the vast majority of the students had not even attempted to work out the integral. One student had obtained the answer-so he thought-using Mathematica. I looked at it carefully and saw that he had gotten the wrong answer. He argued with me-asserting that the computer cannot make a mistake-but we brought the TA over and it turned out he had entered the integral incorrectly. I had obtained the right answer by working it out by hand.

The student in question had thought he was interested in physics but didn't want to bother with the work of physics-which involves diving into the mathematics. But to become a good physicist-or a solid engineer-you need to bite the bullet and become a master of mathematics. It doesn't matter if you're going to be an astronomer, experimentalist, or engineer-in my view if you want to be the best at what you do in these fields, you should have a solid command of math. So if you are interested in physics but aren't a mathematical hot shot, how can you pull yourself to the top of the field? In my view, the answer is to view mathematics the way you would athletics. A friend of mine who shared this view coined the term "mental gymnastics" to characterize his outlook and study habits.

We all aren't Math Genuises

While for some students thinking mathematically comes natural, most of us aren't ready to master the intricacies of studying proofs when we're college freshmen. This article is written for those of us who aren't automatic math whiz kids. If you are a mere mortal who finds math a bit of work, don't be discouraged. It's my belief that average people can raise themselves up to become very good mathematicians with a little bit of hard work. What we need is some training--we need to train our minds to think mathematically. The best way to think about how you can get this done is to draw an analogy between math and athletics.

To master a sport you have to build your muscles and train your body to react in certain ways. For example, if you want to become a great basketball player, you could be lucky enough to be born Michael Jordan. But more likely, you'll have to work at building a basic skill set, and the truth is even players like Michael Jordan put extra work into their craft. Some activiities you might consider that could make you a better basketball player are

Lifting weights to build muscle mass
Run sprints to improve your ability to run up and down the court without getting tired
Spend a large amount of time shooting free throws, doing layups and practicing basic skills like passing

It turns out that becoming a successful physicist or engineer is in many ways similar to athletics. OK, so suppose you want to study Hawking radiation and string theory, but you are not a hot shot mathematician and weren't the best student. Instead of just reading a bunch of books or lamenting the fact we aren't an Einsteinian genius, what are the mathematical equivalents to lifting weights or running sprints we can do to improve our mathematical ability? In my view, we can begin by following two steps

Learn the basic rules first-and don't focus on trying to learn proofs or do the hardest problems.
Repeat, repeat, repeat. Do similar types of problems over and over until they are second nature. Only after a topic becomes second nature calculationally do we consider reading the proofs or theorems in detail.

That is do tons of problems. In my view a student should start off simple. Don't try to understand the proofs. For example, in my recent book, "Calculus In Focus", I take the perspective that students need to learn math by following the formula: show, repeat, try it yourself. That is

Show the student a given rule, like the product rule for derivatives
Focus on mastering calculational skills first. Do this by showing the student how to apply the rule with multiple examples.
Repeat, repeat, repeat. Do a given type of problem multiple times so that it becomes second nature.

Once the "how" to solve problems is second nature, then go back for a deeper look at the material. Then learn the "why" and start learning the formality of mathematics through proofs and theorems. I use this approach to drill the central ideas of calculus in my book Calculus in Focus. More information can be found at http://www.quantumphysicshelp.com/calculus.htm [Broken].

In addition to the basic approach, a certain baseline has to be established if you want to build yourself up for a formal career in math, physics, or engineering. Let's build up a fundamental skill set that is going to build your fundamental math skills and help you master any subject. A few key areas I think students should focus on are outlined below.

The Importance of Algebra
If you study physics or engineering, algebra never goes away. So the first step on the road to becoming the next Stephen Hawking is to master this tedious yet fundamental subject. Do yourself a favor and pick up a decent algebra book and work through it. Do every problem so that by the end of the book, factoring equations, logarithms and other math basics are second nature for you. In the same way that lifting weights is going to make a football or basketball a better athlete when the games are actually played, mastering algebra will pay off later when you're doing your homework in dynamics or quantum theory.

If you go on to become an electrical engineer and study circuit analysis or decide to master black hole physics, one fundamental area of business you'll have in common with your colleagues is trigonometry. Make sure you know your trig inside and out, learn what the trig functions really mean and master those pesky identities. Also don't over look this one crucial fact-trigonometry also provides a simple arena where you can learn how to prove and/or derive results. We all know that later, when you take advanced physics courses, you're going to see the words "show that" pop up frequently in your homework problems. This is sure to cause headaches among the mere mortals amongst us, but it turns out you can improve your skills in this area in a non-threatening way by deriving trig identities. Instead of viewing the derivation of trig identities as a tedious obstacle, start to look at this as an opportunity. All trig books have homework problems where you have to derive an identity so pick up a trig book and do it until your blue in the face. Take it seriously and write up each proof as if you were submitting a short paper to a major journal. This will teach you how to go from point A to point B mathematically and how to write up a derivation in a formal way that will allow someone else to understand what's going on. If you do, later it will be easier to get through homework in advanced classes, you'll get better grades, and you'll develop a good foundation for writing up theoretical derivations for research papers.

Graphing Functions
While any function can be graphed easily on the computer or on a graphing calculator, it is very important to be able to graph a function on the fly with nothing more than a pencil and paper. The key abilities you want to focus on are developing an intuitive sense for how functions behave and learning how to focus on how functions behave in various limits. That is, how does a function look when the argument is small? How does it behave as the argument goes to infinity? Dig out your calculus book and review techiques that use the first and second derivative to graph a function. I review these extensively in my recent book "Calculus in Focus".

Series and Complex Numbers
In my opinion, understanding the series expansion of functions and the behavior of complex numbers can't be underestimated. If you want to understand physics, you need to master the use of series. Start by learning how to expand a function in a series. Some series should be second nature ('oh yeah, that's cosine"). Learn about convergence. Get a copy of Arfken and review the solution of differential equations using series. Try to get an intuitive feel for cutting a series off at a given term while retaining the essential behavior of the function. These are tools that are important when studying theoretical physics or advanced engineering.

- David McMahon
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