Course 18.085: Mathematical Methods for Engineers I
(Fall 2008)

Department of Mathematics
Massachusetts Institute of Technology

Lecturer:   Gilbert Strang (gs@math.mit.edu), room 2-240

Lectures:   Room 4-370, MWF 11-12  
Weekly homework review WED 4-5 in 4-370 EXCEPT Mon Oct 6 in 1-190 at 4pm before the Tuesday evening exam

Teaching Assistants:  

• Ramis Movassagh (ramis.mov@gmail.com, Room 2-342)
• Peter Buchak (pmb@math.mit.edu, Room 2-331, office hours Fridays 12:00-1:00 in 4-370 and by appointment Fridays)

Main Topics:  

• Applied Linear Algebra (so important!) 
• Applied Differential Equations (for engineering and science) 
• Fourier Methods 
• Algorithms (lu, qr, eig, svd, finite differences, finite elements, FFT) 

Course outline

My Goals for the Course:   I hope you will feel that this is the most useful math course you have ever taken. I will do everything I can to make it so. This will not be like a calculus class where a method is explained and you just repeat it on homework and a test. The goals are to see the underlying pattern in so many important applications—and fast ways to compute solutions.

Textbook:   Computational Science and Engineering (Wellesley-Cambridge, 2007). I do think you will need to have this textbook. I did everything I could to control the cost and believe that you will use it for a long time.

Grades:   Let me try to say this clearly: my life is in teaching, to help you learn. Grades have come out properly for 20 years (almost all A-B). I will NOT spend the semester thinking about grades. I hope you don't either. The homeworks will be important and I plan 2-3 evening exams and NO FINAL. Those exams are open book and a chance for you to bring key ideas together.

Videos:   The special event for Fall 2008 is that the lectures will be recorded for OpenCourseWare You will already find a partial earlier set on the website but the course has since evolved. The videos of 18.06 Linear Algebra have been successful on OCW (please use them for help!!). The Lord Foundation gave a new grant for 18.085.

Normally the lead TA runs a weekly review session. This time I plan to do that (and announce the time). It is a good chance for informal questions, which are great in class too (I will repeat them for the recording).

Homework:   I will collect homework once a week. I want to ask your help in preparing solutions to go on math.mit.edu/cse . We will choose the clearest (and NEATEST) solutions to scan or to upload (if they are electronic and especially if they are in LaTeX). For that reason we can try two approaches:

1. Use a number like *78* (assigned from the class list) instead of your name
2. Put problems on separate pages, each with your number (paper-clipped, not stapled)

I will use MATLAB notation to describe algorithms---that is a good choice for homework (Mathematica and Maple both OK, Octave/Python/Sage are growing).

For a class of this size it's impossible to grade every problem in detail. You may discuss them with others before writing the solution yourself. This class is for learning!! I hope very much that you will enjoy it.

Evening Quizzes:   7:30 to 9:00

1. Tuesday October 7 in 54-100
2. Wednesday November 5 in 54-100 (changed)
3. Thursday December 4 in Walker

Quizzes are always open book and open notes. No calculators (and they are not needed). I plan to make the questions straightforward, a chance to review and bring ideas together.

HOMEWORK 1 FOR MONDAY, SEPTEMBER 8: Any 3 problems from section 1.1 and from 1.2. Please PRINT your name so that we can make a class list. Not graded in detail—you may always discuss with others!

Solutions to Homework 1

Solution 1

Solution 2

Homework 2 for MONDAY September 15 (and please see the notes below):

• 1.3 2, 7, 11, 16
• 1.4 2, 7, 12
• 1.5 4, 9, 12, 18

Important:

1. You can ask about all topics and all text problems, especially at the review sessions WEDNESDAYS at 4 in 4-370. No reason for learning to be limited to specific homework problems.
2. If your homework is typed (TEX or plain text) send me pdf if you can.
3. Peter Buchak will have office hour Friday Sept 12 at 12:00 in 2-331 (see under Teaching Assistants above).

Solutions to Homework 2

Solution 1

Solution 2

Solution 3

Solution 4

Solution 5

Homework 3 for WED SEPT 24 // There will soon be a separate MATLAB question for that Friday the 26th

• 1.6 15, 20, 24, 27
• 1.7 4, 5, 18
• 2.1 4, 8

You may discuss with friends/ write your own solutions/ email separately to me if they are typed (especially in TEX)/ late hwk in my 2-240 envelope

I learned that there is an exam in 2.25 conflicting with the plan for our second exam --- so I will look for another date for that exam.

Solutions to Homework 3

Solution 1

Solution 2

Solution 3

Solution 4

Matlab 1: This MATLAB homework for FRIDAY 9/26 is separate from the text homeworks. It is about the convection-diffusion equation -u'' + Vu' = load A symmetric part K/h^2 comes from diffusion -u'' plus an antisymmetric part if we use centered differences for the convection term Vu'. As V increases this becomes convection dominated and u changes.

The homework to turn in asks for V=3 and V=12 each with n=5 and n=21.

1. Solve the equation for u(x) with point load at x=0.5 by hand
2. Solve by finite differences for two choices of n
   (Include 4 plots of finite difference solution superimposed on exact solution.)
3. Find the eigenvalues for both choices of V (remarkable)
4. Your experiment: try a larger V or larger n or more eig (<1 page!!)

Here is the Matlab code that will help with this.

Peter Buchak wrote the code and can help with Matlab questions at his regular Friday noon session (might as well stay in 4-370 unless he needs a computer!) or contact him.. My review sessions are Wed at 4 in 4-370.

Solutions to Matlab 1

Solution 1

Solution 2

Homework 4 for WED OCT 8 (after the 7:30 pm exam on TUES OCT 7 in 54-100)
I will try to move my review session from WED OCT 8 to MON OCT 6

• 2.2 6, 7
• 2.3 7, 18, 24
• 2.4 1, 3, 7, 19

One more question: Find 4 solutions to the 2 by 2 system Mu''+Ku = 0 The solutions have the form cos(wt)x and sin(wt)x. First find M and K: Fixed-fixed / spring constants c1=1 c2=2 c3=4 / masses m1=1 m2=2 Look for eigenvectors and eigenvalues (by hand) of K x = w^2 M x.

**The exam will include the start of Section 2.4 and I will give more detail in class**

Solutions to Homework 4

Solution 1

Solution 2

Solution 3

Prof. Strang's Solution to 2.2.6

Solutions to Quizzes

Quiz 1

Videos of Professor Strang's Lectures (Lincoln Lab, Spring 2001)

18.085 Course information page from Fall 2007.

Movie of elimination   moe.m   (also need realmmd.m )

Code to create K,T,B,C as sparse matrices

MATLAB's backslash command to solve Ax = b   (ps, pdf)

Getting started with Matlab: http://ocw.mit.edu/OcwWeb/Mathematics/18-06Spring-2005/RelatedResources/

Exams and Solutions (Fall 2006)

Quiz 1 ( ps | pdf ) and Solutions ( ps | pdf )
Quiz 2 ( ps | pdf ) and Solutions ( ps | pdf )
Quiz 3 ( ps | pdf ) and Solutions ( ps | pdf )

Exams and Solutions (Fall 2005)

Exam 1 (pdf) and solutions (pdf)
Exam 2 (pdf) and solutions (pdf) and supplementary solutions (pdf)
Exam 3 (pdf) and solutions (pdf)

Exams and Solutions (Fall 2004)

Exam 1 (pdf) and solutions (pdf)
Exam 2 (ps) and solutions (pdf)
Exam 3 (ps) and solutions (ps)

Exams and Solutions (Fall 2003)

Exam 1 (pdf) and solutions (pdf)
Exam 2 (pdf) and solutions (pdf)
Exam 3 (pdf) and solutions (pdf)

Exams and Solutions (Fall 2002)

Exam 1 (ps) (pdf) and solutions (ps) (pdf)
Exam 2 (ps) (pdf) and solutions (ps) (pdf)
Exam 3 (ps) (pdf) and solutions (ps) (pdf)

Exams and Solutions (Fall 2001)

Exam 1 (ps) (pdf) and solutions (ps) (pdf)
Exam 2 (pdf) and solutions (pdf)
Exam 3 (ps) (pdf) and solutions (pdf)

More Exams and Matlab Homeworks from previous years

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