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1、综合练习一.填空题1.函数fxy,的偏导数fxy,和fxy,在区域D内连续,是fxy,在xyfD内可微的______条件;此时对于非零向量lab,,有______.lfab,afxbfy解.充分条件;gradfef,f.lxyla2b2a2b2y2.设A0,0,B1,0,C0,1,则dxxarctanxdy______,其中21xLL为三角形ABC的正向边界.QP1解.PdxQdydxdy1dxdy.xy2LDD33
2、2222xyxyxy3.设曲面:z,:z,:z,它们在xOy面上的12332222投影区域均为xy1,则它们的面积SSS,,的大小关系为______.1232244解.S1zzdxdy1xydxdy,1xy2222xy1xy12222S1zzdxdy1xydxdy,2xy2222xy1xy1222442S1zzdxdy1xyxydxdy,故SSS.3xy3122222xy1xy11n4.幂级数nx1的收敛区间是______.n1
3、n21n1an121n1解.limlimR2,即x123x1.nan12nnn2122225.设L:x1y216,则xy2x4yds______.L22解.x1y25ds11ds112488.LLL二.设fu的导函数连续,且f1f11,令zlnfxyxy,其中xydzyyx是方程exy0确定的隐函数,求.dxx0dzfxydydy解.1yx,而x0时y
4、1,又dxfxydxdxxyxydydydy1yedyeyx102,故xydxdxdx1xedxx0dzf112102.dxx0f1222x2y3z20dydz三.设,求,.22zxydxdxdydzdydz2x4y6z02y3zxdxdxdxdx解.,解得dzdydydz2x2y2y2xdxdxdxdxdyx2x3zx16zdzx2xx,.dx2y2y3z2y1
5、3zdx3z13z122222四.求IzxydV,其中:xyz1.21121z222解.I2dzzxydxdy2dzdzd;0x2y21z200062221xy211222I2dxdyzxydz2ddzd;x2y21000062xrsincos221222或者,yrsinsin,I2dsindrcosrsinrdr.6000zrcos2222五.设:z3
6、xyxy3,面密度为,求对z轴的转动惯量.222222解.IxydSxy1zzdxdyzxyx2y2323222xy13dxdyd2d9.x2y230022六.求Ix2xydydzy2yzdzdx12xydxdy,其中为22z4xy,取上侧.z0解.取:下侧,则I2x2zdv122xy411212xydxdy2zdv1dxdy
7、2dzzdxdy4x2y24x2y240x2y24z2222z4zdz4844.0n11七.设0,讨论级数11cos的收敛性.n1n211111解.1cos,故时,级数绝对收敛;2n2n2n211注意到1cos单调减少趋向于零,故0时,级数条件收敛.n2八A.设向下凸的曲线L经过0,1,且在该点有水平的切线,L上任意xy,处的曲率半径等于该点到x轴距离的平方,求L的方程.3y1解.设Ly:yx,则
8、,注意到曲线下凸,故y0,于是3222y1y3122dydpdxyy21y,令p,则y,代入