4xarcsin431?x?y422x?y?z?y??f?????y??xln(x?y)(1?x?y)(x?y)222244
4y434?f?????y2=
?1??2?y1?x?y422?arcsin??x?y=
4yarcsin431?x?y42x?y?yln(x?y)(1?x?y)(x?y)22224.
3. (1)
?u?x?u?x?u?x?u?x?z?x=2xf1?yexyf2,
?u?y=?2yf1?xexyf2.
(2) =
1y?f1,
?u?y=?xy2?f1?1zf2,
?u?z=?yz2?f2.
(3)
=f1?yf2?yzf3,
?u?y?u?y=xf2?xzf3,
?u?z=xyf3.
(4)
=2xf1?yf2?f3=2yf1?xf2?f3,
?u?z=f3.
4 .(1)
?yf1,
?z?y?xf1?f2,
?z?x22?y?f1?x??y?y?f11?y??yf11,
2?z?x?y?z?y?z?x222??yf1??f1?y?f1?y?f1?y(f11?x?f12)?f1?xyf11?yf12,
?f2?y2???y?xf1?f2??x?f1?y??x(f11?x?f12)?f21?x?f22?xf11?2xf12?f222(2)
?yf1?2xyf2,
2?z?y?2xyf1?xf2,
?z?x22???x2?y2f1?2xyf22??y2?f1?x?2yf2?2xy?f2?x2
?y(f11?y?f12?2xy)?2yf2?2xy(f21?y?f22?2xy)?2yf2?yf11?4xyf12?4xyf22?z?x?y
24322.
???y?y2f1?2xyf2??2yf1?y2?f1?y?2xf2?2xy6
?f2?y
?2yf1?y(f11?2xy?f12?x)?2xf2?2xy(f21?2xy?f22?x)?2yf1?2xf2?2xyf11?2xyf22?5xyf12?z?y222223322
???y?2xyf1?xf2?2xf1?2xy22??f1?y2?x2?f2?y
2?2xf1?2xy(f11?2xy?f12?x)?x(f21?2xy?f22?x)?2xf1?4xyf11?4xyf12?xf222234
5 ??u?s??u?x?x?s??u?y?y?s?1?u2?x?3?u?u?u?x?u?y3?u1?u, ,?????2?y?t?x?t?y?t2?x2?y(?u?s?u?s)2?1?u23?u?u3?u2?u23?u23?u?u1?u2()??(),()?()??(), 4?x2?x?y4?y?t4?x2?x?y4?y?u?t)2?()?(2?(?u?x)?(2?u?y).
2?(x?y?z)6 (1) 设F(x,y,z)?x?y?z?e?(x?y?z), Fx?1?e,Fy?1?e?(x?y?z),
Fz?1?e?(x?y?z),
?z?yFyFz2?z?x??FxFz??1,????1
(2)设F(x,y,z)?z?xx?y22x?yz2tanzx?y22,zx?y22Fx??tanx?y22?x?y22sec2(?12)(x?y)22?32 2xz=?xx?y22tanzx?yzx?y2222?xzx?y2222sec2zx2?y2,
32 Fy?yx?y22tan?x?ysec22zx?y2(?12)(x?y)22?(?2yz)
=
yx?y22tanzx?y22?yzx?yz22sec2zx2?y2,
Fz?1?x2?y2sec212x2?yx?y22=?tan22zx?y2,
7
?z?x??FxFzFyFz??xx?y22cotzx?yzx?y2222?xzx?yyz22csc22zx?y2,
?z?y???yx?y22cot?x?y22csc22zx?y2.
(3) 设F(x,y,z)?x?2y?z?2xyz,Fx?1??z?xyzx Fy?2?xzyFx?1?xyz,
??FxFz=
yz?xyzxyz?xy?yxz?lnzy?,
?z??FyFz=
xz?2xyzxyz?xy.
(4) 设F(x,y,z)?xz?lnz?lny,Fx?21z,Fy?1yFz??xz2?1z,
?z?x??FxFz?zx?z?y,
?z??FyFz?zy(x?z),
7.设F(x,y,z)?x?2y?3z?2sin(x?2y?3z),Fx?1?2cos(x?2y?3z),
?Fy?2?4cos(x?2y?3z),Fz??3?6cos(x?2y?3z),
?z?x?z?x? ??FxFz?Fy1?z2?, ,??Fz33?y? ??z?y?1.
8.设F(x,y,z)??(cx?az,cy?bz),Fx?c?1,Fy?c?2,Fz??a?1?b?2,
?z?x??FxFz?c?1a?1?b?2?b,
?z?y??FyFz?c?2a?1?b?2,
? a?z?x?z?y?c.
9. (1)方程两边同时对x求导得
x(6z?1)dy?dy?dz??,?2x?2y,??dx?dx2y(3z?1)dx解之得? ?dydzdyx?2x?4y??6z?0,??dxdx?3z?1?dx(2) 方程两边同时对z求导得
8
?dxdy?dz?dz?1?0, ?解之得
dydx?2x?2y?2z?0dzdz?y?z?dx?,??dzx?y ?dyz?x??.?dzx?y? (3) 方程两边同时对x求偏导得 ?u1?e? ??0?eu??usinv??u?,?sinv?ucosv,u??xe(sinv?cosv)?1?x?x?x解之得? ?u?u?u?v?vcosv?e??cosv?usinv,?.u?x?x?x?u[e(sinv?cosv)?1]??x?u?v同理方程两边同时对y求偏导得
?cosv??u?u?v?u?u?,0?e?sinv?ucosv,u??x?e(sinv?cosv)?1???y?y?y ?解之得? u?u?u?v?vsinv?e??1?eu?cosv?usinv,?.u??y?y?y?u[e(sinv?cosv)?1]???x
习题1－4
1． 求下列函数的方向导数
?u?lPo
22（1）u?x?2y?3z,P0?1,1,0?,l??1,?1,2?
解：
?u?xP0?2xP0?2
?u?y?u?zP0P0?4yP0?4
?6zP0?0
l0?(16,?16,26)
??u?lP0?2*y16z?4*(?16)??26.
（2）u?(),P0(1,1,1),l?(2,1,?1);
x解：
?u?x?u?yP0P0yz?1y?z()(?2)xxyz?11?z()()xxP0??1,
P0?1,
9
?u?zP0yzy?()ln()xxP0?0,
l0?(26,16,?1626)
??u?lP0?(?1)*?1*16??16.
（3）u?ln(x2?y2),P0(1,1),l与ox轴夹角为解：
?u?x?u?yP0P0?3;
?2xx?y2yx?y22P022P0?1,
??1,
由题意知???3,则???6,
??1 l0?(cos,cos)?(,36232)
??u?lP0?1*12?1*32?1?23.
（4）u?xyz,P0(5,1,2),P1(9,4,14),l?P0P1.
?u?xP0?yzP0?2,
?u?y?u?zP0P0?xzP0?10,
?xyP0?5,
0l?(4,3,12),?l??u?lP0?(313312,),
131313,?5*1213?9813.
4?2*413?10*
2． 求下列函数的梯度gradf
（1）f(x,y)?sin(xy)?(cos(xy);
22 10
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