Course Description

Honors Calculus - Course Description

The following are quotations from our text book Who is Fourier? A Mathematical Adventure, (WIF), and these should be our guidelines for this course!

"Use ordinary, everyday language when you analyze or debate a concept. When the image that emerges becomes so clear that anyone can understand it, that is the time to start thinking of an applicable formula for it." Dr. Werner Heisenberg, Nobel Laureate Physicist

"The ones who make little progress are those who succumb to an adult's self-consciousness and are afraid they will make mistakes or sound foolish." page xvi of WIF

Calculus is one of the greatest inventions of the human race. It has become a required course for all science, engineering, and engineering technology majors, as well as for business and other majors in higher education worldwide.Those who learn it easily appreciate the beauty and the elegance of the theory and the method. For others it is a constant struggle and embarrassment.

Simultaneously with the revolution of modern technology in the last decades there have been many attempts to reform calculus nationwide as well as at IPFW. The purpose of the reform has been to make it more understandable for students, prepare them better for the job market, integrate modern technology, and bring it closer to real-life applications. Graphing calculators and personal computers in the classroom have been an integral part of the reform. The teaching methods have been changed from the standard classroom setting to a laboratory setting, and from primarily individual work to individual and group work. The precision of the definitions, the accuracy of the theorems presented in the classroom, and the level of the analytical presentation versus the intuitive presentation are equally important. Therefore, neither of them can replace the other, and this should not be the purpose of the reform.

The text book of the course is "WHO IS FOURIER? A MATHEMATICAL ADVENTURE," published by the Language Research Foundation, Boston. The book was created by a group of scientists, educators, and artists in Japan and in the USA. The ISBN number of this book is 0-9643504-0-8.

The text book is ideal for this innovative approach and serves mostly the intuitive part of this course. It starts with an informal motivation for the need for trigonometry and calculus. It raises the question of solving the riddle of human speech, that is, analyzing the human voice. In order to solve this mystery, first we have to ask "What is the voice?" With a voice analyzer, that the book calls FFT (short for Fast Fourier Transform) it can be determined that the sounds of the voice are typical complex, periodic waves. Then it explains that the insight ofJean Baptiste Joseph, Baron de Fourier (1768-1830), was that no matter how complicated a wave is that is periodic -- with a pattern that repeats itself -- may be viewed as the sum of many simple waves (sine and cosine waves) with different frequencies and amplitudes. Therefore to study complicated waves it is sufficient to study simple waves.

Keeping this goal in mind, and the goal to solve the original problem, here and throughout the book, the authors do a superior job to motivate all of the concepts of mathematics that are usually covered in any standard calculus book or course.

Fourier series and Fourier and inverse Fourier Transforms are used in many higher-level mathematics, science and engineering classes, such as partial and ordinary differential equations, physics, chemistry, biology, engineering, medicine, speech sciences, communication science, broadcasting, photography, linguistics, and many more courses. The Fast Fourier Transform method is used in many areas of research as well as everyday applications in industries. In physics, it is used in solid-state physics, quantum mechanics, and wave mechanics. Chemistry, biology, and geology, electron maps and other applications use it a lot. In X-ray crystallography this method is the backbone of the whole study. In engineering, for instance, devices engineered on the basis of Fast Fourier Transforms are used in the design and construction of smooth-running cars, trains, and airplanes. These devices use Fast Fourier Transforms to eliminate noises and vibrations.

In photography this method makes it possible to convert blurred pictures into sharp images. In broadcasting and communication noise screening is very important; it also uses this method. In speech and communication as well as in linguistics, voice recognition and voice analysis are crucial. Without the Fast Fourier Transform method these studies cannot be conducted. I could go on and on mentioning a lot more areas and crucial applications of these topics.

In this course the focus will be on the mathematical foundation of the Fourier series, the Fourier and the inverse Fourier Transform, and the theoretical background of the Fast Fourier Transform. In single projects that will be part of the course as well as the grading, the students conduct actual applications from their own areas of studies. They will be conducted by the physics, chemistry, biology, geology, speech and communication, linguistics, and psychology departments, as well as in the School of Engineering, Technology, and Computer Science.

The book WIF is not analytical enough for a calculus MA 163 - 164 course. The lack of complex mathematical formulas, clear and rigorous definitions, explanations, and proofs, and the lack of some mathematical computations make the Lecture Notes (LN) necessary. It is also lacking in standard exercises. LN is the supplement to the book WIF, and thus to the course. Their parallel use makes the course complete and equivalent to the regular calculus MA 163 - 164 course. This explains Heisenberg's quotation: we will use the book WIF as the ordinary, everyday language, and we will just turn to LN when the time is ready to find applicable formulas, methods, computations, and to practice with exercises, etc....