## Contents

- SOME DIFFERENTIAL CALCULATIONS
- INTEGRAL CALCULATION
- EXTENDED COMPUTATION
- UNCLASSIFIED SERVERS
- INTEGER
- DIFFERENT BACKS
- VOTER CALCULATION
- USERS AND DEFINITIONS
- VECTOR SPACE
- VARIOUS ACTIVITIES
- Introduction
- OD Es and constant coefficients
- ODEs with differential coefficients: first series
- OD Es and coefficients of variation: quadratic and homogeneous
- Frobenius method for simple coefficients
- Various sizes
- One- and two-dimensional wave equation
- Temperature measurement on one and two levels
- Green mode
- Summary
- ANSWERS
- SEE

## Preface

This book is based on a course I prepared several years ago and have since taught at Yale. It is a required course for physics majors, and students who wish to skip it must provide documentation to the Dean of Undergraduate Studies that they are familiar with the content. Although it is largely physics-related, since the basic concepts of physics derive largely from Newtonian mechanics, I can see that many of them will be useful to anyone in the physical sciences.

The reason for this book and this course is as follows.

Although I have taught a number of core undergraduate courses, I have had to go back and remove some of the core mathematics that was a pain in the neck for some classes. For example, I remember trying to figure out how to coordinate oscillators and showing that the Taylor series of all energy functions could be stopped at a time determined by the coordinate function of the quadratic function. At this point, some students wanted to know what Taylor series were. This was followed by retreats to discuss Taylor’s position. When I next tried to show that the quadratic equation could be divided into an independent oscillator by changing the coordinates if the potential was metallically coordinated, I had to use some matrix definitions and matrix theory, and some other students got upset.

Once again, we were disappointed. I no longer reject the idea that while teaching physics you should also teach new mathematics. For example, electricity and magnetism courses are very good subjects where we learn Legendre polynomials. However, this is not the place to learn for the first time what a complex representation like eim¢ means. Likewise, when learning the special relationship, you don’t want to introduce sinh and cosh, but you do want to use them and enjoy them naturally serving our purposes. Nowadays, explaining what this practice is like explaining a sentence. In other words, some mathematical explanations were undesirable and confusing for both teachers and students.

This problem, of course, diminished as the students progressed through the system, because they were taking first-year courses in the math department at the time and could tell you some funny things about the edge wedge theorem. But you want students to have a certain level of understanding of the basics of each major subject from the start, so teachers can continue their work with as little disruption as possible. According to Student, this provided more time to think about proper education and the freedom to pursue higher courses.

Many people shared my feelings when it came to teachers. Therefore, it was decided to design and teach courses that would cover topics related to differential calculus of one or more variables (including trigonometric, hyperbolic, logarithmic and exponential functions), exponential calculus of one or more variables, power series, complex numbers. and complex functions, vector calculus, matrices, linear algebra and finally different elements.

Unlike the math courses students usually take in their senior year, this course covers each topic in an easy-going manner. For example, the matrix would be two by two unless a larger matrix is absolutely necessary (e.g. to define decomposition). On the other hand, the treatment of this simple case will be complete and invisible. Courses last one semester and are independent. It was mostly for sophomores, although were recruited by freshmen, upperclassmen, and students from other majors.

This book is a work. Each department needs to decide whether to offer courses on this subject in the second year. My personal opinion (based on our experience at Yale) is that the preventative approach that requires only one class per semester is worth the hours of recovery afterwards. Since mathematics is nature’s chosen language and spans the full spectrum of knowledge, I can’t think of another course that would be more productive in hours for students beginning careers in science. The difference between strong and weak in mathematics will change after distinguishing between success and failure in science.

As usual, I address the student directly, expecting the usual questions, assuming he is standing before me. The book is therefore good for self-study. For this reason, even a department that does not have a course yet can guide its first- or second-year students through this book. They can go there if they encounter a problem with the mathematical methods used in different subjects.

Thanksgiving I would like to thank all students who took the Physics 30 course for their comments, and Ilya Grunberg and Senthil Todaro for their comments on this article.

As always, it was a pleasure working with the Plenum writing team. Special thanks to editor Amelia McNamara, her assistant Ken Howell, and editor-in-chief Joseph Hertz linger.

We thank Meera and AJ Shankar for their help with the index. But my biggest debt is to my wife Uma. Over the years, my children and I have prospered thanks to the efforts of their parents, who gave him a lot of money. This book is another example of what he has been able to achieve through his tireless contributions as a whole community. He dedicated it to her, and I hope it will be one of the most original accounts of his numerous efforts.

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