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SYLLABUS Fall 2011 MATH 2250 Section 2 & Section 5 Lecture Course (Math 2250-2): Title: Differential Equations & Linear Algebra Schedule: MTWF 7:30 AM – 8:20 AM in WEB L103 Instructor: Victor Camacho Office: LCB 317 Email: [email protected] Office Hours: TH 11:00 AM - 12:00 PM Webpage: http://math.utah.edu/~camacho/teaching/fall2011/ Textbook: Differential Equations & Linear Algebra by Edwards, Henry C. & Penney, David E. (3rd ed.) Recitation Section Course (Math 2250-5): Schedule: H 7:30 AM – 8:20 AM in WEB L103 Instructor: Nik Aksamit Office: LCB Loft Email: [email protected] Office Hours: TBA Prerequisites: A grade of "C" or better in Math 2210 or Math 1260 or Math 1280 or (Math 1220 and Physics 2210 or Physics 3210). You are expected to have learned about vectors and parametric curves. Course Outline: This course is an introduction to diﬀerential equations, and how they are used to model problems arising in engineering and science. Linear algebra is introduced as a tool for analyzing systems of diﬀerential equations, as well as standard linear equations. Computer projects will be assigned to enhance the material. We will cover most of chapters 1-10 in the Edwards-Penney text, in addition to two supplementary sections which cover RLC electrical circuits and additional Laplace transform material. The course begins with ﬁrst order diﬀerential equations, a subject which you touched on in Calculus. Recall that a diﬀerential equation is an equation involving an unknown function and its derivatives, that such equations often arise in science, and that the order of a diﬀerential equation is defined to be the highest order derivative occurring in the equation. The goal is to understand the solution functions to diﬀerential equations since these functions will be describing the scientific phenomena which led to the diﬀerential equation in the ﬁrst place. In chapters one and 1 two of Edwards-Penney we learn analytic techniques for solving certain ﬁrst order DE’s, the geometric meaning of graphs of solutions, and the numerical techniques for approximating solutions which are motivated by this geometric interpretation. We will carefully study the logistic population growth model from mathematical biology and various velocity-acceleration models from physics. We will take a four week digression from diﬀerential equations, to learn the fundamentals of linear algebra in chapters 3-4. You need a basic understanding of this ﬁeld of mathematics in order to talk meaningfully about the theory of higher order linear DE’s and of systems of linear DE’s. Chapter 3 starts out with matrix equations and the Gauss-Jordan method of solution. When you see such equations in high-school algebra you might be thinking of intersecting lines in the plane, or intersecting planes in space, or ways to balance chemical reactions, but the need to understand generally how to solve such equations is pervasive in science. From this concrete beginning we study abstract vector spaces in chapter 4. This abstract theory is useful not only to the understanding of Euclidean space Rn, but it is also the framework which allows us to understand solution spaces to systems of linear diﬀerential equations. In Chapter 5 we will study the theory of second order and higher-order linear DE’s, and focus on the ones which describe basic mechanical vibrations and electrical circuits. You may have been introduced to these equations in Calculus, and we will treat them more completely now, including a careful study of forced oscillations and resonance. After Chapter 5 we will jump ahead to Chapter 10, the Laplace transform. This “magic” transform converts linear diﬀerential into algebraic equations, which you can solve before “inversetransforming”. You will have to see it in action to appreciate it. You will see, for example, that this method gives a powerful way to study forced oscillations in the physically important cases that the forcing terms are square waves or other discontinuous functions. Electrical engineers often use Laplace transform techniques to understand circuits. To successfully model more complicated physical systems like shaking in multi-story buildings, one needs the framework of linear systems of diﬀerential equations. The linear algebra of eigenvalues and eigenvectors is developed in chapter 6 as a tool for studying systems of DE’s in chapter 7, where for example eigenvalues and eigenvectors will be related to fundamental modes for mass-spring systems. We will also understand these systems of DEs from the point of view of Laplace transforms. Chapter 8, about matrix exponentials, shows the matrix algebra leads to beautiful formulas for solutions to constant coeﬃcient linear systems of diﬀerential equations. We will only cover this chapter if time permits. Our final course topic is Chapter 9, nonlinear systems and phenomena. This topic ties together many diﬀerent ideas, as we discuss the stability of equilibria, based on linearization, and consider the geometry of solution trajectories in the phase portrait of autonomous systems. Many of the most interesting (complicated) dynamical systems on earth are non-linear; perhaps the most famous current example of this is global climate change. (We won’t be able to study that topic in this course. 2 Recitation Sections: The purpose of these extra sessions on Thursday is to work through examples and to try to clear up any other misunderstandings in the material. In addition, the two course midterms will be administered during these sessions. Homeworks/Quizzes: There will be a homework assignments will all be done through WeBWork and will be due every Thursday at 11:00 PM. The lecture notes and textbook will be very helpful in solving the homework problems. I will assign additional practice problems that will not be graded but I highly recommend you work through on your own. I suggest that you have access to a copy of the textbook through the entire semester. You are encouraged to make friends and study groups for discussing homework and practice problems, although you will each be responsible for knowing how to complete each problem on your own (copying someone else's work will not be productive to your learning, nor will it help you on the quizzes or exams). I will not accept late homework under any circumstances. However, the two lowest homework grades will be dropped from your grade. There will be a 10 minute quiz administered at the beginning of class every Friday to test how well you grasped the homework. In general the quiz problems will be easier than the homework problems as they won't require as much busy work or imagination. If you anticipate not being in class on the day of the quiz then you MUST arrange to take the quiz early. This includes athletes and those with any other official obligations such as jury duty etc. I will not accept late quizzes except in the event of unforeseeable medical or family emergencies for which you did not have enough advance notice to arrange to take the quiz early. Even in these extreme cases I will still require WRITTEN documentation. However, the two lowest quiz scores will be dropped from your grade. The Thursday problem sessions should be helpful to many of you. Note also that the Math Department Lab is located in the Rushing Student Center between JWB and LCB. Initial tutorial times will be announced. More information about the lab can be found at the Math Department web page for undergraduates at www.math.utah.edu/ugrad. The Math Tutoring Center is in the Rushing Student Center, in the basement between LCB and JWB on President's Circle. You will be able to find tutors there who can help with your math homework. The tutoring center is open from 8:00 AM - 8:00 PM Monday through Thursday, and 8:00 AM to 4:00 PM on Fridays. For more information about the math tutoring center visit http:// math.utah.edu/ugrad/mathcenter.html. Computer Projects: There will be a few computer projects assigned during the semester related to the classroom material. They will be written in the software package MAPLE. In addition, you will be asked to use this computer software to check various homework calculations from throughout the course. There is a Math Department Computer Lab at which you all automatically have 3 accounts, and there are other labs around campus where MAPLE is also available, for example, at the College of Engineering and the Marriot Library. There will be tutoring center support for these projects. There will be introductory sessions to the Math Lab and to MAPLE in LCB 115 at times which will be announced during the first week of classes. Exams: In this course you will take two in-class midterms and one final exam. Each exam is close book though you will be allowed to use a scientific calculator. Each midterm will be taken during your recitation section so as to not interrupt the flow of the lectures. The exam schedule is as follows: Midterm #1: Thursday, September 29th from 7:30 AM - 8:20 AM in WEB L103 Midterm #2: Thursday, November 10th from 7:30 AM - 8:20 AM in WEB L103 Final Exam: Thursday, December 15th from 8:00 AM - 10:00 AM in WEB L103 Grading policy: Your grade will be based on: Homework/Projects: Quizzes: Midterms: Final Exam 20% 10% 40% 30% (the two lowest scores will be dropped) (the two lowest scores will be dropped) (20% each) [Note: In order to receive a grade of at least "C" in the course you must earn a grade of at least "C" on the final exam.] Important Dates: Last day to add: Last day to drop: Last day to withdraw: August 28th August 31st October 24th Tutoring: The Rushing Math Center offers free drop-in tutoring, a computer lab, and study areas for undergraduates. The Rushing Student Center is in the basement adjacent to LCB and JWB. The hours for the fall semester are: 8:00 AM – 8:00 PM Monday through Thursday 8:00 AM – 4:00 PM on Friday Rules: Please do not behave in a way that will lessen my ability to teach or another student’s ability to learn. Be honest about your own academic work. 4 ADA Statement: The Americans with Disabilities Act requires that reasonable accommodations be provided for students with physical, cognitive, systemic learning, and psychiatric disabilities. Please contact me at the beginning of the semester to discuss any such accommodations that you may require for this course. 5