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1、有定義有極限極限等于該點(diǎn)的函數(shù)值例7討論函數(shù)?xy22?22,x?y?0f(x,y)??x?y?22?0,x?y?0在(0,0)的連續(xù)性．解取y?kxxy2kkxlim22?lim?2x?0x?yx?0x2?k2x21?ky?0y?kx其值隨k的不同而變化，極限不存在．故函數(shù)在(0,0)處不連續(xù)．§2.偏導(dǎo)數(shù)?yf(x??x)?f(x)00導(dǎo)數(shù)定義f?(x0)?lim?lim?x?0?x?x?0?x一、偏導(dǎo)數(shù)的定義及其計(jì)算定義設(shè)函數(shù)z?f(x,y)在點(diǎn)(x0,y0)的某一鄰域內(nèi)有定義，當(dāng)y固定在y0而x在x0處有增量?x時(shí)，相應(yīng)地函數(shù)有增量f(,xx00??y)?fxy(,)00f(
2、,xx???yf)(x,y)如果lim0000存在，??x0?x則稱此極限為函數(shù)z?f(x,y)在點(diǎn)(x0,y0)處對(duì)x的偏導(dǎo)數(shù)，記為?z?f，zx?x0?xx?x0xy?y，?xx?x0或fx(x0,y0)0y?yy?y00一、偏導(dǎo)數(shù)的定義及其計(jì)算z?f(x,y)在點(diǎn)(x0,y0)處對(duì)x的偏導(dǎo)數(shù)，?zfx(,000??xy)?fxy(,0)，lim???x0?x?xx?x0y?y0同理可定義函數(shù)z?f(x,y)在點(diǎn)(x0,y0)處對(duì)y的偏導(dǎo)數(shù)：fxy(,??y)?fxy(,)lim0000??y0?y?z?f記為：,zyx?x0,或fy(x0,y0)?yx?x0y?y0?yx?x0
3、y?y0y?y0一、偏導(dǎo)數(shù)的定義及其計(jì)算z?f(x,y)在點(diǎn)(x0,y0)處對(duì)x的偏導(dǎo)數(shù)，?zfx(,000??xy)?fxy(,0)，lim?x?x??x0?x?x0y?y0如果函數(shù)z?f(x,y)在區(qū)域D內(nèi)任一點(diǎn)(x,y)偏導(dǎo)函數(shù):對(duì)x的偏導(dǎo)數(shù)都存在，那么這個(gè)偏導(dǎo)數(shù)就是x、y的函數(shù)，它就稱為函數(shù)z?f(x,y)對(duì)x的偏導(dǎo)函數(shù)，?z?f記作z，x或fx(,),xy?x?x一、偏導(dǎo)數(shù)的定義及其計(jì)算z?f(x,y)在點(diǎn)(x0,y0)處對(duì)y的偏導(dǎo)數(shù)，f(x,y??y)?f(x,y)?zlim0000??y?0?y?yxx?0yy?0偏導(dǎo)函數(shù):z?f(x,y)對(duì)自變量x的偏導(dǎo)函數(shù)，?z?f
4、記作，zx或fx(x,y)，?x?x同理可以定義函數(shù)z?f(x,y)對(duì)自變量y的偏導(dǎo)函數(shù)，記作：?z?fz，，y或fy(x,y)偏導(dǎo)函數(shù)簡稱為偏導(dǎo)數(shù).?y?y一、偏導(dǎo)數(shù)的定義及其計(jì)算fx(,???xy)fxy(,)0000fxy(,)?limx00??x0?x偏導(dǎo)數(shù)的概念可以推廣到二元以上函數(shù)如u?f(x,y,z)在(x,y,z)處f(x??x,y,z)?f(x,y,z)f(x,y,z)?lim,x?x?0?xf(x,y??y,z)?f(x,y,z)f(x,y,z)?lim,y?y?0?yf(x,y,z??z)?f(x,y,z)f(x,y,z)?lim.z?z?0?z一、偏導(dǎo)數(shù)的定義
5、及其計(jì)算22例1求z?x?3xy?y在點(diǎn)(1,2)處的偏導(dǎo)數(shù)．?z?z解?2x?3y;?3x?2y.?x?y?z?x?1?2?1?3?2?8,?xy?2?z?3?1?2?2?7.x?1?yy?2一、偏導(dǎo)數(shù)的定義及其計(jì)算y例2設(shè)z?x(x?0,x?1)，x?z1?z求證??2z.y?xlnx?y?zy?1?zy證?yx,?xlnx,?x?yx?z1?zxy?11y??yx?xlnxy?xlnx?yylnxyy?x?x?2z.原結(jié)論成立．一、偏導(dǎo)數(shù)的定義及其計(jì)算有關(guān)偏導(dǎo)數(shù)的幾點(diǎn)說明：?udy１、偏導(dǎo)數(shù)是一個(gè)整體記號(hào)，不能拆分;可拆?xdx２、求分段點(diǎn)、不連續(xù)點(diǎn)處的偏導(dǎo)數(shù)要用定義求；例如,
6、zfxy??(,)xyf,求(0,0),f(0,0).xyfx(,???xy)fxy(,)解0000fxy(,)?limx00??x0?x
7、(0????x)0
8、0fx(0,0)?lim?0?fy(0,0).??x0?x一、偏導(dǎo)數(shù)的定義及其計(jì)算?xy?(,)(0,0)xy?例322fxy(,)??xy???0(xy,)?(0,0)求f(,)xy的偏導(dǎo)數(shù)fx(,???xy)fxy(,)0000fxy(,)?limx00解??x0?x當(dāng)(x,y)?(0,0)時(shí),fxy(,???yfxy)(,)0000fxy(,)?limy00??y0?yfxf(0??,0)?(0,0)0f(0,0)?li
9、m?lim?0,x??x0?x??x0?xfy(0,0???)f(0,0)0f(0,0)??lim?0,ylim??y0?y??y0?y例3?xy?(,)(0,0)xy?22fxy(,)??xy?f(0,0)?0x??0(xy,)?(0,0)f(0,0)?0y求f(,)xy的偏導(dǎo)數(shù)解(,)(0,0)xy?時(shí)2222y()xy??xy?2xy(y?x)fx(x,y)??222,()xy22?2(x?y)22x(x2?y2)x(x?y)?2y?xyfy(x,y
