source: University of Nottingham 上次更新日期：2014年6月24日
This module introduces mathematical analysis building upon the experience of limits of sequences and properties of real numbers and on calculus. It includes limits and continuity of functions between Euclidean spaces, differentiation and integration. A variety of very important new concepts are introduced by investigating the properties of numerous examples, and developing the associated theory, with a strong emphasis on rigorous proof. This module is suitable for study at undergraduate level 2. Dr Joel Feinstein is an Associate Professor in Pure Mathematics at the University of Nottingham. After reading mathematics at Cambridge, he carried out research for his doctorate at Leeds. He held a postdoctoral position in Leeds for one year, and then spent two years as a lecturer at Maynooth (Ireland) before taking up a permanent position at Nottingham. His main research interest is in functional analysis, especially commutative Banach algebras. Dr Feinstein has published two case studies on his use of IT in the teaching of mathematics to undergraduates. In 2009, Dr Feinstein was awarded a University of Nottingham Lord Dearing teaching award for his popular and successful innovations in this area.
Additional materials for this module are available at: http://unow.nottingham.ac.uk/resources/resource.aspx?hid=... and
http://itunesu.nottingham.ac.uk/albums/71.rss
See also Dr Feinstein's blog at http://explainingmaths.wordpress.com/.
Workshop 1: 22:54
Revision Quiz : 21:23
Lecture 1: 46:56
Lecture 2a: properties of the Euclidian norm 18:19
Lecture 2b: open balls and closed balls 19:11
Workshop 2: Why do we do proofs? 30:40
Lecture 3: Bounded sets 45:16
Lecture 4a: Examples of bounded and unbounded d-cells 18:19
Lecture 4b: Bounded and unbounded d-cells continued 23:20
Workshop 3: Examples Class 1 25:45
Lecture 5: Interior and non-interior points 33:28
Lecture 6: interior points/ non-interior points 42:40
How do we do proofs? Part I - Dr Joel Feinstein 28:54
Lecture 7: Topology of d-dimensional Euclidian space 37:15
Lecture 8a: Closed sets 29:10
Lecture 8b: Sequences in d-dimensional Euclidian Space 8:10
Lecture 9: Absorption of sequences by sets 40:43
Workshop 5: Examples Class 2 24:00
Lecture 10a: Proof of the sequence criterion for closedness 26:13
Lecture 10b: Subsequences and Sequential Compactness 14:59
How do we do proofs? Part II - Dr Joel Feinstein 25:58
Lecture 11: Subsequences of sequences 44:46
Lecture 12a: Proof of Bolzano-Weierstrass theorem 36:56
Lecture 12b: Functions, Limits and Continuity 9:10
Lecture 13a: Continuation of Functions, Limits and Continuity 27:53
Lecture 13b: Continuous Functions 14:52
Lecture 14a: Sequence definition of continuity 32:20
Lecture 14b: Further theory of function limits and continuity 12:59
Workshop 8: Examples Class 4 26:49
Lecture 15: Sandwich theorem for real-valued function limits 47:27
Lecture 16: Application of the sandwich theorem 41:17
Lecture 17a: The boundedness theorem 24:20
Lecture 17b: Pointwise convergence: definition and examples. 10:38
Lecture 18: Sequences of functions 46:26
Lecture 19a: Uniform convergence 21:54
Lecture 19b: Rigorous Differential Calculus 21:36
Lecture 20: Fermat's Theorem, Rolle's Theorem and the Mean Value Theorem 51:51
Lecture 21: An introduction to Riemann integration 53:35
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