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Department of Mathematics MATHS 253: Study Guide for Semester 1, 2017. Welcome

Department of Mathematics MATHS 253: Study Guide for Semester 1, 2017. Welcome to MATHS 253, which is the standard sequel to MATHS 250, available in semesters 1 and 2 on the City Campus. It covers topics in linear algebra, multi-variable calculus and their applications. It lays a foundation for a large number of stage 3 courses in pure and applied mathematics and statistics and also for many advanced courses in physics and various other applied sciences. 1. Title: MATHS 253 is also known as Advancing Mathematics 3. 2. Lecturer & course coordinator: Prof. Arkadii Slinko. Oﬃce: Room 203, ﬂoor 2, Maths-Physics building 303, phone extn 85749. Oﬃce hours: Monday, Thursday, Friday 2-3pm, other times by appointment. E-mail: a.slinko@auckland.ac.nz Lecturer: Dr. Jeroen Schillewaert Oﬃce: Room 213, ﬂoor 2, Maths-Physics building 303, phone extn TBA Oﬃce hours: TBA Email: TBA 3. Times and rooms: (a) Lectures: Monday 4pm in 105S-039 (Clock Tower), Thursday & Friday 3pm in 303-101. (b) Tutorials: Tuesday: 1pm in 303-B07 & 4pm in 303-B11; Wednesday: 9am in 303-B07; Thursday 12pm in 303-B05. There will be no tutorial in week 1. 4. Description: (a) Points: This is a 15-point course. (b) Pre-requisites: 250, or 152 or Physics 112 or Physics 210. Students wanting to take 253 directly from 208 must have an A+ in 208 and must discuss the situation with the course coordinator, such students will need to do some extra work to catch up, especially over the ﬁrst 3 weeks. In general, students from 208 should take 250 before 253. (c) Restrictions: You cannot include MATHS 253 along with MATHS 230 (if you took 230 prior to its termination) or Physics 211 in your degree. 1 (d) Aims: The main aim is to complete the students preparation prior to commencing the various advanced and more theoretically rigorous level 300 papers. Stu- dents will be required to reach a good technical skill level in linear algebra and in calculus. There will be emphasis on combining multi-variable calculus with linear algebra in a way that is suitable for further advanced study. 5. Expectations of students: (a) Pre-requisite knowledge: In addition to the pre-requisite papers noted above, it is also essential that students have an ability and a desire to follow reasoning. (b) Recommended preparation: MATHS 250 (or MATHS 130 if you took it prior to its termination). (c) Study expectations: All students are expected to do 10 hours per week on this course. This com- prises: 3 hours of lectures, 1 hour of tutorial and 6 hours of individual work. This may take the form of preparation for lectures, reading the textbook, doing assignments, preparing for tutorials, visiting the lecturer during oﬃce hours and so on. 6. Resources: (a) Texts: Linear Algebra, (Poole) and Calculus, (Stewart). They are available new at both the University Bookshop in the Student Commons area and VOL 1 Bookshop, 33 Symonds St. These texts are also the texts for 150/108/250/208. (b) Lecture Notes: The lecture notes are available in electronic form on Canvas and will be available from SRC as a booklet. Excellent explanations and worked examples are also available directly from the textbooks. 7. Technology: The Department of Mathematics is using the software package Matlab for all undergraduate courses. Matlab knowledge will be useful, especially in visualization of graphs of functions of several variables, however familiarity with Matlab is not mandatory in this course. Wolfram Alpha can also be used. 2 8. Lecture topics. This is an approximate schedule, it may change slightly as the course progresses. lecture Topic 1.1 Vector spaces over R and C and their subspaces. 1.2 Bases & dimension. The coordinate mapping. 1.3 Linear transformations and their matrices. 2.1 Change of basis matrices and their properties. Algebra of linear operators. Matrices of linear operators and change of basis. 2.2 Eigenvectors and eigenvalues. Algebraic and geometric multiplicities of eigenvalues. Invariant subspaces. 2.3 Diagonalisation of operators. 3.1 Applications of diagonalisation. Discrete time system evolution. 3.2 Projection formula. Gram-Schmidt orthogonalisation. Projection matrix. 3.3 Inner products and real inner product spaces. Orthogonality in inner product spaces and their orthogonal bases. 4.1 Orthogonal bases for vector space of polynomials and trigonometric polynomials. 4.2 Projections as best approximations. Polynomial and Fourier approximations. 4.3 Complex inner products. Orthogonal matrices. 5.1 Adjoint of an operator. 5.2 Eigenvalues of Hermitian operators. Orthogonal diagonalisation of Hermitian operators and symmetric matrices. Spectral decomposition. 5.3 Quadratic forms, their matrices. Change of basis. Principal axes theorem. 6.1 Conics and Quadrics. 6.2 Positive deﬁnite quadratic forms. Sylvester’s criterion. 6.3 No lecture due to holiday. Mid-semester break 7.1 Partial derivatives. Higher order partial derivatives. Symmetry of the Hessian matrix. 7.2 First order approximations. Diﬀerentiability. 7.3 The chain rule and applications. 8.1 Gradient, tangent planes, directional derivatives. 8.2 Taylor series. Best quadratic approximation. 8.3 Maxima and minima. Critical points. 9.1 Constrained and unconstrained optimisation. 9.2 Double integrals over rectangles. 9.3 Fubini’s theorem. 10.1 Double integrals over general domains. 10.2 Change of variables in double integrals. 10.3 Triple integrals. 11.1 Space curves. Arc length parametrisation. 11.2 Surfaces and their areas. 11.3 Vector ﬁelds. Conservative vector ﬁelds. 12.1 No lecture due to holiday. 12.2 Line integrals. 12.3 Green’s theorem. 3 Planner. Week Mon Tue Wed Thur Fri 1 6-3 7-3 8-3 9-3 10-3 lect 1 lect 2 lect 3 2 13-3 14-3 15-3 16-3 17-3 lect 4 tut 1 tut 1 lect 5 & tut 1 lect 6 3 20-3 21-3 22-3 23-3 24-3 lect 7 tut 2 tut 2 lect 8 & tut 2 lect 9 ass 1 due 4 27-3 28-3 29-3 30-3 31-3 lect 10 tut 3∗ tut 3∗ lect 11 & tut 3∗ lect 12 5 13-4 4-4 5-4 6-4 7-4 lect 13 tut 4 tut 4 lect 14 & tut 4 lect 15 ass 2 due 6 10-4 11-4 12-4 13-4 14-4 lect 16 tut 5∗ tut 5∗ lect 17 & tut 5∗ Good Friday 17-4 18-4 19-4 20-4 21-4 mid sem break this week 24-4 25-4 26-4 27-4 28-4 mid sem break this week 7 1-5 2-5 3-5 4-5 5-5 lect 18 tut 6 tut 6 lect 19 & tut 6 lect 20 semester test 8 8-5 9-5 10-5 11-5 12-5 lect 21 tut 7∗ tut 7∗ lect 22 & tut 7∗ lect 23 9 15-5 16-5 17-5 18-5 19-5 lect 24 tut 8 tut 8 lect 25 & tut 8 lect 26 ass 3 due 10 22-5 23-5 24-5 25-5 26-5 lect 27 tut 9∗ tut 9∗ lect 28 & tut 9∗ lect 29 11 29-5 30-5 31-5 1-6 2-6 lect 30 tut 10 tut 10 lect 31 & tut 10 lect 32 ass 4 due 12 5-6 6-6 7-6 8-6 9-6 Queen’s birthday lect 33 lect 34 * Tutorials in week 4, 6, 8, 10 are assessed (2%). 9. Assessment: The assessment formula will be either • 12% assignments, 8% tutorials, 25% test, 55% exam, or • 35% test, 65% exam, whichever gives the better result. 4 Note there is no option of just sitting the ﬁnal exam only! (a) Assignments: There will be 4 assignments with the following due dates: March 24, April 7, May 19, June 2. No extensions are allowed. Each assignment counts for 3% of your assess- ment. Assignment questions are intended to extend your understanding of this course. (b) Tutorials: Tutorials are a very important part of this course. There will be 10 of them. Each tutorial will be devoted to the material of the previous week of study. Tutorials will be posted on Canvas several days in advance and students are advised to start thinking about the problems before the tutorial. During the tutorial students will be able to get help with the completion of the tutorial problems. Students will be encouraged to work in groups. Your work in tutorials in weeks 4, 6, 8, 10 will be assessed. Each assessed tutorials counts for 2%. (c) Test: The test will be 1,5 hours long. The date for the test is 2 May at 18:15 – 19:45. It will take place in rooms 206-209, 206-315, 105-018, 105- 029. The distribution of students to the rooms will be made in alphabetical order but the exact cut-oﬀs will be announced later. Clashes in semester tests should be reported to the course organiser early so that appropriate arrangements could be made. (d) The exam: The exam will be for 2 hours long. (e) Aegrotates: Students who cannot sit either the test or the ﬁnal examination due to some approved medical or other emergency, should complete an aegrotate appli- cation form in the normal manner. However, an aegrotate pass cannot be granted to any student purely on the completion of assignments and tutorials only. You must sit at least one of either the test or the examination to be considered for an aegrotate pass. 10. Calculators: Calculators will not be permitted in the mid-term test and the ﬁnal examination. 11. Canvas: Canvas is the prime means of information about the running of the course. All announcements made in lectures will also be made on Canvas. Students are re- uploads/Litterature/ department-of-mathematics-maths-253-study-guide-for-semester-1-2017.pdf