Department of Mathematics Syllabus

This syllabus is advisory only. For details on a particular instructor's syllabus (including books), consult the instructor's course page. For a list of what courses are being taught each quarter, refer to the Courses page.

MAT 25: Advanced Calculus
Approved: 2013-07-01, John Hunter

Suggested Textbook: (actual textbook varies by instructor; check your instructor)
Stephen Abbott, Understanding Analysis, $53; Alternate text: Kenneth Ross, Elementary Analysis, ISBN 146146703, $50
Search by ISBN on Amazon: 1441928669

Prerequisites:

MAT 21B

Course Description:

Students develop skills required to understand rigorous definitions in analysis and to construct and write proofs. They learn the axiomatic definition of the real numbers as a complete ordered field and the basic topology of the real numbers. They study the convergence of sequences and series.

Suggested Schedule:

Lecture(s)

Sections

Comments/Topics

3

1.1, 1.2

Sets and functions. Logic and proofs. Proof by induction.

1

1.2, 8.4

Algebraic and order axioms for the real numbers. Absolute values.

1

1.3

Completeness axiom for the real numbers. Suprema and infima.

2

1.4

Archimedian property of the real numbers. Density of the rational numbers.

3

2.1, 2.2

Sequences. Definition of the limit of a sequence.

3

2.3

Algebraic and order limit theorems.

2

2.4

Monotone convergence. The limsup and liminf.

2

2.5

Subsequences. Bolzano-Weierstrass theorem.

1

2.6

Cauchy sequences.

3

2.7

Infinite series. Absolute convergence. Comparison test. Alternating series test.

1

2.8

Double summations. Products of infinite series.

3

3.1, 3.2

Topology of the real numbers. Open and closed sets. Accumulation, boundary, and interior points.

2

3.3

Compact sets of real numbers. Heine-Borel theorem. Finite intersection property.

1

3.4

Connected and disconnected sets of real numbers.

Additional Notes:

(1) The sections listed in the topics refer to the text by Abbott. Chapters 1 and 2 of Ross cover similar material, except that Ross discusses the topology of metric spaces, rather than the topology of the real numbers.
(2) A goal of this class is to ensure students learn to write rigorous proofs and communicate mathematical concepts using language. Have students regularly practice writing formal proofs that emphasize course content and mathematical thinking.