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)


Understanding Analysis by Stephen Abbott

Analysis on Manifolds by James Munkres

Notes on Lagrange Multipliers by Kumaresan

Visual Complex Analysis by Tristan Needham


Midterm exam: 30%
Final exam: 30%
Weekly Homeworks: 30%
Class participation: 10%

Homework Policy:

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.

Homework sets so far:

Homework #1 due on Tuesday January 9 in class.
Homework #2 due on Tuesday January 16 in class.
Homework #3 due on Tuesday January 23 in class.
Homework #4 due on Tuesday Feb 6 in class.
Homework #5 due on Thursday Feb 15 in class.
Homework #6 due on Tuesday March 13 in class.
Homework #7 due on Tuesday March 27 in class.
Homework #8 due on Thursday April 5 in class.
Homework #9 "due" on Tuesday April 9, but it won't be collected.
Homework #10 "due" on Thursday April 19, but it won't be collected.

Other material:

Tutorial #1 from Friday January 6.
Tutorial #2 from Tuesday January 23.
Tutorial #3 from Friday February 9.
Tutorial #4 from Friday March 23.

Lecture Schedule:

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)