一注基礎高等數學知識總結

1、高等數學知識總結一、空間解析幾何...................................................................................................................................................31向量代數...................................................

2、..................................................................................................32曲面及其方程..................................................................................................................

3、...........................43空間曲線及其方程.....................................................................................................................................54平面及其方程..........................................

4、...................................................................................................55空間直線及其方程...............................................................................................................

5、......................5二、極限和連續(xù).......................................................................................................................................................71數列極限.................................

6、....................................................................................................................72函數極限..................................................................................................

7、...................................................73幾個重要極限.............................................................................................................................................74無窮小量..............

8、.......................................................................................................................................75連續(xù)函數...............................................................................

9、......................................................................7三、一元函數的微分學...........................................................................................................................................

10、81導數的定義.................................................................................................................................................82導數運算..............................................................

11、.......................................................................................83常數和基本初等函數的導數：.................................................................................................................84微分概

12、念及其運算法則.............................................................................................................................85Lagrange中值定理..........................................................................

13、...........................................................86函數的單調性與曲線的凹凸性.................................................................................................................97函數的極值與最大值最小值...................

14、..................................................................................................98Cauchy中值定理..............................................................................................................

15、..........................99法則：型未定式或型未定式(不是未定式不能用洛必達法則)............................9LHospital?00??10泰勒(Tayl)公式——用多項式近似表示函數..................................................................................9四、多元微分學........

16、.............................................................................................................................................101極限與連續(xù)性......................................................................

17、.....................................................................102微分和偏導數...........................................................................................................................................103

18、復合函數的微分法...................................................................................................................................114方向導數和梯度.......................................................................

19、................................................................115空間曲線的切線與法平面.......................................................................................................................126曲面的切平面與法線方程.........

20、..............................................................................................................127Tayl公式.....................................................................................................

21、..........................................138多變量函數的極值...................................................................................................................................13五、一元函數的不定積分.......................

22、..............................................................................................................141不定積分.......................................................................................................

23、............................................142基本積分表——（求導的逆運算）.......................................................................................................143不定積分的性質.............................................

24、..........................................................................................144換元法............................................................................................................................

25、...........................145分部積分法...............................................................................................................................................14六、定積分...................................

26、..........................................................................................................................151定積分定義（分割，近似，求和，取極限）............................................................................

27、...152牛頓－萊布尼茲公式...............................................................................................................................153定積分的性質（設所列定積分都存在）..........................................................

28、.....................................154廣義積分...................................................................................................................................................15一、空間解析幾何1向量代數?向量的線性運算向量加法：三

29、角形法則三角形法則或平行四邊形法則：1)交換律a?b?b?a?2)結合律(a?b)?c?a?(b?c)?實數與向量的運算法則：設、為實數，則有：??bac??1)結合律?(?a)??(?a)?(??)a；2)分配律(???)a??a??a；?(a?b)??a??b??空間直角坐標系?)(zyxzyxOMM??????kjir設a?(ax?ay?az)?b?(bx?by?bz)，則有1）a?b?(ax?bx?ay?by?az?bz)?2）

30、a?b?(ax?bx?ay?by?az?bz)?3）?a?(?ax??ay??az)?4）ba?b??a?(bx?by?bz)??(ax?ay?az)??zzyyxxababab??5）向量模：6）兩點間的距離：222||zyx???r?212212212)()()(||||zzyyxxABAB???????7）方向角方向角：非零向量r與三條坐標軸的夾角?、?、?稱為向量r的方向角方向余弦方向余弦：???||cosrx??||cosry

31、??||cosrz???向量的數量積：ab?|a||b|cos?幾何意義幾何意義：數量積ab等于a的長度與b在a的方向上的投影的乘積。||a?cos||b1）aa?|a|2??）a?b?ab?012120xxyy???3）交換律?ab?ba4）分配律?(a?b)?c?a?c?b?c?5）(?a)b?a(?b)??(ab)?(?a)(?b)???(ab)??、?為數?6）ab?axbx?ayby?azbz?222222||||coszyx

32、zyxzzyyxxbbbaaabababa?????????baba??向量的向量積：c?a?bc的模|c|?|a||b|sin??其中?為a與b間的夾角c的方向垂直于a與b所決定的平面?c的指向按右手規(guī)則右手規(guī)則從a轉向b來確定?幾何意義幾何意義：以a與b為兩鄰邊的有向面積有向面積。1）a?a?0?2）ab?a?b?03）交換律a?b??b?a?4）分配律?(a?b)?c?a?c?b?c??）(?a)?b?a?(?b)??(a?b)6