Ch0101 - Chapter 1 Mathematical Prelude 1 Mathematical...

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Chapter 1 Mathematical Prelude 2 1 Mathematical Prelude Just below the title of each chapter is a tip on what I perceive to be the most common mistake made by students in applying material from the chapter. I include these tips so that you can avoid making the mistakes. Here’s the first one: The reciprocal of y x 1 1 + is not x + y. Try it in the case of some simple numbers. Suppose x=2 and y=4. Then 4 3 4 1 4 2 4 1 2 1 1 1 = + = + = + y x and the reciprocal of 4 3 is 3 4 which is clearly not 6 (which is what you obtain if you take the reciprocal of 4 1 2 1 + to be 2+4). So what is the reciprocal of y x 1 1 + ? The reciprocal of y x 1 1 + is y x 1 1 1 + . This book is a physics book, not a mathematics book. One of your goals in taking a physics course is to become more proficient at solving physics problems, both conceptual problems involving little to no math, and problems involving some mathematics. In a typical physics problem you are given a description about something that is taking place in the universe and you are supposed to figure out and write something very specific about what happens as a result of what is taking place. More importantly, you are supposed to communicate clearly, completely, and effectively, how, based on the description and basic principles of physics, you arrived at your conclusion. To solve a typical physics problem you have to: (1) form a picture based on the given description, quite often a moving picture, in your mind, (2) concoct an appropriate mathematical problem based on the picture, (3) solve the mathematical problem, and (4) interpret the solution of the mathematical problem. The physics occurs in steps 1, 2, and 4. The mathematics occurs in step 3. It only represents about 25% of the solution to a typical physics problem. You might well wonder why we start off a physics book with a chapter on mathematics. The thing is, the mathematics covered in this chapter is mathematics you are supposed to already know. The problem is that you might be a little bit rusty with it. We don’t want that rust to get in the way of your learning of the physics. So, we try to knock the rust off of the mathematics that you are supposed to already know, so that you can concentrate on the physics. As much as we emphasize that this is a physics course rather than a mathematics course, there is no doubt that you will advance your mathematical knowledge if you take this course seriously. You will use mathematics as a tool, and as with any tool, the more you use it the better you get at using it. Some of the mathematics in this book is expected to be new to you. The mathematics that is expected to be new to you will be introduced in recitation on an as-needed basis. It is anticipated that you will learn and use some calculus in this course before you ever see it in a mathematics course. (This book is addressed most specifically to students who have never had a
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Chapter 1 Mathematical Prelude 3 physics course before and have never had a calculus course before but
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Ch0101 - Chapter 1 Mathematical Prelude 1 Mathematical...

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