多元函数可微性的研究.doc

目 录中文摘要··············································································································2ABSTRACT············································································································20引引言言·········································································································································21 1预备知预备知 识识····························································································································· ····31.1 多元函数全微分的定义·································································································31.2 函数在点沿方向 可微的定义·····································3,f x y00(,)xyg2 2元函数可微的充分条元函数可微的充分条件件··········································································································43 3 多元函数可微性的充要条多元函数可微性的充要条 件件··························································································74 4定理的应定理的应 用用··················································································································· ········14 4.1 定理 2.1，定理 2.2 的应 用··············································································14 4.2 定理 3.1，定理 3.3 的应 用··············································································15 4.3 定理 3.2，定理 3.4 的应用···············································································15参考文 献················································································································163多元函数可微性的研究多元函数可微性的研究摘要：摘要： 本文针对多元函数可微性的充分条件和充要条件进行了研究。

第一，对 Henle 定理（二元函数可微的充分条件）的充分条件的证明进行了改进，并将其充分条件推广到 n 元，得出了从降低偏导连续的条件的多元函数可微的充分条件；第二，从多元函数可微的定义和方向导数的定义出发，并使用拼凑发得到了多元函数可微的的充要条件关键词：关键词：多元函数；可微；充分条件；充要条件；偏导数；连续Study of the Differenability of a Function Many VariblesAbstract:Abstract: Based on multivariate function differentiable sex sufficient conditions and sufficient condition is studied. First, for Henle theorem (dual function of differentiable sufficient conditions of sufficient conditions of proof), and improvements will be generalized to the sufficient condition is obtained, n-gram from reducing partial derivative continuous conditions of differentiable multiple function fully conditions; In the second place, from multiple function of differentiable definition and directional derivative definition, and use of multivariate function together hair gets the sufficient and necessary conditions of differentiable. KeyKey wordswords：： function of many variables; differentiable; sufficieny; necessary and sufficient conditions, partial derivative, continuity0引言引言众所周知，一元函数中，可微与可导是一回事，但在多元函数中情况就不同了，以二元函数为例，在现行的数学分析教材中给出了二元函数可微的充分条件和必要条件。

若函数在点 P处可微，则函数在点处连续，且在该点处yxf,00,xyyxf,00,xy存在但存在，且在点处连续仅是函数在点处,xyff,xyffyxf,00,xyyxf,00,xy可微的必要但非充分的条件例如：222222,0,00,xy xyxyf x yxy在点处存在，且连续，但在处不可微0 , 0,xyffyxf,yxf,0 , 0若函数的偏导数在点的某领域内存在，且在该点连续，则yxf,00,xy,xyff在点处可微但偏导数在点连续仅是函数在点yxf,00,xy,xyff00,xyyxf,处可微的充分但非必要条件例如：00,xy4222222221sin,0,00,xyxyxyf x yxy在点不连续，但在处可微这说明在点处连续作为,xyff0 , 0yxf,0 , 0,xyff00,xy函数在点可微的条件较严格在一般的教材中，对可微的充要条件也未yxf,00,xy涉及本文的目的在于探究函数在点处可微的较弱的充分条件和充要条yxf,00,xy件。

1 1预备知识预备知识1.1 多元函数全微分的定义函数在点全微分的定义为：),(yxfz yx,设函数在点的某一领域内有定义，若全增量yxfz,yx,,,zf xx yyf x y 可表示为 ， zA xB y    其中、不依赖于、，而仅与、有关，，且，ABxyxy22yx 0lim 0 则称函数在点可微分，而称为函数在点的全微分，记作yxfz,yx,yBxAyx,即dzyBxAdz1.21.2 函数在点沿方向 （ 为单位向量）可微的定义,f x y00(,)xygg如果存在有限极限：，则称100 00000(,)lim,,f xyfxygf xyg 为在沿方向的方向导数，且有：00(,)f xy g ,f x y00(,)xyg，其中00 000000(,),,((,), , )f xyfxygf xyo xygg00((,), , )0(0 )o xyg52 2元函数可微的充分条件元函数可微的充分条件HenleHenle 定理定理 如果函数在点处的偏导数存在，至少有一偏导数在点[6]yxf,00,xy的一个领域内存在，且在点处连续，则函数在点可微。

00,xy00,xyyxf,00,xy证明的主要依据是引理引理函数在点可微的充要条件是曲面在处有切[7]yxf,00,xy,zf x y00,xy平面下面我将对 Henle 定理的充分条件的证明进行改进，不必用上面的引理，并将其充分条件推广到 n 元2.12.1 HenleHenle 定理充分条件证明的改进定理定理 2.12.1 若在点处某个领域 U内偏导数存在，且其中有一个偏yxf,00,xy 0p导数在点处连续，则在点处可微00,xyyxf,00,xy证明： 不妨设在点处连续，函数在点处的全增量为：xf00,xyyxf,0p 000000000000,,,,,,f xx yyf xyf xx yyf xyyf xyyf xy 应用拉格朗日中值定理得：000000,,,, 01xf xx yyf xyyfxx yyx 由于在点处连续，所以有：xf00,xy，其中0000,,xxfxx yyxfxyxx    00lim0 xy   因为在点处存在，因而有：yf00,xy000000,,,yf xyyf xyfxyyy  所以0000,xyfxyxfxyyxy      这里当时，满足.由函数在点处可微的定义便知在022yx0, 0f0pf可微。

0p2.2定理 2. 1 的推广6例 1：函数2222222/,0( , , ) 0,0xyxyzxyuf x y z xy 在点的偏导数为(0,0,0)o0(0,0,0)(0,0,0)(0,0,0)lim0xxfxffx 0(0,0,0)(0,0,0)(0,0,0)lim0yyfyffy 0(0,0,0)(0,0,0)(0,0,0)lim0zzfzffz 所以，，，都存在，但，不连续，这是因(0,0,0)xf(0,0,0)yf(0,0,0)zf(0,0,0)xf(0,0,0)yf为当时，，当时， 220xy(0,0,0)0xf220xy222222222( , , )(/)()xfx y zxyxyzx yyxxyxyxy 当时，(0)ykx k2222(1)( , , ) (1) 1xkkkfx y z kk 与有关，所以在原点处的极限不存在，可见在原点k( , , )xfx y z(0,0,0)o。