Math 205a: Multivariable Calculus and Linear Algebra
Fall 2007
10:10 - 11:00 MWF
10:00 - 10:50
R
Instructor: Prof.
Bruce Hughes
Office Hours: MWR
Office: SC 1528
Phone: 2-6660
Electronic mail: bruce DOT hughes AT vanderbilt DOT edu
Web page: http://math.vanderbilt.edu/~hughescb/Math205a.html
Teaching Assistant: Mr. Vladimir Chaynikov
Office Hours: MR 2:30 – 3:30
Office: SC 1232 B
Phone: 2-2025
Electronic mail: vladimirDOTvl DOT chaynikov AT vanderbilt
DOT edu
Catalog description:
Mathematics 205a - 205b. Multivariable calculus and linear algebra. Vector algebra and geometry; linear transformations and matrix algebra. Real and complex vector spaces, systems of linear equations, inner product spaces. Functions of several variables and vector-valued functions: limits, continuity, the derivative. Extremum and nonlinear problems, manifolds. Multiple integrals, line and surface integrals, differential forms, integration on manifolds, the theorems of Green, Gauss, and Stokes. Eigenvectors and eigenvalues. Emphasis on rigorous proofs. Enrollment limited to first-year students with test scores of 5 on the Calculus BC advanced placement examination or the approval of the director of undergraduate studies. 205a is a prerequisite for 205b. Credit is not given for both 205a-205b and 170b, 175, 194, or 204. [4-4] Hughes.
Prerequisites: This course is for highly motivated first-year students with a solid background in single-variable calculus (demonstrated by a 5 on the Calculus BC advanced placement exam) and a strong interest in constructing and understanding proofs.
Who should take Math 205ab? All qualified prospective mathematics majors are encouraged to take this course. Other qualified science and engineering students with a strong interest in mathematics should consider it. The main criteria are that the student has a 5 on the Calculus BC advanced placement exam, an excellent background in single-variable calculus, and a fascination with the theoretical side of mathematics. Math 205ab is an alternative to Math 175 (Second Year Accelerated Calculus) and Math 204 (Linear Algebra). All of the material from Math 175 and Math 204 will be covered in Math 205ab, but Math 205ab will go much further and deeper.
Textbook: Multivariable Mathematics: Linear Algebra, Multivariable
Calculus, and Manifolds by Theodore Shifrin
(published by John Wiley & Sons, Inc. 2005).
Syllabus: We will cover chapters 1 through 5 of the textbook this semester (the rest of the book will be covered in Math 205b). Here is a tentative day-by-day plan for the course:
1. Overview
Chapter 1: Vectors and matrices
2. Vectors in Rn
3. Dot product
4. Linear subspaces of Rn
5. Linear transformations and matrices
6. Algebra of linear transformations and matrix algebra
7. The transpose
8. The cross product and introduction to determinants
9. Abstract vector spaces
10. Proofs based on axioms
11. Review
12. Test 1
Chapter 2: Functions, limits and Continuity
13. Parameterized curves in Rn
14. Scalar-valued functions of several variables
15. Vector-valued functions of several variables
16. Topology in Rn: balls and open sets
17. Convergence of sequences in Rn (more on proofs)
18. Closed sets (more on proofs)
19. Limits of functions
20. Continuity
21. Partial derivatives
22. Review
23. Test 2
Chapter 3: The derivative
24. Differentiability and the Jacobian
25. More on differentiability
26. Differentiation rules (including the chain rule)
27. The gradient
28. More on curves: arclength and parameterization by arclength
29. Motion and curvature
30. Kepler’s laws
31. Higher order partial derivatives
Chapter 4: Implicit and explicit solutions of linear systems
32. Gaussian elimination and the theory of linear systems
33. Consistency of systems
34. Existence and uniqueness of solutions; rank of matrix; nonsingular matrices
35. Review
36. Test 3
37. Elementary matrices
38. Calculating matrix inverses
39. Linear independence
40. Basis and dimension
41. More on abstract vector spaces
42. Column space and row space of a matrix
43. Null space of a matrix; the nullity-rank theorem
44. Nonlinear equations: implicit functions and manifolds
Chapter 5: Extremum problems
45. Compactness and the maximum value theorem
46. Maximum/minimum problems
47. Review
48. Test 4
49. Quadratic forms
50. Second derivative test
51. Lagrange multipliers
52. Projections and least squares; orthogonal bases and the Gram-Schmidt process
53. Inner product spaces
54. Review
55. Review
56. Final Exam
Mid-Term Exams: Four tests will be given during the semester on the following dates:
Wednesday, September 19
Wednesday, October 10
Wednesday, October 31
Friday, November 16
Final Exam: A final examination will be given on Friday, December 21 from 9:00 - 11:00 a.m.
Homework: Homework is an important component of this course and will be assigned daily. Some assignments will be collected and graded, others will be discussed in class by students and the instructor. On some assignments students will not be allowed to work together. On others, students are encouraged to work together. However, a student should not present as their own work solutions to which they did not make a substantial contribution. Collaborators in a solution should be acknowledged. Avoid looking up solutions in books. If you do use a book, be sure to site the source (including page number). You may not consult solution sheets distributed in previous years.
Grades: Your final grade will be determined from a total of 1000 possible points as follows:
Attendance: Attendance is expected for each class
meeting. More than two absences will be considered "excessive" on the
mid semester progress reports. See the
Honor System: Vanderbilt's Honor Code governs all work in this course. It is a violation of the Honor Code to consult homework solution handouts, tests, or test solution handouts from previous semesters.
Web Resources for Undergraduate Mathematics Majors: The American Mathematical Society maintains a very useful page for undergraduate mathematics majors at http://www.ams.org/outreach/undergrad.html It includes information on summer programs (Research Experiences for Undergraduates), semester programs, graduate studies in mathematics, clubs, undergraduate journals, competitions, careers, jobs and much more.
Notes on Proofs by Greg Friedman: Click here for pointers for beginners on how to do proofs.
Shifrin's list of errata for his book: Click here.
A guide to using
For all other links for
Digital Resources related to Multivariable Mathematics:
Mathematical Association of America's Digital Classroom Resources: Click here.
For a Collection of Tools for Multivariable Calculus: Click here.
Updated