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University of Waterloo Faculty of Mathematics MATH 137: Calculus 1 Physics-based Section 8 Fall 2010 Lecture Notes E.R. Vrscay Department of Applied Mathematics c circlecopyrt E.R. Vrscay 2010 1
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Lecture 1 Introduction As written in the information sheet for this course, the purpose of MATH 137 is “to deepen your un- derstanding of calculus that you began to learn in high school.” More specifically, we shall investigate in more detail the ideas of “limit”, “derivative” and “integral”. But this particular “Physics-based” section is somewhat special, which is probably why you are enrolled in this section. (Presumably, you like Physics.) Unlike the other sections of MATH 137, we’ll apply these ideas to situations encountered in Science and Engineering, especially in Physics. You can read more on the philosophy of this section in the Course Information Sheet for this section (found in the UW-ACE site for this section). But let’s step back for a moment and ask the question, “Why study Calculus?” Here are a couple of possible answers: 1. Because it’s fun. You may well enjoy “doing mathematics” for “mathematics’ sake,” manip- ulating equations, combining them, etc., or proving theorems. This is the attitude of a pure mathematician. Starting with Calculus, one may proceed to the more advanced subject of Analysis . 2. Because Calculus is the natural language of the sciences. It is remarkably effective as a tool for (a) solving scientific problems, e.g., trajectories of particles, fluid motion, (b) formulating scientific theories of the natural world. This is the attitude of a physicist, an engineer or an applied mathematician. The two viewpoints given above may be viewed as complementary. Mathematics is deductive in nature: You start with some axioms or definitions and proceed, using mathematical logic, to make other conclusions. On the other hand, science is inductive : You observe some phenomena and try to formulate a model that explains these phenomena. In the case of a physical theory, it is then 2
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hoped that you can predict some kind of hitherto-unobserved behaviour that can then be observed experimentally. The key word above is “modelling” – the job of the scientist is to model physical or biological phenomena. (By the way, this is why theoretical physicists and applied mathematicians have been the most sought out people by financial institutions. These institutions want people who know how to “model” phenomena.) In this course, most of the phenomena will involve classical dynamics, i.e., particles moving under the influence of forces. In these cases, the “model” is simply Newton’s Second Law, i.e., F = m a , (1) where F is the force vector, m is the mass and a is the acceleration vector. We’ll return to this equation shortly.
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