ENGS 92 Home Page

Course information

Schedule

Waterloo Maple, makers of MAPLE V

The MathWorks, makers of MATLAB

Thayer School of Engineering

Dartmouth College

Engineering Sciences 92

Course Information

Syllabus

1. Fourier transforms: Orthogonal function expansions. The four kinds of Fourier transform. Fourier symmetries and theorems. Physical interpretations and applications. Generalized functions and sampling theory. (6 weeks)
2. Functions of a complex variable: Continuity, differentiation, analyticity, Cauchy-Riemann equations. Singularities. Multivalued functions, branch points and branch cuts. (1 week)
3. Complex integrals: Line integrals in the complex plane. Cauchy's integral theorem and formula. Laurent series and residues. Applications to Fourier, Hilbert, Laplace, and Z transforms. (2.5 weeks)

Prerequisite knowledge

The stated prerequisites for Engs 92 are Math 33, Engs 22, or equivalent. In terms of specific knowledge and skills, I shall assume that you know:

• Properties of elementary functions, including polynomials, ratios of polynomials, trigonometric, exponential, and logarithmic functions.
• Elementary properties of complex numbers; the complex exponential. How to calculate the real/imaginary and magnitude/phase representations of a complex algebraic function (like the transfer function of a circuit).
• Matrices and vectors. How to add and subtract vectors, how to express a vector as a sum of other vectors (components). What the dot product and norm are, and how to calculate them. How to multiply a matrix and a vector, or a matrix and another matrix.
• How to determine if a function is continuous, differentiable, or integrable. How to differentiate and integrate functions of one or more real variables, including polynomials and transcendental functions. How to use change of variables and integration by parts. How to do a Taylor expansion of a function.
• How to use the Laplace transform to solve a differential equation with initial conditions and a driving function.

Learning objectives

Engs 92 is a course in applied mathematics. I will endeavor to strike an appropriate balance between the intrinsic elegance of the mathematics and techniques for applications. At the end of Engs 92, I expect you to know:

1. How to define, calculate and manipulate several important transforms: Fourier (four kinds), Laplace, Z, and Hilbert.
2. How to apply these transforms to linear systems, wave propagation, and signal analysis.
3. How to use the computer (particularly, Matlab) to perform Fourier calculations.
4. How to define, manipulate and analyze functions of a complex variable.
5. How to integrate functions of a complex variable, and how to apply complex integration to calculate transforms.

Organization

Schedule

See attached

Grading

This course is graded on the basis of weekly homeworks, quizzes, and a final exam.

• Homeworks (9): 30%
• Quizzes (6): 40%
• Final exam: 30%

Readings

You are expected to keep up with the reading and come to class prepared to discuss the assigned reading.

Homeworks

Homework assignments will be due by 4:30 PM each Wednesday. Drop your homework in the box at Kathy Crowe's desk. Graded papers will be returned to you the following Monday.

Quizzes

There are six weekly quizzes, substituting for midterm examinations. These will be taken in-class, during Thursday X-hours, and will cover material from the most recently graded homework (not the one you just turned in!).

Final exam

The final exam will be a timed (8-hour) take-home, covering the entire course. The final exam is open-book.

Honor Principle

The Honor Principle is an integral part of the academic program at the Thayer School. Consult of the Dartmouth College ORC or the Thayer School graduate student handbook for a discussion of Academic Honor at Dartmouth College. Specifically, in Engs 92:

1. You are encouraged to study together and permitted to discuss the homework, with this understanding: when you turn in a paper with your name on it you are saying to me that the final work is yours, that you understand your solution, and that you can explain it, to me or to the class. Copying other people's answers is a violation of the Honor Principle as well as being a poor way to learn. In the end, the strongest measure of what you have learned (70% of the grade) will be your scores on quizzes and exams, which you do by yourself.
2. Acknowledge, by means of footnotes, all sources of help outside the usual course readings | titles of books or journal articles, computer programs, study partners. Failure to do so constitutes plagiarism and violates the Honor Principle. It may be possible for you solve a problem by looking it up in a book (e.g., another text or a table of integrals) or by applying a mathematics program (e.g., Matlab, Maple). Some homework problems will require use of these tools, but you should otherwise apply them with discretion, since you will not be able to use them for the quizzes or final exam. In all cases, they will be helpful for checking your homework, computing graphs, and performing routine numerical manipulations (e.g., root finding).
3. It is not permitted to receive aid from any unauthorized source during a quiz or exam, nor is it permissible to give aid to a classmate during a quiz or exam.
4. The Honor Principle extends in the obvious way to library reserve books, computer resources, and other facilities associated with the course. To hoard, "borrow" without authorization, damage, or in any other way restrict the access of your fellow students to these public resources is clearly in violation of the spirit of the Honor Principle.

Disabilities

Students with disabilities enrolled in this course and who may need disability-related classroom accommodations are encouraged to make an appointment to see me, ideally, before the end of the second week of the term. All discussions will remain confidential, although the Student Accessibility Services office may be consulted to discuss appropriate implementation of any accommodation requested.

Absence

Being absent from a quiz or the final exam, or handing in an assignment late, is excused in case of illness or family emergency. If you are part of a sport travel team, please let me know your schedule so that we can work around it.

Reference Books (on reserve in Feldberg Library)

• R.N Bracewell, The Fourier Transform and its Applications — An excellent, though somewhat dated, classic text on Fourier transforms. Has a very useful pictorial dictionary of transforms at the back of the book. This has been used as a text for Engs 92 in the past.
• A. Wunsch, Complex Variables with Applications — A very good book, with many examples. This has been used as a text for Engs 92 in the past.
• E.O. Brigham, The Fast Fourier Transform and Applications — A comprehensive introduction to the discrete Fourier transform and various applications in electrical engineering. Good for understanding the details of the FFT algorithm.
• R.V. Churchill & J.W. Brown, Complex Variables and Applications — A classic text on complex variables.
• F.J. Flanigan, Complex Variables: Harmonic and Analytic Functions — Does a very nice job of connecting 1D calculus, 2D calculus, and complex variables.
• W.R. LePage, Complex Variables and the Laplace Transform for Engineers — An old book (reprinted by Dover), which provides a more extensive treatment of the Laplace transform, its foundations in complex analysis, and its applications, mostly in electrical engineering.
• M.J. Lighthill, Introduction to Fourier Analysis and Generalised Functions — A beautiful, concise explanation for the more mathematically inclined student.
• A.V. Oppenheim & A. Willsky, Signals and Systems — A sophomore-level introduction to systems and transforms, both continuous- and discrete-time.

In the Reference section of Feldberg Library

• M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions — An excellent compendium of information about all kinds of familiar and obscure functions, including: graphs, series representations, derivatives and integrals, real and complex arguments, and numerical tables.
• I.S. Gradshteyn and I.M Ryzhik, Table of Integrals, Series, and Products — Also known affectionately as "The Russian Book", this is one of the most extensive tables of integrals ever compiled. "If they don't have what you're looking for, you can probably get along without it" (or, you probably can't integrate it anyway).

In the Mathematics Library (Baker Berry Cook)

• A. Erdélyi (editor), Tables of Integral Transforms and Higher Transcendental Functions — volumes in Caltech's famous Bateman Manuscript Project. A wealth of information, including integral transform tables not found anywhere else.

Computing in Engs 92

Many of the real applications of this course are facilitated by mathematical software packages. Two packages commonly available at Dartmouth are Maple, a product of Waterloo Maple, and Matlab, from The MathWorks. Maple manipulates mathematical expressions symbolically, e.g., it can do algebra, derivatives, Taylor series expansions, indefinite integrals, etc. Maple can also do numerical computations (such as solving differential equations and inverting matrices) and graphics, but these are secondary to its symbolic power. Matlab, on the other hand, is a superlative numerical and graphics engine. Matlab is used in industry as well as academia, and in my opinion is an indispensable tool for the modern engineer. I will be using Matlab extensively in this course, and you will need to develop a certain level of proficiency with it.