Calculus III – Multivariable Calculus
- Course ID:MATH 537/538
- Teachers:Miguel Hernandez
Description and Objectives
The students attending this class, having completed two full years of calculus, are ready for a course of study that relies on calculus and extends it in a meaningful way. i.e. Differential equations (DE). Our course will enable a deeper understanding of and appreciation for the basic precepts of physics and chemistry studied in previous years. We will concentrate on methods of solving, analyzing and approximating solutions for DE that prove useful in a wide variety of applications (real world). Much effort will be spent in solving equations and learning techniques for doing so. When we become strong in dealing with elementary DE, we will examine the most famous one now extant—Schrödinger’s equation. This equation is a second order partial DE used to solve the energy levels and wave functions for all of the elements. Knowledge gained here will allow us to move gently into quantum mechanics in a somewhat qualitative way. We progress on into relativity and move on out into space where we can examine some interesting concepts such as string theory, worm holes, black holes dark matter and so on. We expect to become quite starry eyed.
Elementary Differential Equations and Boundary Value Problems. ISBN 0-471-08955-9 by William E Boyce and Richard C DiPrima. Chapters one, two, and three are of prime importance and constitute the main body of the course. There are 201 pages of excellent instructional material on DE and a plethora of problems and examples. In addition to this source we will use Schaums’ Outline on DE, and Differential Equations by Ralph Palmer Agnew Published in 1942 By McGraw Hill Book co.
We will use Quantum Mechanics by Leonard Schiff as well. For the more advanced theoretical topics there are quite a few sources available to us from the internet and magazines to a colleague of mine who has agreed to come and give the school some talks on worm holes and string theory.
Differential Equations is a fairly difficult topic and requires serious effort and dedication on the part of the student and teacher. It is advisable to examine the material in the book carefully and attempt to memorize theorems and axioms and other definitive statements. The primary text is excellent and offers solutions for all problems presented. Each problem is a precious gem to be admired and contemplated. A well organized notebook is a valuable study asset. I will make every effort to attempt to carefully examine your sons work and recommend corrections where needed. A test will be given at the close of each major topic.
We begin by examining ways to classify (DE). This will provide organizational structure for the remainder of the course involving DE. We will also spend some time examining the history of this topic and learn about the extraordinary individuals who have made major contributions over the past three hundred years. We begin by examining ordinary DE which depend on a single variable and then progress to solving partial DE which depend on more than one variable.
(An important example of applying DE to a real world problem arises in Newtons’ second law; This states that the force applied to a body is given by the mass of the body times its acceleration, where the acceleration is the second time derivative of the position), order of a DE, linear and nonlinear DE; Direction fields and use of a calculator in DE. Homework will be assigned in order to reinforce retention of concepts examined.
In the second phase of our DE study, we will begin with first order DE and linear equations. When equations are separable solutions are relatively straightforward; when they are not, then it is necessary to employ a variety of techniques to assist in obtaining solutions. We will learn how to model with linear equations,solve problems in population dynamic learn how to differentiate between linear and nonlinear DE (no pun intended here) work problems in mechanics, learn about exact equations and integrating factors, determine which equations are homogeneous and which are not, and solve many problems that will help to illuminate these techniques. By November we will be ready to examine homogeneous equations with constant coefficients, fundamental solutions to homogeneous equations, linear independence, complex roots of what are called the characteristic equation (a simplistic way to solve what appears to be a complex equations), non homogeneous equations and the method of undetermined coefficients. There will be a final exam to determine how well we have mastered the essentials of a reasonably complex field of study.
Schrodinger’s Equation and Quantum Mechanics
Schrodinger’s Equation, for which Erwin Schrodinger, an Austrian physicist, won the Nobel prize, is fundamental in that the solution of this second order differential equation, when applied to particular atomic systems, yields the secrets we need to enable us to use matter and bend it to our will. The solution of the equation for even relatively simple atomic systems, like sodium, for example, was a monumental task of extreme drudgery before the advent of computers. This difficulty did, however, give rise to some very ingenious methods for solving DE and other mathematical, non analytic solution equations. We hope to examine some of these techniques as time permits. Numerical solutions to DE, for example.
As it happens, the study of DE can help lay the foundation for a deeper understanding of what God hath wrought.
Relativity, Time Travel, and Some Oddities of the Universe
Another topic of great importance (from the point of view of having a lot of fun) is, and you might have guessed this, relativity and time travel. We may not get too far here, but my goal of turning your sons into physicists might be helped along. Since the first part of this course has been tough we will make the second part less so, more imaginative, and more exciting. We will take a leap into space and examine black holes, dark matter and dark energy, worm holes, and string theory.
Success in this course is somewhat akin to success in studying a language. Can one understand the language and use it freely and easily? Mathematics is, after all, another language. Knowing a language means one can easily express ideas in it and understand and evaluate statements and subtle phrases and thoughts. Prompt memorization of necessary concepts and regular attempts to solve all problems assigned would help insure mastery of the topic. The ability to focus ones mind on the subject matter and ignore other thoughts is not easy to achieve and will require some effort. Not rushing for solutions is helpful. A calm, thoughtful approach to problem solving is recommended.
I encourage students to contact me when problems arise. I should be available at any time during the school day. Parents can contact me through telephone or email if there are particular concerns I may be able to help with. I hope to be a successful teacher and to inspire your son to achieve greater intellectual success.