Syllabus

Materials science and engineering is a discipline that combines knowledge of chemistry, mechanics, and physics and then applies them to the study of materials and their properties. It is a challenging and diverse enterprise--obtaining expertise in a large set of diverse subjects--but a career devoted to diverse study and applications will be very rewarding and fulfilling.

Mathematics is the language that binds together disparate topics in physics, engineering, and chemistry. While it is possible to become an excellent materials science and engineer without some working knowledge of a large subset of mathematical topics, it is much easier to master this large range of topics with mathematics to guide you. Through mathematics, you will discover similarities between topics that are not obvious--and not taught to you as being similar. Such similarities and analogies will make learning much, much easier--and I think much more enjoyable.

MIT's Department of Materials Science and Engineering has determined that students benefit from a background and a working knowledge of many more mathematics topics that pertain specifically to MS&E than are taught in a one semester subject in the mathematics department. It is reasonable to ask, "Is this subject a substitute for a mathematics subject such as linear algebra or partial differential equations taught by a mathematics professor as part of the mathematics department?'' No, this is not a replacement for such a subject and I encourage you to take subjects in mathematics in the future. This subject is designed to be very broad in scope and therefore its depth in any one topic is limited.

I do believe very strongly that you will enjoy studying math more after taking this introduction and that the mathematical background you will receive this semester will make your materials science education richer and more rewarding.

I have designed this subject to help you learn as much essential mathematics as possible in a short time. To this end, there are several aspects to the course that you will need to know.

Mathematical Software

Symbolic mathematical computer software is a tool used by almost every applied scientist. Such software helps produce results quickly, visualize them, document them, and minimizes the silly errors that creep into complicated mathematical manipulations. Although there are many other good choices, I have decided to use Mathematica^® as a vehicle for learning and doing mathematics. It has a fairly steep learning curve, but it probably repays the time investment with powerful (once learned) language syntax and packages.

Mathematica^® is available for all MIT students on Athena. How to get Mathematica^® to run on Athena will be explained to you. MATHEMATICA licenses for personal laptop and desktop machines are available for free to all MIT students. How to download a license will be explained to you. You will need Mathematica^® for your first homework set; you should try and get it working someplace very soon.

Homeworks should be turned in as printed a printed document. I hope you will also email homework (you can embed extra graphics and text if you wish) your homeworks to the TA as an electronic document (preferably a Mathematica^® notebook or an HTML document.) I will (with permission) publish (on the website) the best homework from each set with attribution.

Examination Philosophy

Tests and exams are powerful motivators to get students to take a subject seriously but I think that working through homework problems better promotes learning if students are self-motivated.

Therefore, there will be no exams, tests, or quizzes in 3.016. Your entire grade will be based on your homeworks. Homeworks will graded carefully (described below) and there will be about one homework set per week while the course is meeting (i.e., no homework will be assigned during the weeks that the laboratory 3.014 meets).

Homeworks

Of course when you do homework, you are not under the potentially menacing eyes of an exam proctor. This means that you can receive help in the form of:

• Books
  Go to the library and find solutions to problems. It is good practice and you will learn quite a bit by doing so. I recommend that you attempt a solution before going to the library--not only will it help you appreciate the solution, it will also make your search for a solution a bit easier!

• Experts
  By all means, consult with experts on your homework. It is a good idea as long as you understand what you turn in.

• Classmates
  This is the best choice of all. I think it is both inevitable and beneficial to give and receive help. Cooperating on homework will help you learn to communicate your ideas and begin to appreciate the difficulties and rewards of teamwork.

As explained below, the homework assignments in 3.016 will be, in part, cooperative.

You will find that you are more busy some weeks than others and relying on a classmate during a busy week can be a life-saver. However, if you start slacking off and don't hold up your end of the bargain when you are able, you will engender resentment and endanger professional and friendly relationships. I leave it to your own conscience to play fairly and contribute when you can--and, while understanding that everyone experiences different kinds of pressures--to be forthright and honest with others who do not contribute consistently.

Homework cooperation has a potential downside because you all receive individual grades. We will attempt to mitigate this downside by dividing the homework into two parts:

• Group
  For each homework set, a few problems will be designated as Group Exercises. For these problems, the entire group will turn in one homework Every member of the group will receive exactly the same credit for the homework grade.

We will begin by allowing you to self-assemble your own homework groups. These groups can be any size that you can manage. During the week of September 29, I will ask whether the groups are functional. If anyone complains or is dissatisfied with your group structure for any reason whatsoever, I will create random subgroups of five or six students for each homework set. I expect the random subgroups would be troublesome for everyone, so I hope this is an incentive to be a cooperative group member.

• Individual
  A few problems will be assigned each week to each student to complete on their own. These problems will come out of the textbook and tend to be a bit easier than the group exercises. They are designed to maintain a sufficient amount of currency and emphasize that reading the textbook is an essential part of this course.

Grading

As stated above, all of the final grade will depend on the homeworks. Homeworks will be graded by ranking them in order from Best Homework to Least Best Homework. A decision will be made regarding how may points out of a possible 100 point scores that the Least Best Homework deserves and the homework scores will be interpolated between a score of 99 for the Best Homework to that of the Least Best.

Homeworks will be evaluated on the basis of:

• Accuracy
  The solution must be a reasonable and correct answer to the homework question.
  
• Exposition
  The solution must clearly show the reasoning that was utilized to find it and the method of solution should be clearly apparent. Exegetic solutions will be ranked higher.
  
• Beauty
  Good solutions will often require graphics and, with care, graphics can often beautifully explain the solution. The layout of the page, the quality of the supporting prose, the clarity of the graphics and all that ``Je ne sais quoi'' is fairly subjective but very important. The reader will include his/her judgment of your art in the ranking of homeworks.
  
• Observation
  Supplemental observations provide aids in understanding and demonstrate mastery of a topic. An example of a supplemental observation might be something like, "Note that the limit of long times, that the total concentration goes to zero. This is sensible because the boundary condition on mass flux is directed outward everywhere on the finite domain.''

Not all homeworks are weighted equally:

Note that there will be times when you have two homework sets pending--this is done so that you can arrange your time conveniently.

Late Policy

Students will be allowed one late homework (up to 3 days) without excuse for the individual portion. No second late homework will be allowed without formal documentation about an unforeseeable emergency. No late group homework portions will be accepted--no exceptions.

Text Book

We will use a fairly general textbook on applied mathematics (E. Kreyszig, Advanced Engineering Mathematics, Eighth ed., J.W. Wiley, ~ 1200 pages). You'll notice that reading assignments do not follow the table of contents--while I like the book, there are pedagogical reasons for studying mathematics in the sequence we will follow in this subject. Extra material pertaining to materials science specifically will be created and placed on the web.

I have identified 66 sections of the book (330 pages in total) as required reading. The readings for each lecture will appear in the Lecture Notes (posted on the web at: http://pruffle.mit.edu/3.016. I hope you will keep up with the reading--I think it would be wise to give the material a cursory reading prior to the lecture and then read it more carefully before starting the homework.

This course is designated as a 8 (3-0-5) unit course. Time spent awake at lectures and recitations is less than half of your job--reading and doing homework is the greater part.

Lecture Notes

Lecture notes will be available for you to print out for each lecture. The lecture notes will be available at http://pruffle.mit.edu/3.016. These will supplement (not replace) the textbook. The lecture notes also serve as a guide to help the student understand what parts of the text are considered more relevant or important.

The specific purpose of the notes is to provide neatly typeset equations and graphics that will be used in the lecture along with a few observations. This will eliminate the time required to write and draw, perhaps a bit sloppily, for you in your notes and for me on the blackboard.

The lecture notes will have reading assignments printed at the beginning of each lecture; they will look like this:

Reading:
Kreyszig Sections: §6.1 (¶304-09) , §6.2 (¶312-18) , §6.3 (¶321-23) , §6.4 (¶331-36) 

Much of what is important is spoken by the lecturer (or in the form of welcome questions and points of clarification by the students) and some explanatory notes will be written upon the blackboard. Mixing projected displays of equations and graphs with blackboard writing will allow incremental adjustments to the best learning pace.

The notes will have places for you to fill in auxiliary discussion and explanation. Those places will look like this:

You can use these notes in several ways. You could print them out before lecture and write your own lecture notes directly during the lecture. You could take lecture notes on your own paper and then neatly copy them onto a printout later. You could print them before lecture and write on them rapidly and then copy--neatly and thoughtfully--notes onto a freshly printed set of lecture. I recommend the latter for effective learning and the creation of a set of notes that might provide future reference material--but do whatever works for you.

The lecture notes will also refer to Mathematica^® notebooks that will also be available at the course website for downloading. These notebooks will be used as Mathematica^® sessions during the lectures to illustrate specific points and provide examples for you to help solve homework problems.