Introduction to Real Analysis II, Jan-April 2018
Vijay Ravikumar (vijayr at cmi dot ac dot in)
Tuesday 10:30 - 11:45 / Thursday 10:30 - 11:45
CMI Lecture Hall 5
('Riemann’s Integral' by Lun Yi Tsai)
('Riemann’s Integral' by Lun Yi Tsai)
Understanding Analysis by Stephen Abbott
Analysis on Manifolds by James Munkres
Notes on Lagrange Multipliers by Kumaresan
Visual Complex Analysis by Tristan Needham
1) Late homework will be accepted at half credit until exactly one week after the due date. No homework will be accepted after that point.
2) If you have difficulty with an assignment, you are encouraged to approach the instructor for help. It is also fine to discuss the problems with other students, but...
3) Your final write-up must be your own. If you have received help solving a problem, then you must cite your source(s). In particular, plagiarism, or any kind of copying, will not be tolerated. Offences will result in serious disciplinary action, up to and including a failing grade in the course.
| date | lecture # | announcements |
| Jan 2 (tues) | 1: the riemann integral | |
| Jan 4 (thu) | 2: integrating functions with discontinuities | |
| Jan 9 (tues) | 3: properties of the integral | homework #1 due |
| Jan 11 (thu) | 4: the fundamental theorem of calculus | |
| Jan 16 (tues) | 5: inner product spaces | homework #2 due |
| Jan 18 (thu) | 6: more review of linear algebra | |
| Jan 23 (tues) | 7: metric spaces | homework #3 due |
| Jan 25 (thu) | 8: topology of R^n | |
| Jan 30 (tues) | 9: compact and connected sets in R^n | |
| Feb 1 (thu) | 10: the derivative | |
| Feb 6 (tues) | 11: partial derivatives, the jacobian | homework #4 due |
| Feb 8 (thu) | 12: criterion for differentiability | |
| Feb 13 (tues) | 13: chain rule | |
| Feb 15 (thu) | 14: applications of chain rule, equality of mixed partials | homework #5 due |
| Feb 27 (tues) | 15: inverse function theorem | |
| Mar 1 (thu) | 16: inverse function theorem | |
| Mar 6 (tues) | 17: implicit function theorem | |
| Mar 8 (thu) | 18: implicit function theorem | |
| Mar 13 (tues) | 19: tangent space and normal space | homework #6 due |
| Mar 15 (thu) | 20: lagrange multipliers | |
| Mar 20 (tues) | 21: bilinear forms | |
| Mar 22 (thu) | 22: diagonalization of symmetric matrices | |
| Mar 27 (tues) | 23: analyzing critical points with hessian | homework #7 due |
| Mar 29 (thu) | 24: taylor's theorem | |
| Apr 3 (tues) | 25: complex functions, euler's formula | |
| Apr 5 (thu) | 26: complex power series, complex derivative | homework #8 due |
| Apr 10 (tues) | 27: properties of complex inversion map | |
| Apr 12 (thu) | 28: properties of stereographic projection | |
| Apr 17 (tues) | 29: classification of mobius transformations |
('Some Quadric Surfaces' by Lun Yi Tsai)