10602 高等微積分二

第17講 8.2 Volume and sets of measure zero

課程影音

L17A


L17_A
        8.2 Volume and sets of measure zero
              (1) Defintion: Let A⊂ R^n be a bounded set 
              (2) Note
              (3) Property: All rectangles B =[a,b]x...x[an,bn] ⊂ R^n
                                  has volume = ∏ (bi-ai) 
              (4) Question:
                 
 1.怎樣的 bounded set 會有體積呢?
                       2.A has volume zero <=> 1A is integreable and ∫1A = 0
 
 
L17_B
        8.2 Volume and sets of measure zero
              (1)Question: A has volume zero <=>
                                  1A is integreable and ∫1A = 0
              (2)Property: A bounded set A in R^n has zero
              (3)Cor: 1.If A has volume zero, then any subset of
                              A has volume zero
                  1.Any finite union of volume zero sets also has volume zero
 
 
L17_C
        8.2 Volume and sets of measure zero
              (1) Property: A bounded set A in R^n has zero
              (2) Definition: A set A ⊂ R^n
                                   (not necessary bounded) has measure zero,...
              (3) Property: 1.Any subset of measure zero set is also measure zero
                   2.A set in R^n having volume is not measure zero.
                   3.If A has volume zero, then A is measure zero
                   4.Any single point set in R^n has volume zero and measure zero
                   5.The real line, regarded as a subset of R^2 has measure zero,
                          but as a subset of R,it does not. 
 
              (4) Thm: Suppose that the sets A1,A2,...have
                             measure zero in R^n. Then the union UAi
                             has measure zero
              (5) Cor:Any countable set in R^n in measrue zero
              (6) Property: measure zero ≠>volume zero
 

L17_D
        8.3 Lebesque's thm
              (1) Question: 衡量函數值
              (2) Definition
              (3) Thm (Lebesque's thm)
              (4) Remark

 

【6/21 @ 109學年度第2學期休學申請截止Last Day to Withdraw from School】   ♠   【6/28 @ 教師送繳應屆畢業生109學年度第2學期成績截止Deadline for 2021 Spring Grade Submission for Graduating Students】   ♠   【6/28 @ 暑假開始Summer Vacation Begins】   ♠   【6/28 @ 暑期班選課及繳費開始(至7月9日止)Course Selection and Fee Paying Begins for Summer Session (6/28-7/9)】   ♠   【7/1 @ 暑期班上課開始Summer Session Begins】   ♠   【7/5 @ 110學年度暑碩專班上課開始、註冊日、休退學及畢業生免繳學雜費截止2021 Summer In-service Master Program Begins; Registration (2021 Summer In-service Master Program)】   ♠   【7/9 @ 教師送繳非應屆畢業生109學年度第2學期成績截止Deadline for 2021 Spring Grade Submission】   ♠   【7/23 @ 110學年度暑碩專班休退學及畢業生退2/3學雜費(學分費)截止Last Day for 2/3 Tuition and Fees Refunded(graduates or full withdrawal of 2021 Summer In-service Master Program)】   ♠   【7/30 @ 109學年度第2學期結束2021 Spring Semester Ends】   ♠   【6/14全球事務處】110年「強化與東協及南亞國 家合作交流計畫」(新南向計畫)開放申請至6月14日!   ♠   【6/5研究發展處】【計畫徵求公告】閎康科技產學合作計畫!   ♠   【清華學院】弦歌不輟,遠距上路!語文中心外語進修暑期班採全遠距教學,防疫同時安心學!   ♠   【6/7計算機與通訊中心】填卷送禮、抽獎!「校園網路與授權軟體服務品質問卷調查活動」,歡迎全校教職員工生登入校務資訊系統填卷 (110/6/7上午9:00 ~110/6/18)   ♠   【6/13清華新聞】清大與英國利物浦大學合作 成功研發新型鈣空氣電池儲存機制!   ♠   【6/13 陽明交大新聞】疫情升溫電信強攻智慧醫療 血氧監測雲端平台吸睛!   ♠   【6/13 成大新聞】攜手成大 大河里福德祠邁向數位營造!   ♠   【7/26 人事室】科技部111年度(第60屆)補助科學與技術人員國外短期研究案!   ♠