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課程概述 |
Chapter 1. Rn and its topology (7 weeks)
x1.1 Introduction
Metric on Rn, Schwartz inequality, limits, Cauchy sequence, series, completeness of
Rn, countable and uncountable sets. Examples.
x1.2 Topology of Rn
1. Open set, interior of a set, closed set, closure, accumulation points, boundary.
Examples.
2. Continuous functions (Examples, including monotone), uniform convergence
and power series.
x1.3 Compact and connected sets. (1 week)
1. Compact sets, The Heine-Borel (Bolzano-Weierstrass) theorem and open covering.
2. Connected set, one-dimensional classi cation, path-connectedness, applications
to continuous functions (Intermediate theorem). Examples. (xx1.1-1.3 for 3
weeks)
x1.4 Metric space
1. Rn (in `p)
2. The space of continuous functions, Ascoli-Arzela theorem, Stone-Weierstrass
theorems (compactness revisited).
3. Fixed point theorem.
(a) Contraction maps.
(b) Applications to O.D.E: existence and uniqueness, continuity on initial
data, local stability in 2 2 system. (2 weeks)
(c) Brouwer xed points (optional).
4. Baire category theorem and its applications.
Chapter 2. Di erentials (6 weeks)
x2.1 1. Overview (one-dimension) Rolle's theorem and mean-value theorem. Applications.
2. Linear transformation, Di erential (de nition) and example (with emphasis on
geometry)
3. Partial derivatives and continuous P.D ! di erentials
4. The chain rule. The determinant of its di erential (for Rn ! Rn)
x2.2 Higher order di erentials. Partial derivatives. Taylor expansions.
x2.3 Implicit function theorem and its variations. Examples.
x2.4 Local extrema and Lagrange multipliers. Applications. Examples.
x2.5 Sard's theorem.
Chapter 3. Riemann Integral of multi-variables (1.5 weeks)
1. De nition, change of variable formulas (for continuous function). Examples.
2. Surface integral and line integral. Stokes' theorem. (Applications)
3. Lebesgue's criterion for the existence of Riemann integral. (1 week, the next
semester) |