Class Notes
These are the notes from first semester.
Sets and maps.
Finite and countable sets.
Cardinals. Topological spaces.
Open and closed sets.
Continuous functions.
Connected and complete spaces.
More complete spaces.
Compact spaces.
The real numbers: definitions.
Review.
Series in
R
.
Rational and irrational numbers.
Functions on
R
.
Ruled functions. Integration of ruled functions.
Differentiation. Fundamental Theorem of Calculus. Mean Value Theorem.
Higher derivatives and Taylor expansions. Applications.
Asymptotic Developments.
L'Hopital's Rule. Improper integrals.
The Gamma function. Stirling's formula.
Continuity and differentiation under the integral. Improper integrals.
Groups and homomorphisms.
Cosets and Lagrange's Theorem. Normal subgroups.
Quotient groups. Examples.
Rings and fields.
Complex numbers. The exponential function.
The Fundamental Theorem of Algebra. More exponentials.
Linear algebra. Definitions. Linear independence.
The matrix of a transformation. Rank.
Endomorphisms and diagonalization. Diagonalization.
Minimal polynomials.
Polynomials and endomorphisms. Triangulation.
Determinants.