南昌大学第六届高等数学竞赛

?u??x1?x?y?1???1?xy????21?xy??x?y?y?1?xy?212=1?x,

?2u?2x221?x?x=

??2.

五、 计算曲线积分

I??Lxdy?ydx4x2?y2，其中L是以点（1，0）为中心，R为半径

的圆周R?0,R?1,取逆时针方向.

P?x,y???y4x2?y2Q?x,y??,

x4x2?y2, 当

?x,y???0,0?时,

?Py2?4x2?Q??2?y?x4x2?y2

??当0?R?1时?0,0??D,由格林公式知,I?0.

???x?cos?C:?2??y??sin?,?从0到2?. 当R?1时, ?0,0??D,作足够小的椭圆曲线

当??0充分小时,C取逆时针方向,使C?D,于是由格林公式得

?L?C?xdy?ydx?0224x?y，

xdy?ydxxdy?ydx??L4x2?y2?C4x2?y2因此

12?2d?=

?02??2

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=? 六、设函数f?x?在?0,???内具有连续的导数，且满足

f?t??2??x2?y2fD???x2?y2dxdy?t4，

?222x?y?tD其中是由所围成的闭区域，求当x??0,???时f?x?的表达式.

f?t??2?2?d??tr200f?r?rdr?t4 t =4??r3f?r?dr?t40, 两边对t求导得

f??t??4?t3f?t??4t3，且f?0??0，

这是一个一阶线性微分方程，解得

f?t??1?t4??e?1?.

???a??n???1?1?七、设n0xsinxdx，求级数n?1??anan?1??的和. n?令x?n??t, 则an??0?n??t?sintdt

n??n? =

0sintdt?an. a?n?

n?n2?0sintdt ?n2??2?tdt?n20sin?2??0sintdt?n2?.

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111??anan?1?1??1???nn?1??.

?11Sn?????aak?1k?1?kn?n1?11?1?1???1???????k?1? =??n?1?, ?=k?1??kS?limn??1?1?11??????n?1??

八、设f?x?在?0,???上连续且单调增加，试证：对任意正数a，b，恒有

?axf?x?dx?ba1bb?0f?x?dx?a?0f?x?dx. 2??令F?x??x则F??x??x0?f?t?dt，

0bx?f?t?dt?xf?x?,

?ba F?b??F??a??F?bx?f?t?dt?xf?x??dx ?x=d?x??a??0????xf?x??xf?x???dx a? =2于是

?baxf?x?dx,

?baa11?b?. xf?x?dx??Fb?Fa??bfxdx?afxdx?????????2??0??0??2?九、设??u,v?具有连续偏导数，由方程??x?az,y?bz?=0确定隐函数

z?z?x,y?，求a?z?z?b. ?x?y???z???z??????2??b??0, ?x???x? 两边对x求偏导得?1???1?a两边对y求偏导得?1????a????z??z?????1?b?2???0, ?y??y?? 7

?z?1??z?2?，, ???xa?1??b?2??xa?1??b?2?a?z?z?b=1. ?x?yxn?2n?11?12???1n，判别数列?xn?的敛散性. uk?xn十、设

?u?x?xx?0kkk?10定义，令，则k?1当n?2时，

n,

un?xn?xn?1?2n?1?2n?1n,

2n?n?1=?1n?n?1??nnn?n?1n??1?n?n?1?2.

un1lim?n??14nn,

由

?nn?1?1n可知n?1?u?n收敛，从而?xn?收敛.

2222x?y?z?R??R?0?上，问0?十一、设半径为r的球面的球心在球面：

当r为何值时，球面?在球面?0内部的那部分面积最大？

由对称性可设?的方程为

x2?y2??z?R??r22，球面?被球面?0所割部分的

222z?R?r?x?y方程为，

?z??xxr2?x2?y2,

?z??xyr2?x2?y2,

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