※ [本文轉錄自 NCCU03_IT 看板]
作者: callala (卡拉拉)
標題: [轉錄][知識] 微積分架構
時間: Thu Feb 5 11:44:22 2004
※ [本文轉錄自 callala 信箱]
作者: venlical.bbs@ptt2.cc ("venlical.bbs@ptt2.cc")
標題: [知識] 微積分架構
時間: Thu Feb 5 11:38:09 2004
作者：realtemper 著色：pico
參考：
Jerrold E. Marsden & Michael J. Hoffman, Elementary Classical Analysis,
2nd edition(1993) 9th printing(2003), New York
微積分學的嚴謹架構 (至微積分學基本定理)
一切均源於實數的定義─Complete Ordered Field
以及有關邏輯、集合、加減乘除的公設（視為trivial）
1.路徑一：實數的性質(數) 2.路徑二：Metric Space Topology(形)
Completeness of R 定義open, closed → （定義interior）
↓ ↓ boundary
Cauchy's Criterion in R A closed ←→ A = A ∪ closure
(convergent ←→ Cauchy sequence) accumulation points of A
↓ ｜
定義一般metric space的completeness ｜
n ↓ ↓
R is complete 定義compact
｜ ｜
｜ ｜
｜ 數形合流區 ↓
｜ 定義 Bolzano-Weierstrass Theorem
｜ sequentially → (在一般metric space中compact ←→ seq. compact)
｜ compact ｜
｜ ↓
╰──────→ n Hiene-Borel Theorem
(A< R is compact ←→ A is closed & bounded)
｜
├──定義connected, path-connected
↓
定義一般metric space ─→ 連續函數值域與定義域的關係
的continuity ︵ △open → ︵ open
值 closed → 定 closed
※compact ← 義 compact
域 ◎connected ← 域 connected
︶ path-conn. ← ︶ path-conn.
｜
Extreme-value Theorem ←──┤※
＆ ｜
Intermediate Value Theorem ←──┤◎
｜ ｜
｜ ｜
｜Inverse Function Theorem ←──┤△
｜ ｜
｜ 定義uniform continuity ｜※
｜ ｜ ｜
Mean-Value ｜ ↓
Theorem ｜ Uniform Continuity Theorem
｜ ╰─→ (cont. & domain compact → uni. cont.)
｜ ｜
｜ ｜
｜ Definite Integral Theorem ←╯
｜ (Theorem 4.8.4)
｜ ↓
╰→Fundamental Theorem of Calculus
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※ 編輯: callala 來自: 61.230.54.152 (02/05 11:47)
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