Математика. Жулева Л.Д - 7 стр.

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1.1. æÕÎËÃÉÉ ÎÅÓËÏÌØËÉÈ ÐÅÒÅÍÅÎÎÙÈ 7
þÁÓÔÎÙÅ ÐÒÏÉÚ×ÏÄÎÙÅ ×ÙÓÛÉÈ ÐÏÒÑÄËÏ×
þÁÓÔÎÙÍÉ ÐÒÏÉÚ×ÏÄÎÙÍÉ ×ÔÏÒÏÇÏ ÐÏÒÑÄËÁ ÎÁÚÙ×ÁÀÔÓÑ ÞÁÓÔÎÙÅ ÐÒÏÉÚ-
×ÏÄÎÙÅ ÏÔ ÞÁÓÔÎÙÈ ÐÒÏÉÚ×ÏÄÎÙÈ ÐÅÒ×ÏÇÏ ÐÏÒÑÄËÁ.
ïÂÏÚÎÁÞÅÎÉÑ:
(z
0
x
)
0
x
= z
00
xx
= f
00
xx
(x, y) =
∂
2
f
∂x
2
=
∂
2
z
∂x
2
,
(z
0
x
)
0
y
= z
00
xy
= f
00
xy
(x, y) =
∂
2
f
∂x∂y
=
∂
2
z
∂x∂y
,
(z
0
y
)
0
x
= z
00
yx
= f
00
yx
(x, y) =
∂
2
f
∂y∂x
=
∂
2
z
∂y∂x
,
(z
0
y
)
0
y
= z
00
yy
= f
00
yy
(x, y) =
∂
2
f
∂y
2
=
∂
2
z
∂y
2
.
òÁÓÐÏÌÏÖÅÎÉÅ ÉÎÄÅËÓÏ× x É y ÉÌÉ ÓÉÍ×ÏÌÏ× ∂x, ∂y ÓÏÏÔ×ÅÔÓÔ×ÕÅÔ ÐÏÒÑÄËÕ
ÄÉÆÆÅÒÅÎÃÉÒÏ×ÁÎÉÑ. ðÒÏÉÚ×ÏÄÎÙÅ ÔÒÅÔØÅÇÏ ÐÏÒÑÄËÁ ÏÂÏÚÎÁÞÁÀÔÓÑ ÔÁË:
(z
00
xx
)
0
x
= z
000
xxx
=
∂
3
f
∂x
3
,
(z
00
yx
)
0
y
= z
000
yxy
=
∂
3
f
∂y∂x∂y
É Ô.Ä.
ðÒÉÍÅÒ 3. îÁÊÔÉ ×ÔÏÒÙÅ ÞÁÓÔÎÙÅ ÐÒÏÉÚ×ÏÄÎÙÅ ÆÕÎËÃÉÉ z = sin(xy).
z
0
x
= y cos(xy),
z
0
y
= x cos(xy),
z
00
xx
= (y(cos(xy))
0
x
= −y
2
sin(xy),
z
00
xy
= (y cos(xy))
0
y
= cos(xy) − xy sin(xy),
z
00
yx
= (x cos(xy))
0
x
= cos(xy) − xy sin(xy),
z
00
yy
= (x cos(xy))
0
y
= −x
2
sin(xy).
âÒÏÓÁÅÔÓÑ × ÇÌÁÚÁ, ÞÔÏ z
00
xy
= z
00
yx
, Ô.Å. × ÓÍÅÛÁÎÎÙÈ ÐÒÏÉÚ×ÏÄÎÙÈ ÐÏÒÑÄÏË
ÄÉÆÆÅÒÅÎÃÉÒÏ×ÁÎÉÑ ÎÅ ÉÇÒÁÅÔ ÒÏÌÉ.
ðÒÉÍÅÒ 4. äÁÎÁ ÆÕÎËÃÉÑ z = e
x
(x cos y − y sin y).
äÏËÁÚÁÔØ, ÞÔÏ
∂
2
z
∂x
2
+
∂
2
z
∂y
2
= 0.
1.1. æÕÎËÃÉÉ ÎÅÓËÏÌØËÉÈ ÐÅÒÅÍÅÎÎÙÈ                                       7

   þÁÓÔÎÙÅ ÐÒÏÉÚ×ÏÄÎÙÅ ×ÙÓÛÉÈ ÐÏÒÑÄËÏ×
   þÁÓÔÎÙÍÉ ÐÒÏÉÚ×ÏÄÎÙÍÉ ×ÔÏÒÏÇÏ ÐÏÒÑÄËÁ ÎÁÚÙ×ÁÀÔÓÑ ÞÁÓÔÎÙÅ ÐÒÏÉÚ-
×ÏÄÎÙÅ ÏÔ ÞÁÓÔÎÙÈ ÐÒÏÉÚ×ÏÄÎÙÈ ÐÅÒ×ÏÇÏ ÐÏÒÑÄËÁ.
   ïÂÏÚÎÁÞÅÎÉÑ:

                                                ∂ 2f    ∂ 2z
                  (zx0 )0x =  00
                             zxx =  00
                                   fxx (x, y) =       = 2,
                                                ∂x2     ∂x
                                                   2
                                                 ∂ f       ∂ 2z
                  (zx0 )0y = zxy
                              00    00
                                 = fxy (x, y) =        =        ,
                                                ∂x∂y ∂x∂y
                     0 0      00    00           ∂ 2f      ∂ 2z
                  (zy )x = zyx = fyx (x, y) =          =        ,
                                                ∂y∂x ∂y∂x
                     0 0      00    00          ∂ 2f    ∂ 2z
                  (zy )y = zyy = fyy (x, y) = 2 = 2 .
                                                ∂y      ∂y

òÁÓÐÏÌÏÖÅÎÉÅ ÉÎÄÅËÓÏ× x É y ÉÌÉ ÓÉÍ×ÏÌÏ× ∂x, ∂y ÓÏÏÔ×ÅÔÓÔ×ÕÅÔ ÐÏÒÑÄËÕ
ÄÉÆÆÅÒÅÎÃÉÒÏ×ÁÎÉÑ. ðÒÏÉÚ×ÏÄÎÙÅ ÔÒÅÔØÅÇÏ ÐÏÒÑÄËÁ ÏÂÏÚÎÁÞÁÀÔÓÑ ÔÁË:

                               00 0     000     ∂ 3f
                             (zxx )x = zxxx =         ,
                                                ∂x3
                               00 0     000        ∂ 3f
                             (zyx )y = zyxy   =
                                                ∂y∂x∂y

É Ô.Ä.
   ðÒÉÍÅÒ 3. îÁÊÔÉ ×ÔÏÒÙÅ ÞÁÓÔÎÙÅ ÐÒÏÉÚ×ÏÄÎÙÅ ÆÕÎËÃÉÉ z = sin(xy).

                  zx0 = y cos(xy),
                  zy0 = x cos(xy),
                  00
                 zxx = (y(cos(xy))0x = −y 2 sin(xy),
                  00
                 zxy = (y cos(xy))0y = cos(xy) − xy sin(xy),
                  00
                 zyx = (x cos(xy))0x = cos(xy) − xy sin(xy),
                  00
                 zyy = (x cos(xy))0y = −x2 sin(xy).

                            00    00
  âÒÏÓÁÅÔÓÑ × ÇÌÁÚÁ, ÞÔÏ zxy   = zyx , Ô.Å. × ÓÍÅÛÁÎÎÙÈ ÐÒÏÉÚ×ÏÄÎÙÈ ÐÏÒÑÄÏË
ÄÉÆÆÅÒÅÎÃÉÒÏ×ÁÎÉÑ ÎÅ ÉÇÒÁÅÔ ÒÏÌÉ.
  ðÒÉÍÅÒ 4. äÁÎÁ ÆÕÎËÃÉÑ z = ex (x cos y − y sin y).
                ∂2z    ∂2z
  äÏËÁÚÁÔØ, ÞÔÏ ∂x 2 + ∂y 2 = 0.

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