Course Outline for: Advanced Linear Algebra and Complex Variables

Transcription

1 NYU Polytechnic School of Engineering, Mathematics Department Course Outline for: Advanced Linear Algebra and Complex Variables Spring section Kl: Tues-Thurs 2-3:20 PM, RGSH 202 Classes: January 24, 2017-May 4, Final Basic dates: First day of classes this term Monday Jan 23, 2017, last day of classes Monday May 8, 2017, Reading day May 9, 2017, final exams, May 10- May 16, 2017 Sunday, Feb 5, 2017 Add drop date Monday, Feb 20, 2017 President's day, no classes Monday, March 13 - Sunday, March 18,2017 Spring Break Friday, March 31, 2017 last day to withdrawal from classes or term this semester This course provides a deeper understanding of topics introduced in MA 2012 and MA 2034 and continues the development of those topics, while also covering functions of a Complex Variable. Topics covered include Elements of Complex Variables, analytic functions, Cauchy-Riemann equations, Integrals and Cauchy integral theorem. Power and Laurent Series, Residue theory. : The Gram-Schmidt Process.inner product spaces and applications, Singular value decomposition, LU decomposition\ Prerequisites: MA-UY 2122 or MA-UY 2114 AND MA-UY 2012 or MA-UY Note: Course not open to students who have taken MA-UY Course Coordinator: Professor Edward Y. Miller, Office: RH 321G em1613@nyu.edu, Phone: Office Hours: Tues & Thurs 12:00-2:00 or by appointment. Student Disabilities: If you are student with a disability who is requesting accommodations, please contact New York UniverJIJy's Moses Center for Students with Disabilities at or mosescsd@nyu.edu.you must be registered with CSD to receive accommodations. Information about the Moses Center can be found at The Moses Center is located at 726 Broadway on the 2nd floor. Course Pre-requisites: You are expected to have mastery of the concepts and skills covered in MA 2122, Multivariable Calculus B. Course Description:_ This course falls into two parts: The first deals with Functions of a complex variable. Derivatives and CauchyRiemann equations. Integrals and Cauchy integral theorem. Harmonic functions, the exponential function, trigonometric functions, logarithmic functions. Contour integrals, antiderivatives, the Cauchy-Goursat theorem, Cauchy integral formula, Liouville's theorem, fundamental theorem of algebra. Power and Laurent Series. Residue theory.

2 The second part continues the study of linear algebra, the gram Schmidt process for inner product spaces, eigenvalues and their applications, the singular value decomposition, and so on. This course will be a rigorous mathematics course where students will be required to understand all of the definitions, theorems and proofs. Students will often be asked to explain mathematical concepts in essay questions on exams, to prove certain facts. Course Objectives: Upon successfully completing this course students will be able to: Use the complex numbers and compute the elementary functions, sine, cosine, exponential, logarithms, and powers extended to the complex domain. Use and understand analytic functions and their primary properties. Their derivatives, the Cauchy Riemann equations, Harmonic functions. Be able to apply Liouville's theorem appropriately, for example to prove the fundamental theorem of algebra. Understand the theory of integration of analytic functions and the applications of the Cauchy-Goursat theorem. For example, to compute residues, integrals, and give power series and Laurant series expansions of these functions. To understand the simpler examples of conformal mappings, product expansions and elliptic integrals To understand the linear algebra associated to Eigenvalue problems, inner product spaces, gram Schmidt, LU and other decomposition theorems and their applications and significance. Course Structure: The 3-credit, semester course meets for lecture twice a week for 80 minutes per class for 14 weeks plus final. You are also expected to study outside of class, a good 'rule of thumb' is three hours of study for each hour of class. Examinations (The grading policy is detailed in a section below). Twice weekly Lecture. First Midterm Exam in class. Second Midterm Exam [final for complex variables part] in class Final Exam Scheduled during final exam week[ cumulative, complex and linear parts]. Textbooks: There are two required texts: #1 : Complex Variables and Applications published by McGraw-Hill.ISBN We will use the 9th Edition, but you may use any edition you wish. # 2: Linear Algebra and Its Applictions, by David Lay, Addison and Wesley, ISBN-13;

3 Grading Policy Course Grode: Final Grades will be calculated according to the rules below. The course grade is determined by the best of your course averages using the table below. The complex part will have two subsections a,b with test on each. Course Grade: Final grades will be calculated according to the rules below. The course grade is determined by the best of your course average using the table below. Average 1 Average 2 Midterm on complex part 65% 55% a and midterm on complex part b Final Exam (cumulative) Complex and Linear parts 25% 40% together Homework 10% 5% Conversion of Course Average to Course Grade Course Average Course Grade A A B B B C c C D D below 55 F Course Lecture Syllabus (The sections referenced are from the 9th Edition of Churchill.) [Lecture 1] Chapter - 1 Complex numbers and their properties. Sectionl Sums and Products Section 2 Basic Algebraic Properties. Section 3 More Algebraic Properties

4 Section 4 Moduli {lengths), Arguments {angles), geometric representation of sums and products, triangular inequality. Section 5 Complex conjugates, inverses. Section 6 Exponential form. [Lecture 2] Some standard methods and representations Section 7 Exponential form, products and quotients, algebraically and geometrically. Section 8 Powers and roots of complex numbers via exponential form. Section 9 Examples, geometric representations, n-th roots of unity and their geometry. Section 10 Regions in the complex plane, bounded, open, and connected. [Lecture 3] Chapter 2-Introduction to Analytic Functions, Examples Section 11 Definition: Functions of a complex variable. Two approaches: Real, Imaginary & Modulus, Argument. Section 12 Mappings, translation, reflection, dilation, z->za2 in detail. Section 13 Mappings by the exponential function, examples.d [Lecture 4] Limits and Continuity Section 14 Limits in the complex setting: limit of f{z) as z7a. Section 15, 17 Theorems on limits and continuity. Section 16 Limits involving the point at infinity, stereographic projection, the projective plane. [Lecture 5] Derivatives of complex functions, analytic functions Section 18, 19 Derivatives, differentiation formulas. Section 20, 21 Cauchy Riemann equations, a sufficient condition. Section 23, 24, 25 Analytic functions, examples, Harmonic functions. [Lecture 6] Chapter 3-Elementary Functions Section 28 Exponential Function exp{z). Section 29, 30, 31 Logarithm Function Log{z), branches and derivatives of the logarithm function, identities. Section 32 Complex exponents, aaz. Section 33 Trigonometric functions, sine, cosine, tangent, etc. Section 34 Hyperbolic functions sinh, cosh, tanh etc. Section 35 Inverse Trigonometric Functions.

5 [Lecture 7] Computation of elementary functions, Examples, Review of chapters 1-3 [Lecture 8] Midterm Exam February 16, 2017 [Lecture 9] Contour Integration Section 36 Derivatives of w(t) complex function of real variable t. Section 37 Review: Definite integral. Section 38, 39 Parametrized paths, contour integration. Section 40, 41 Examples, Basic inequality for contour integration. Section 42,43 Antiderivatives, examples, the complex analog of the theorem of calculus. fundamental [Lecture 10] Cauchy-Goursat Theorem for simply closed curves D Section 44 Statement of The Cauchy Goursat theorem. SecTion 46 Simply and multiply connected regions, the main trick in evaluation integrals in This special analytic setting, Examples. Section 47 Cauchy Integral Theorem, an application of The Cauchy Goursat Theorem. Section 48 Formulas for all derivatives, an application of The Cauchy Goursat Theorem. Section 49 Liouville' s Theorem, an application of The Cauchy Goursat Theorem. [Lecture 11] Applications of the Cauchy Integral Theorem Section 50 Cauchy's Inequality, the Maximal modulus principle. Section 51 Expansion of an analytic function converging on a disk as a convergent power series. Examples. Section 52,65 Laurant expansions, three Types of isolated singular points. Section 62, 63, 66 Integrals by method of residues. Residues at poles. [Lecture 12] Computation of integral via method of residues Section 71, 72 Evaluation of some improper integrals. Review by examples of the method of residues to do integrals. [Lectures 13-14] Added examples of uses and properties of Residues to compute integrals, proper and improper.

6 [Lectures 15-16] Mappings by Elementary Functions and fractional linear transformations, chapter 8, Conformal mappings, physical examples, the Schwarz-Christoffel transformations. chapter 9. [Lecture 17]: Introduction to Product expansions, elliptic integrals, doubly periodic mappings, Riemann surfaces, branched coverings. [Lecture 18): Review of Complex variables part: [Lecture 19] Second Midterm on complex variables April 4,2017. [Lectures 20-2n; Linear Algebra Component, will be cover chapters 5-7 of Lay's text. [Lecture 28] Review for final [cumulative] : May 4, 2014 Additional Learning Resources: General Evening Math Workshops Days Hours Location Mon-Thurs Check JAB 373 hours Room 2C General Math Workshops Days Hours Location Friday Saturday Check hours Check hours Internet Resources

7 Math Department Web site. See also /web.mit.edu/ /www /Video/video-f al html Paul's Online Math Notes, Choose Class Notes and then the course you want. There are many good lectures, the Khan Academy is a favorite for many students. Important: General Exam Policies Valuables (especially your laptop!): Please do not bring your laptop or any other valuable items to the exam. You are required to leave your bags and books at the front of the exam room. Time and Place: It is your responsibility to consult the web site to know when and where an exam is being held. You will not receive any special consideration for being late or missing an exam by mistake. Identification: You are required to bring your Polytechnic ID to the exam. If for any reason you are unable to do so, another photo ID, such as a driver's license, is acceptable. Before the Exam: You must wait outside the exam room before the start of an exam. You must sit only in seats where there is an exam for your course. You must not move the exam to a different seat. Neatness and Legibility: You are expected to write as neatly and legibly on your exam. Your final answer must be clearly identified (by placing a box around it). Points will be deducted if the grader has difficulty reading or finding your answer. Missed Exams: If you missed an exam due to a medical reason, then University policy requires you to provide written documentation to the Office of Student Development (JB158). It is University policy

8 that the Mathematics Department may not give make-up exams without prior authorization by the Office of Student Development. Academic Integrity: Any incident of cheating or dishonesty will be dealt with swiftly and severely. The University does not tolerate cheating. (There is no such thing as "a little bit of cheating.") During an exam you are not allowed to borrow or lend a calculator; borrowing or lending a calculator will be considered cheating. TI-30 is the only calculator allowed! No Exceptions. No cell phones may be used during the exam period, no exceptions.