| 04/02/2019: |
Applications of the integral I: Taylor's
formula; indefinite integrals; Uniform convergence revisited:
exchange of integration and differentiation (see,
e.g., this
handout, Theorem 5.5)
|
| 05/02/2019: |
Applications of the integral II: Laplace's
method and Stirling's formula (see O.A. Ivanov, Easy as
pi? An introduction to higher mathematics, Springer,
1999, Section 2.5)
|
| 11/02/2019: |
Laplace's method (finish); Uniform convergence of series,
Cauchy's criterion (Rudin, 7.7-7.10); uniform limits of
continuous functions are continuous (Rudin, 7.11-7.12)
|
| 12/02/2019: |
Point-set topology in metric spaces I:
open and closed sets (Rudin, 2.15-2.24)
|
| 18/02/2019: |
Point-set topology in metric spaces II:
compact sets (Rudin, 2.31-2.40)
|
| 19/02/2019: |
Point-set topology in metric spaces II:
more about compactness (Rudin, 2.41, 4.16 with alternative
proof); continuous pre-images of open sets are open (Rudin,
4.8)
|
| 25/02/2019: |
Monotonically convergent sequences of functions on
a compact set are uniformly convergent (Rudin, 7.13);
convergence, integration, and differentiation revisited.
|
| 26/02/2019: |
Equicontinuity and the Arzela-Ascoli theorem
|
| 04/03/2019: |
Stone-Weierstrass theorem
|
| 05/03/2019: |
Power series, radius of
convergence
|
| 11/03/2019: |
Curves in Rn
|
| 12/03/2019: |
Midterm Review
|
| 18/03/2019: |
Midterm Exam
|
| 19/03/2019: |
Linear transformations, norm of a linear
transformation, invertibility
|
| 25/03/2019: |
Total derivative
|
| 26/03/2019: |
Directional derivative, partial
derivatives
|
| 01/04/2019: |
Chain rule
|
| 02/04/2019: |
Taylor's formula in Rn
|
| 08/04/2019: |
Contraction mapping theorem
|
| 09/04/2019: |
Inverse function theorem
|
| 23/04/2019: |
Implicit function theorem I
|
| 29/04/2019: |
Implicit function theorem II
|
| 30/04/2019: |
Riemann integral in Rn I: definition
and elementary properties
|
| 06/05/2019: |
Riemann integral in Rn II: iterated
integrals and Fubini's theorem
|
| 07/05/2019: |
Riemann integral in Rn III: change of
variables and polar coordinates
|
| 13/05/2019: |
Riemann integral in Rn IV: the
divergence theorem with elementary applications
|
| 14/05/2019: |
Review for final exam
|
| TBA: |
Final Exam
|