Contents
- SOME DIFFERENTIAL CALCULATIONS
- INTEGRAL CALCULATION
- EXTENDED COMPUTATION
- UNCLASSIFIED SERVERS
- INTEGER
- DIFFERENT BACKS
- VOTER CALCULATION
- USERS AND DEFINITIONS
- VECTOR SPACE
- VARIOUS ACTIVITIES
- Introduction
- ODEs and constant coefficients
- ODEs with differential coefficients: first series
- OD The coefficients of variation are quadratic and homogeneous.
- Frobenius method for simple coefficients
- Various sizes
- One- and two-dimensional wave equations
- We measure the temperature on both the first and second 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 the course primarily focuses on physics, given that the fundamental concepts of physics are largely derived from Newtonian mechanics, I believe that many of these concepts will be beneficial to anyone in the physical sciences.
This book and this course are being offered for the following reasons.
Despite having taught several core undergraduate courses, I have had to revisit and simplify some core mathematics that caused difficulty in certain classes. For instance, I recall attempting to understand the coordination of oscillators and demonstrating that the coordinate function of the quadratic function could determine the time at which the Taylor series of all energy functions would halt. At this point, some students expressed a desire to understand the definition of the Taylor series. Retreats ensued to discuss Taylor’s position. Next, I attempted to demonstrate that if the potential was metallically coordinated, we could divide the quadratic equation into an independent oscillator by altering the coordinates. However, this required the application of some matrix definitions and matrix theory, which infuriated some other students.
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 excellent 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. These days, elucidating this practice resembles explaining a sentence. To put it another way, both teachers and students found some mathematical explanations undesirable and confusing.
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 strange 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 the student, this gave them more time to consider proper education and the freedom to pursue higher courses.
Many people shared my feelings when it came to teachers. Therefore, we 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 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 comprehensive and transparent. The courses span one semester and are conducted independently. The majority of the students were sophomores, but freshmen, upperclassmen, and students from other majors also participated.
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, and I assume he is standing before me. The book is therefore useful 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. Students can refer to this book if they encounter difficulties with the mathematical methods used in various subjects.
Thanksgiving I would like to thank all students who took the Physics 30 course for their comments, as well as 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. I hope his many efforts will be one of the most original. He dedicated it to her.
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