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2022年02第二节多元函数的基本概念 .pdf

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1、第二节多元函数的基本概念分布图示领域平面区域的概念二元函数的概念例1 例2 例3 二元函数的图形二元函数的极限例4 例5 例6 例7 例8 例 9 例10 二元函数的连续性例11 二元初等函数例12-13 闭区域上连续函数的性质内容小结课堂练习习题 6-2 内容提要一、平面区域的概念：内点、外点、边界点、开集、连通集、区域、闭区域二、二元函数的概念定义 1 设 D 是平面上的一个非空点集，如果对于D内的任一点),(yx，按照某种法则 f，都有唯一确定的实数z与之对应，则称f 是D上的二元函数，它在),(yx处的函数值记为),(yxf，即),(yxfz，其中 x，y 称为

2、自变量，z称为因变量.点集 D 称为该函数的定义域，数集),(),(|Dyxyxfzz称为该函数的值域.类似地，可定义三元及三元以上函数.当2n时,n 元函数统称为多元函数.二元函数的几何意义三、二元函数的极限定义 2 设函数),(yxfz在点),(000yxP的某一去心邻域内有定义，如果当点),(yxP无限趋于点),(000yxP时，函数),(yxf无限趋于一个常数A，则称A 为函数),(yxfz当),(yx),(00yx时的极限.记为Ayxfyyxx),(lim00.或Ayxf),(（),(),(00yxyx）也记作APfPP)(lim0或APf)()(0PP二元函数的极限与一元函数

3、的极限具有相同的性质和运算法则，在此不再详述.为了区别于一元函数的极限，我们称二元函数的极限为二重极限.四、二元函数的连续性定义 3 设二元函数),(yxfz在点),(00yx的某一邻域内有定义，如果),(),(lim0000yxfyxfyyxx,则称),(yxfz在点),(00yx处连续.如果函数),(yxfz在点),(00yx处不连续，则称函数),(yxfz在),(00yx处间断.与一元函数类似，二元连续函数经过四则运算和复合运算后仍为二元连续函数.由x和y的基本初等函数经过有限次的四则运算和复合所构成的可用一个式子表示的二元函数称为二元初等函数.一切二元初等函数在其定义区域内是连续的.

4、这里定义区域是指包含在定义域内的区域或闭区域.利用这个结论，当要求某个二元初等函数在其定义区域内一点的极限时，只要算出函数在该点的函数值即可.特别地，在有界闭区域D上连续的二元函数也有类似于一元连续函数在闭区间上所满足的定理.下面我们不加证明地列出这些定理.定理 1（最大值和最小值定理）在有界闭区域D 上的二元连续函数,在 D 上至少取得它的最大值和最小值各一次.定理 2（有界性定理）在有界闭区域D 上的二元连续函数在D 上一定有界.定理 3（介值定理）在有界闭区域D 上的二元连续函数,若在 D 上取得两个不同的函数值,则它在 D 上取得介于这两值之间的任何值至少一次.例题选讲多元函数的概念例

5、 1（E01）某公司的总体成本（以千元计）为)1ln(245),(2wzyxwzyxC，其中x是员工工资，y是原料的开销，z是广告宣传的开销，w是机器的开销，求)10,0,3,2(C。解用2替换x，3替换y，0替换z，10替换w，则)110ln(03425)100,3,2(2，C6.29（千元）。例 2（E02）求二元函数222)3arcsin(),(yxyxyxf的定义域.解013222yxyx22242yxyx所求定义域为.,42|),(222yxyxyxD文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G

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10、:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5

11、I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N1

12、0 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4例 3（E03）已知函数,),(2222yxyxyxyxf求),(yxf.解设,yxu,yxv则,2vux,2vuy故得),(vuf22222222vuvuvuvu,222vuuv即有.2),(22yxxyyxf二元函数的极限例 4（E04）求极限2222001sin)(limyxyxyx.解令

13、,22yxu则uuyxyxuyx1sinlim1sin)(lim0222200=0.例 5 求极限.)sin(lim22200yxyxyx解22200)s i n(l i myxyxyx,)s i n(l i m2222200yxyxyxyxyx其中yxyxyx2200)sin(limyxu2uuusinlim0,1222yxyxxyxxy22221x21,00 x所以.0)sin(lim22200yxyxyx例 6（E05）求极限22limyxyxyx.解当0 xy时，22220yxyxyxyxxyyx2),(02121yxxy文档编码:CK8H5I5E8N10 HL4G4J1K1O10 Z

14、L6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9

15、Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档

16、编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8

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18、N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL

19、4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K

20、1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4所以.0lim22yxyxyx例 76 求极限.2lim424300yxxxyyx解422242220yxxxyxyxy242422)(21xyxyyx02212xy(当)0,0 yx所以.02lim424300yxxxyyx例 8

21、求.)(lim2200 xyyxyx解.)(lim)ln(lim22002200yxxyxyyxyxeyx因为)ln()(0)ln(22222222yxyxyxxyyxxy.)ln()(2222yxyx而22)l n()(l i m222200yxtyxyxyx令,0lnlim0ttt所以,0)l n(l i m2200yxxyyx故.1)(lim02200eyxxyyx例 9（E06）证明2200limyxxyyx不存在.证取kkxy(为常数),则,1limlim222202200kkxkxkxxyxxykxyxyx易见题设极限的值随k 的变化而变化,故题设极限不存在.例 10 证明263

22、00limyxyxyx不存在.文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL

23、6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z

24、3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编

25、码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H

26、5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N

27、10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4

28、G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4证取,3kxy626330263003limlimxkxkxxyxyxkxyxyx,12kk其值随 k 的不同而变化,故极限不存在.二元函

29、数的连续性例 11(E07)讨论二元函数)0,0(),(,0)0,0(),(,),(2233yxyxyxyxyxf在)0,0(处的连续性.解由),(yxf表达式的特征，利用极坐标变换：令,sin,cosyx则)cos(sinlim),(lim330)0,0(),(yxfyx),0,0(0f所以函数在)0,0(点处连续.例 12 求.1)ln(lim210 xyxyyx解210011)01l n(1)l n(l i mxyxyyx.1例 13 求.lim10yxyexyx解因初等函数yxyeyxfx),(在)1,0(处连续，故.2101lim010eyxyexyx课堂练习1.设,22yxxyyx

30、f求).,(yxf2.若点),(yx沿着无数多条平面曲线趋向于点),(00yx时,函数),(yxf都趋向于A,能否断定?),(lim),(),(00Ayxfyxyx3.讨论函数0,00,),(2222422yxyxyxxyyxf的连续性.文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E

31、8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 H

32、L4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1

33、K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10

34、ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F

35、9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文

36、档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4文档编码:CK8H5I5E8N10 HL4G4J1K1O10 ZL6W5F9Z3J4

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