高等数学复旦大学出版第三版下册课后答案习题全(陈策提mai供huan)

25. 设z?yf(x2?y2)，其中f(u)为可导函数，验证： 1?z1?zzx?x?y?y?y2. 证明：∵

?zyf??2x2xyf??x??f2??f2, ?zf?y?f??(?2?y?y)f2?f?2y2f?f2, ∴1?zx?x?1y??z?y??2yf?f?2y2f?1y1zf2?yf2?yf?f?y2?y2 22?2z?226. z?f(x?y),其中f具有二阶导数，求z?2?x2,?x?y,z?y2. 解：

?z?x?2xf?, ?z?y?2yf?, ?2z?x2?2f??2x?2xf???2f??4x2f??,?2z ?x?y?2xf???2y?4xyf??,由对称性知，?2z?y?2f??4y22f??.

27. 设f具有二阶偏导函数，求下列函数的二阶偏导数： (1)z?f??x??x,y??;

(2)z?f?xy2,x2y?;(3)z?f?sinx,cosy,ex?y?. 解：(1)

?z?x?f111??1?f2??y?f1??yf2?,

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1??2z11???21??????????f?f??f?f???f?f??2f22??,212211121112??2y??xyy?yy1?2z1x??x?1?x??1?f12?????2??2f2??f22?????2???2?f12???f22????2f2?,y?x?yyy??y?y?y??y,

?zx?x??f2???2???2f2?,?yy?y?2x?2z2x?x2xx???3f2?2f22?????2??3f2??4f22??.2?yyyy?y?y(2)

?z?f1??y2?f2??2xy?y2f1??2xyf2?, ?x?2z?y2f11???y2?f12???2xy?2yf2??2xyf21???y2?f22???2xy2?x

?2yf2??y4f11???4xy3f12???4x2y2f22??,?????2z?2yf1??y2f11???2xy?f12???x2?2xf2??2xyf21???2xy?f22???x2?x?y????

?2yf1??2xf2??2xy3f11???2x3yf22???5x2y2f12??,?z?f1??2xy?f2??x2?2xyf1??x2f2?,?y?2z?2xf1??2xyf11???2xy?f12???x2?x2f21???2xy?f22???x22?y?????2xf1??4x2y2f11???4x3yf12???x4f22??.(3)

?z?f1??cosx?f3??ex?y?cosxf1??ex?yf3?, ?x?z??sinxf1??cosxf11???cosx?f13???ex?y?ex?yf3??ex?yf31??cosx?f33???ex?y2?x?ex?yf3??sinxf1??cos2xf11???2ex?ycosxf13???e2(x?y)f33??,?????2zx?y??ex?yf??ex?y???????cosx?f?(?siny)?f?ef32???(?siny)?f33???ex?y?31213?????x?y?ex?yf3??cosxsinyf12???ex?ycosxf13???ex?ysinyf32???e2(x?y)f33??,?z?f2?(?siny)?f3?ex?y??sinyf2??ex?yf3?,?y?2z??siny?f??(?siny)?f???ex?y??ex?yf3??ex?y?f??(?siny)?f???ex?y???cosyf22333?22??32??y2?ex?yf3??cosyf2??sin2yf22???2ex?ysinyf23???e2(x?y)f33??.28. 试证：利用变量替换??x?1y,??x?y,可将方程 3159

?2u?2u?2u?4?32?0 ?x2?x?y?y?2u?0. 化简为

????证明：设u?f(?,?)?f?x???1?y,x?y? 3??u?u???u???u?u???????x???x???x?????2u?2u???2u???2u???2u???2u?2u?2u?2??????2??2?2?22?x???x?????x?????x???x?????????2u?2u?1??2u?2u??(?1)??????x?y??2?3??????????u?u?1??u1?u?u???????(?1)????y???3???3?????2u1?2u?1?1?2u?2u?????(?1)?22????y3???3?3?????????2u?2u?2u?4?32?x2?x?y?y?1?2u4?2u?2u?2u?2u?2u??1?2u2?2u?2u??2?2??4????2??3???2?22????????23??3??????9??3??????????4?2u???0.3?????2u?0. 故

????29. 求下列隐函数的导数或偏导数：

21?2u2?2u?2u?1??u?????2?(?1)???229??3???????3???21?2u4?2u?2u?1??u?????2?(?1)????22??3??3???????3?dy; dxydy22(2)lnx?y?arctan，求；

xdx(1)siny?e?xy?0，求

x2(3)x?2y?z?2xyz?0，求

?z?z,; ?x?y?z?2z,(4)z?3xyz?a，求. ?x?y233解：(1)[解法1] 用隐函数求导公式，设F(x,y)=siny+ex-xy2, 则 Fx?e?y ,Fy?coys?xy2 ,

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x2

dyFxex?y2y2?ex故dx??F???cosy?2xy. ycosy?2xy[解法2] 方程两边对x求导，得

cosy?y??ex??y2?x?2yy???0

故 y??y2?excosy?2xy. (2)设F(x,y)?lnx2?y2?arctanyx?12ln?x2?y2??arctanyx, ∵F12x1x?x?2x2?y2?1???y?2?????y?x2???yx2?y2, ?x??F12y1y?2x2?y2?y2?1x?y?xx2?y2, 1?????x??∴

dydx??FxF?x?yy. yx?(3)方程两边求全微分，得

dx?2dy?dz?2(yzdx?xzdy?xydz)2xyz?0,

xyz?xy2xyzxyzdz?yz?xyzxyzdx?xz?xyzdy,则 dz?yz?xyzdx?xz?2xyzxyz?xyxyz?xydy,

故

?zyz?xyz?x?xyz?xy, ?zxz?2xyz?y?xyz?xy. (4)设F(x,y,z)?z3?3xyz?a3,

Fx??3yz, Fy??3xz, Fz?3z2?3xy,

则

?z?x??Fx?3yzyzF??2xy?z2?xy, z3z?3?z?y??FyF???3xzxz3z2?3xy?z2?xy, z

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