数学分析试题集锦（一）：16

? E Bcê? ?D?±s x1êêê??? ? ? ?DDD???±±±sss￥￥￥àààQQQ 1
p/ f ?￥ê? ?
 (1) u = x2 ln(x2 +y2)
(2) u = (x+y)cos(xy)
(3) u = arctan xy
(4) u = xy + xy
(5) u = xyesin(xy)
(6) u = xy +yx. 2
 ! f(x;y) = 8 < : ysin 1x2+y2; x2 +y2 6= 0; 0; x2 +y2 = 0: I3f ?(0,0)?￥ê? ?. 3
￡ üf ?u = px2 +y2(0,0)? ???ê? ??i. 4
p/ f ?￥?±s
 (1) u = px2 +y2 +z2
(2) u = xeyz +e?x +y. 5
p/ f ?ó??￥?±s
 (1) u = xpx2+y2?(1,0)?(0,1)
(2 ) u = ln(x+y2)?(0,1)?(1,1)
1 (3) u = qx y?(1,1,1)
(4) u = x+(y ?1)arcsin qx y?
0
1
. 6
 I3f ?f(x;y)(0,0)?￥ V±?
? f(x;y) = 8 < : xysin 1x2+y2; x2 +y2 6= 0; 0; x2 +y2 = 0: 7
￡ üf ? f(x;y) = 8 < : x2y x2+y2; x 2 +y2 6= 0; 0; x2 +y2 = 0: (0,0)? ??Oê? ?i
?N?? V±b 8
￡ üf ? f(x;y) = 8< : (x2 +y2)sin 1x2+y2; x2 +y2 6= 0; 0; x2 +y2 = 0: ￥ê? ?i
?ê? ?(0,0)?? ??
O(0,0)?￥?? #×?í ?
7fe?(0,0) V±b 9
 ! f(x;y) = 8 < : x2y2 x2+y2; x 2 +y2 6= 0; 0; x2 +y2 = 0: ￡ ü@f@x?@f@y(0,0)? ??. 10
 ! f(x;y) = 8< : 1?ex(x2+y2) x2+y2 ; x 2 +y2 6= 0; 0; x2 +y2 = 0: ￡ üf(x;y)(0,0)? V±
ipdf(0;0). 11
 ! f(x;y) = 8 < : x3 x2+y2; x 2 +y2 6= 0; 0; x2 +y2 = 0: 2 (1) x = x(t);y = y(t) ^YVe?￥?i V±wL
'x2(0) + y2(0) = 0;t 6= 0 H
x2(t)+y2(t) 6= 0;x(t)ay(t) V±
.p￡f(x(t);y(t)) V±. (2) f(x;y)(0,0)? V±. 12
 !jxj;jyj?l
 ?¨?±sw/ ò T￥í ? T
 (1) (1+x)m(1+y)n; (2) arctan x+y1+xy. 13
 !u = f(x;y) ?
a < x < b;c < y < d= V±
O?±sdu? 1 ,
ùf(x;y)? ?= ^??|è ?′
$￡ üF￥2 ?. 14
 !@f@x(x0;y0)i
@f@y(x0;y0) ??
p￡f(x;y)(x0;y0) V±. 15
p/ f ?￥ ?μ=¨ê? ?
 (1) u = lnpx2 +y2
(2) u = xy + yx
(3) u = xsin(x+y)+ycos(x+y)
(3) u = exy. 16
p/ f ?·?¨￥ê? ?
 (1) u = x3 siny +y3 sinx,p@6u@x3@y3
(2) u = arctan x+y1?xy
p ?μ?¨ê? ?
(3) u = sin(x2 +y2),p@3u@x3
@3u@y3
(4) u = xyzex+y+z,p@p+q+ru@xp@yq@zr
(5) u = x+yx?y (x 6= y)
p@m+nu@xm@yn
3 (6) u = ln(ax+by),p@m+nu@xm@yn. 17
￡/ f ? ?@ @2u @x2 + @2u @y2 = 0: (1) u = ln(x2 +y2)
(2) u = x2 ?y2
(3) u = ex cosy
(4) u = arctan yx. 18
 !f ?u = ’(x+?(y))
￡ ü @u @x @2u @x@y = @u @y @2u @x2: 19
 !fx;fy?(x0;y0)￥ #×=iO?(x0;y0) V±
5μ fxy(x0;y0) = fyx(x0;y0): x2ˉˉˉ???fff ? ? ?DDD???fff ? ? ?±±±sssEEE 1
p/ f ?￥ ?μ=¨ê? ?
 (1) u = f(ax;by)
(2) u = f(x+y;x?y)
(3) u = f(xy2;x2y)
(4) u = f(xy; yz)
(5) u = f(x2 +y2 +z2)
4 (6) u = f(x+y;xy; xy). 2
 !z = yf(x2?y2)
?f ^ V±f ?
￡ 1 x @z @x + 1 y @z @y = z y2: 3
 !v = 1rg(t ? rc)
c1è ?
f ?g=¨ V?
r = px2 +y2 +z2b ￡ ü@2v@x2 + @2v@y2 + @2v@z2 = 1c2 @2v@t2 . 4
?f ?f(x;y;z)?i? L ?t ?@1" f(tx;ty;tz) = tnf(x;y;z) 5?f(x;y;z)1nQ Qf ?. !f(x;y;z) V±
 k￡ üf(x;y;z)1nQ Q f ?￥ 1Hq ^ x@f@x +y@f@y +z@f@z = nf(x;y;z): 5
￡/ ò T
 (1) u = ’(x2 +y2),5y@u@x ?x@u@y = 0
(2) u = y’(x2 ?y2),5y@u@x +x@u@y = xuy
(3) u = x’(x+y)+y?(x+y),5@2u@x2 ?2 @2u@x@y + @2u@y2 = 0
(4) u = x’(yx)+?(yx),5x2@2u@x2 +2xy @2u@x@y +y2@2u@y2 = 0. 6
 !u = f(x;y) V±
USMDx = rcos
y = rsin /
￡ ü (@z@x)2 +(@z@y)2 = (@z@u)2 +(@z@v)2: ? H?(@z@x)2 +(@z@y)2 ^B?? T?M . 8
 !f ?u = f(x;y) ?@ ?? ? ?Z? @2u @x2 + @2u @y2 = 0 5 ￡ ü/ MD/???M
'ˉμ@2u@s2 + @2u@t2 = 0. (1) x = ss2+t2,y = ts2+t2
(2) x = es cost;y = es sint
(3) x = ’(s;t);y = ?(s;t) ?@@’@s = @?@t ; @’@t = ?@?@s.?FZ??1 O
 ó ￡Z?. 9
T1M ￥MD
|?;·;?1?1M 
 (1) ? = x;· = x2 +y2,MDZ?y@z@x ?x@z@y = 0
(2) ? = x;· = y ?x;? = z ?x,MDZ?@u@x + @u@y + @u@z = 0. 10
T1M ?yM ￥MD
|u;v1?￥1M 
w = w(u;v)1 ?￥yM 
 (1) !u = x+y;v = yx;w = zx
MDZ? @2z @x2 ?2 @2z @x@y + @2z @y2 = 0 (2) !u = xy;v = x;w = xz ?y
MDZ? y@ 2z @y2 +2 @z @y = 2 x: 11
p/ Z? ???￥f ?z = f(x;y)￥B¨?=¨ê? ?
 (1) e?xy ?2z +ex = 0
(2) x+y +z = ex+y+z
(3) xyz = x+y +z
(4) x2 +y2 +z2 ?2x+2y ?4z ?5 = 0. 12
p?/ Z? ???￥f ?￥?±sdz
6 (1) z = f(xz;z ?y)
(2) F(x?y;y ?z;z ?x) = 0
(3) f(x+y +z;x2 +y2 +z2) = 0
(4) f(x;y)+g(y;z) = 0. 13
 !z = z(x;y)?Z?x2 + y2 + z2 = yf(zy) ???
￡ ü(x2 ? y2 ? z2)@z@x +2xy@z@y = 2xzb 14
 !z = x2 + y2
?y = f(x)1?Z?x2 ?xy + y2 = 1 ???￥? f ?
pdzdx?d2zdx2. 15
 !u = x2 +y2 +z2
?z = f(x;y)1?Z?x3 +y3 +z3 = 3xyz ? ??￥?f ?
p@u@x
@2u@x2. 16
p/ Z?F ???￥f ?￥? ??ê? ?
 (1
 8 < : x2 +y2 +z = a2; x2 +y2 = ax; pdydx; dzdx
(2) 8 < : x2 ?u2 ?yv = 0; y ?v2 ?xu = 0; p@u@x; @v@x; @u@y; @v@y
(3) 8 < : u2 ?v = 3x+y; u?2v2 = x?2y; p@u@x; @u@y; @v@x; @v@y
(4) 8 < : u = xyz; x2 +y2 +z2 = 1; p@2u@x2; @2u@y2; @2u@x@y. 17
/ Z?F?lz1x;y￥f ?
p@z@x
@z@y. 7 (1) 8 >>> < >>> : x = cos cos’; y = cos sin’ z = sin ; ; (2) 8 >>> < >>> : x = u+v; y = u2 +v2 z = u3 +v3: ; x3+++??????¨¨¨ 1
p/ wL ? U?)￥MLZ??Eü ?Z? (1) x = asin2 t;y = bsintcost;z = ccos2 t,?t = …4
(2) 2x2 +3y2 +z2 = 9;z2 = 3x2 +y2,?(1,-1,2)
(3) x2 +y2 +z2 = 6;x+y +z = 0,?(1,-2,1)
(4) x = t?cost;y = 3+sin2 t;z = 1+cos3t,?t = …2. 2
p/ w ? ? U?)￥Mü ?Z??ELZ?
 (1) y ?e2x?z = 0
?
1,1,2

(2) x2a2 + y2b2 + z2c2 = 1?( ap3; bp3; cp3)
(3) z = 2x2 +4y2?(2,1,12)
(4) x = ucosv;y = usinv;z = av?P0(u0;v0). 3
￡ üwLx = aet cost;y = aet sint;z = aet ?x2 + y2 = z2￥L M??]B?. 4
pü ?wLx2=3 + y2=3 = a2=3(a > 0) ?B?￥MLZ?
i￡ ü ?tML$USà ??|￥L ?é. 5
pw ?x2+2y2+3z2 = 21￥Mü ?
 P ?ü??ü ?x+4y+6z = 0. 6
￡ ü
w ?F(x?az;y ?bz) = 0￥Mü ?DB?°Lü?
 ?a;b1è ?. 8 7
￡ üw ?z = xexy￥ ?BMü ??YVe?. 8
p w ? F(x;y;z) = 0;G(x;y;z) = 0 ￥?LOxyü ? ￥g?wL￥MLZ?. x4ZZZ___??? ? ? ? 1
 !f(x;y;z) = x+y2 +z3
pf?P0(1;1;1)
??l = (2;?2;1)￥Z _? ?. 2
pf ?u = xyz?A(5;1;2))
??B(9;4;14)￥Z_?!AB ￥Z_ ? ?. 3
p@u@lflfl (x0;y0) ; (1) u = ln(x2 +y2)
(x0;y0) = (1;1)
lDxà?_￥C?160–; (2) u = xexy,(x0;y0) = (1;1), lD_ (1;1)]_. 4
 !f ?f(x;y)(x0;y0) V±
?ê_ l1 = ( 1p2; 1p2)
l2 = (? 1p2; 1p2)
@f(x0;y0)@l1 = 1
@f(x0;y0)@l2 = 0
??l P¤ @f(x0;y0) @l = 7 5p2: 5
 !fP0(2;0) V±
f(x;y)P0·_P1 = (2;?2)￥Z_? ? ^1
 ·_e?￥Z_? ? ^
3
 kís
 (1)·_P2 = (2;1)￥Z_? ? ^ 
$ (2)·_P3 = (3;2)￥Z_? ? ^ 
$ x5 ü ü ü à à à T T T 9 1
/ f ?·??￥ ü à T
 (1)f(x;y) = 2x2 ?xy ?y2 ?6x?3y +5
(1,-2)?. (2) f(x;y) = x2 +xy +y2 +3x?2y +4
(-1,1)?. 2
pf ?f(x;y) = xy(1,1)? #×￥n¨{ ?ì μ°?[￥ ü à T. 3
pf ?f(x;y) = y2x2(1,-1)? #×￥=¨ ü à T
i ?ì μ °?[. 4
p/ f ?(0;0)? #×￥ 1¨ ü à T
 (1) f(x;y) = sin(x2 +y2)
(2) f(x;y) = ex ln(1+y)
(3) f(x;y) = p1+x2 +y2; (4) f(x;y) = ex cosyb 5
￡ ü ü à T￥·B?
?nP i+j=0 Aijxiyi +o(‰n) = 0 (‰ ! 0)
 ?‰ = px2 +y2.p￡Aij = 0
i;j1dμ? ?
i+j = 0;1;...;n

i ?¨ ·B?pf(x;y) = ln(1+x+y){ ?ì μ°?[￥n¨ ü àZ 7 T. 6
YVf(x;y) = sinxcosy¨?′? ?
￡ üi 2 (0;1)
 P 3 4 = … 3 cos … 3 cos … 6 ? … 6 sin … 3 sin … 6 : 7
 !f(x;y)u×D=μê? ?i
Ofx(x;y) = fy(x;y) · 0.￡ üf(x;y)D1è ?. 8
?jxj;jyj ^?l￥ 
?/ f ???=Q[￥í ? T
 (1) cosxcosy
(2) arctan 1+x+y1?xy . 10 9
 !f ?f(x;y)μ°?n¨ ??ê? ?
 k￡u(t) = f(a + ht;b + kt)￥n¨? ? u(n)(t) = (h @@x +k @@y)nf(a+kt;b+kt): 10
 !f(x;y)1nQ Qf ?
￡ ü (x @@x +k @@y)mf = n(n?1):::(n?m+1)f: 11
 !f(x;y) = ?(ax + by)
?a;b1è ?
ce?￥ #× =
?μq¨ ??? ?.p￡
(0,0)? #×￥ ü à T ^ f(x;y) = q?1X k=0 ?(k)(0) k! kX j=0 Cjk(ax)j(by)k?j +Rq(x;y): 11