Dates:

Topics:

Advanced Calculus/Ordinary Differential Equations (``ODE'')

Sept 1

Review of the real number system

G1.1

Sept 3

Review of the real number system

G1.2

Sept 8

Metric spaces, continuity, uniform convergence

G1.3

Sept 10

Metric spaces, continuity, uniform convergence

G1.4-1.5

Sept 15

Contraction mapping principle Existence and uniqueness theorems for ODE

G2.1-2.2

Sept 17

Existence and uniqueness theorems for ODE

G2.2

Sept 22

Global existence theorem for linear ODE

G2.2

Sept 24

Linear transformations, orthogonal projections and matrix exponential

G2.3

Sept 29

Tue. Sept 29 CORRECTION DAY:U Classes follow a Monday schedule.

Oct 1

Linear transformations, orthogonal projections and matrix exponential

G2.3, 3.1-3.2

Oct 6

Linear transformations, orthogonal projections and matrix exponential

Linear systems of ODE with constant coefficients

G3.2-3.3

Oct 8

Linear systems of ODE with constant coefficients

Derivatives in Rⁿ and in Banach spaces

G3.3-3.4

Oct 13

Derivatives in Rⁿ and in Banach spaces

G3.5,3.6

Oct 15

Derivatives in Rⁿ and in Banach spaces Newton's method and the inverse function theorem

G3.6-3.7

Oct 20

Newton's method and the inverse function theorem

G3.7

Oct 22

The implicit function theorem

G3.8

Oct 27

Midterm

Measure theory

Oct 29

Riemann integral in Rⁿ

G4.1

Nov 3

Cantor-type sets, dyadic decompositions in Rⁿ

G4.2

Nov 5

Measures arising from volume functions on open sets

G4.3-4.4

Nov 10

Measures arising from volume functions on open sets

G4.4-4.5

Nov 12

Basic properties of the Lebesgue measure

G4.6-4.7

Nov 17

Measurable and integrable functions

G5.1-5.2

Nov 19

Measurable and integrable functions Convergence theorems for Lebesgue integrals: monotone and dominated convergence theorems and Fatou's lemma

5.2-5.3

Nov 24

Convergence theorems for Lebesgue integrals: monotone and dominated convergence theorems and Fatou's lemma

G5.4-5.6

Nov 26

 Thanksgiving Break Nov. 25-29 Thanksgiving Break – NO CLASSES IN-SESSION

Dec 1

Integration of complex functions

G5.7-5.8

Dec 3

Criterion for Riemann integrability

G6.1-6.2

Dec 8

Criterion for Riemann integrability

G6.3-6.4

Dec 10

Review