5.2 二元函數(shù)的偏導(dǎo)數(shù)與全微分PPT課件

,5.2二元函數(shù)的偏導(dǎo)數(shù)與全微分,一、偏導(dǎo)數(shù)二、高階偏導(dǎo)數(shù)三、全微分四、全微分在近似計(jì)算中的應(yīng)用,1,5.2二元函數(shù)的偏導(dǎo)數(shù)與全微分,一、偏導(dǎo)數(shù),1、偏導(dǎo)數(shù)的定義,2,5.2二元函數(shù)的偏導(dǎo)數(shù)與全微分,3,5.2二元函數(shù)的偏導(dǎo)數(shù)與全微分,4,5.2二元函數(shù)的偏導(dǎo)數(shù)與全微分,偏導(dǎo)數(shù)的概念可以推廣到二元以上函數(shù),如函數(shù)在點(diǎn)處,5,5.2二元函數(shù)的偏導(dǎo)數(shù)與全微分,例1求,解法1,解法2,在點(diǎn)(1,2)處的偏導(dǎo)數(shù).,6,5.2二元函數(shù)的偏導(dǎo)數(shù)與全微分,例2設(shè),證,例3求,的偏導(dǎo)數(shù).,解,求證:,7,5.2二元函數(shù)的偏導(dǎo)數(shù)與全微分,偏導(dǎo)數(shù)記號(hào)是一個(gè),例4已知理想氣體的狀態(tài)方程,求證:,證,說明:,(R為常數(shù)),不能看作,分子與分母的商!,此例表明,整體記號(hào),8,5.2二元函數(shù)的偏導(dǎo)數(shù)與全微分,2.偏導(dǎo)數(shù)的幾何意義,如圖,9,5.2二元函數(shù)的偏導(dǎo)數(shù)與全微分,（1）幾何意義:,10,5.2二元函數(shù)的偏導(dǎo)數(shù)與全微分,（2）偏導(dǎo)數(shù)存在與連續(xù)的關(guān)系,？,但函數(shù)在該點(diǎn)處并不連續(xù).,偏導(dǎo)數(shù)存在連續(xù).,一元函數(shù)中在某點(diǎn)可導(dǎo)連續(xù)，,多元函數(shù)中在某點(diǎn)偏導(dǎo)數(shù)存在連續(xù)，,11,則稱它們是z=f(x,y),5.2二元函數(shù)的偏導(dǎo)數(shù)與全微分,二、高階偏導(dǎo)數(shù),設(shè)z=f(x,y)在域D內(nèi)存在連續(xù)的偏導(dǎo)數(shù),若這兩個(gè)偏導(dǎo)數(shù)仍存在偏導(dǎo)數(shù)，,的二階偏導(dǎo)數(shù).,按求導(dǎo)順序不同,有下列四個(gè)二階偏導(dǎo),數(shù):,12,5.2二元函數(shù)的偏導(dǎo)數(shù)與全微分,類似可以定義更高階的偏導(dǎo)數(shù).,例如，z=f(x,y)關(guān)于x的三階偏導(dǎo)數(shù)為,z=f(x,y)關(guān)于x的n1階偏導(dǎo)數(shù),再關(guān)于y的一階,偏導(dǎo)數(shù)為,第二、三個(gè)偏導(dǎo)數(shù)稱為混合偏導(dǎo)數(shù).,二階及二階以上的偏導(dǎo)數(shù)統(tǒng)稱為高階偏導(dǎo)數(shù).,13,5.2二元函數(shù)的偏導(dǎo)數(shù)與全微分,解,14,5.2二元函數(shù)的偏導(dǎo)數(shù)與全微分,例6求函數(shù),解,注意:此處,但這一結(jié)論并不總成立.,的二階偏導(dǎo)數(shù)及,15,5.2二元函數(shù)的偏導(dǎo)數(shù)與全微分,問題,例如,對(duì)三元函數(shù)u=f(x,y,z),當(dāng)三階混合偏導(dǎo)數(shù),在點(diǎn)(x,y,z)連續(xù)時(shí),有,16,5.2二元函數(shù)的偏導(dǎo)數(shù)與全微分,證,17,5.2二元函數(shù)的偏導(dǎo)數(shù)與全微分,例8證明函數(shù),滿足,證,利用對(duì)稱性,有,方程,18,5.2二元函數(shù)的偏導(dǎo)數(shù)與全微分,三、全微分,全增量,19,5.2二元函數(shù)的偏導(dǎo)數(shù)與全微分,定義2如果函數(shù)z=f(x,y)在點(diǎn)(x,y),可表示成,其中A,B不依賴于x,y,僅與x,y有關(guān)，,稱為函數(shù),在點(diǎn)(x,y)的全微分,記作,若函數(shù)在域D內(nèi)各點(diǎn)都可微,則稱函數(shù),f(x,y)在點(diǎn)(x,y)可微，,的全增量,則稱此函數(shù)在D內(nèi)可微.,20,5.2二元函數(shù)的偏導(dǎo)數(shù)與全微分,證,“可微”與“連續(xù)”的關(guān)系？,21,5.2二元函數(shù)的偏導(dǎo)數(shù)與全微分,“可微”與“偏導(dǎo)數(shù)存在”的關(guān)系？,22,5.2二元函數(shù)的偏導(dǎo)數(shù)與全微分,同樣可證,證由全增量公式,得到對(duì)x的偏增量,因此有,23,5.2二元函數(shù)的偏導(dǎo)數(shù)與全微分,反例:函數(shù),易知,但,注:定理3的逆定理不成立.,偏導(dǎo)數(shù)存在函數(shù)不一定可微!,因此,函數(shù)在點(diǎn)不可微.,24,5.2二元函數(shù)的偏導(dǎo)數(shù)與全微分,定理4(可微的充分條件),若函數(shù),的偏導(dǎo)數(shù),則函數(shù),在點(diǎn),連續(xù)，,在該點(diǎn)可微.且,全微分的定義可推廣到三元及三元以上函數(shù),.,例如,三元函數(shù),的全微分為：,25,5.2二元函數(shù)的偏導(dǎo)數(shù)與全微分,例9計(jì)算函數(shù),在點(diǎn)(2,1)處的全微分.,解,例10計(jì)算函數(shù),的全微分.,解,26,5.2二元函數(shù)的偏導(dǎo)數(shù)與全微分,可知當(dāng),*四、全微分在數(shù)值計(jì)算中的應(yīng)用,近似計(jì)算:,由全微分定義,較小時(shí),及,有近似等式:,(可用于近似計(jì)算;誤差分析),(可用于近似計(jì)算),27,5.2二元函數(shù)的偏導(dǎo)數(shù)與全微分,例11計(jì)算,的近似值.,解設(shè),則,取,則,28,5.2二元函數(shù)的偏導(dǎo)數(shù)與全微分,半徑由20cm增大,解已知,即受壓后圓柱體體積減少了,例12有一圓柱體受壓后發(fā)生形變,到20.05cm,則,高度由100cm減少到99cm,體積的近似改變量.,求此圓柱體,29,5.2二元函數(shù)的偏導(dǎo)數(shù)與全微分,偏導(dǎo)數(shù)的定義,偏導(dǎo)數(shù)的計(jì)算、偏導(dǎo)數(shù)的幾何意義,高階偏導(dǎo)數(shù),（偏增量比的極限）,純偏導(dǎo),混合偏導(dǎo),（相等的條件）,內(nèi)容小結(jié),30,5.2二元函數(shù)的偏導(dǎo)數(shù)與全微分,思考練習(xí),則（）,(A),(C),為,曲線在點(diǎn),的切向量,為,31,5.2二元函數(shù)的偏導(dǎo)數(shù)與全微分,思考練習(xí),(D),曲線在點(diǎn),的切向量,為,答案（C）,32,5.1多元