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§3. þÁÓÔÎÙÅ ÐÒÏÉÚ×ÏÄÎÙÅ ×ÙÓÛÉÈ ÐÏÒÑÄËÏ× 3
òÅÛÅÎÉÅ.
z
0
x
= (x
2
+ y
2
)
0
x
e
xy
+ (x
2
+ y
2
)(e
xy
)
0
x
=
= 2xe
xy
+ (x
2
+ y
2
)e
xy
y = (2x + x
2
y + y
3
)e
xy
z
0
y
= (x
2
+ y
2
)
0
y
e
xy
+ (x
2
+ y
2
)(e
xy
)
0
y
=
= 2ye
xy
+ (x
2
+ y
2
)e
xy
x = (2y + x
3
+ xy
2
)e
xy
.
§3. þÁÓÔÎÙÅ ÐÒÏÉÚ×ÏÄÎÙÅ ×ÙÓÛÉÈ ÐÏÒÑÄËÏ×
ïÐÒÅÄÅÌÅÎÉÅ. þÁÓÔÎÙÍÉ ÐÒÏÉÚ×ÏÄÎÙÍÉ ×ÔÏÒÏÇÏ ÐÏÒÑÄËÁ ÎÁÚÙ×ÁÀÔÓÑ
ÞÁÓÔÎÙÅ ÐÒÏÉÚ×ÏÄÎÙÅ ÏÔ ÞÁÓÔÎÙÈ ÐÒÏÉÚ×ÏÄÎÙÈ. 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 ÓÉÍ×ÏÌÏ× x É y ÉÌÉ ∂x, ∂y ÓÏÏÔ×ÅÔÓÔ×ÕÅÔ ÐÏÒÑÄËÕ ÄÉÆÆÅÒÅÎ-
ÃÉÒÏ×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 1. îÁÊÔÉ ×ÓÅ ×ÔÏÒÙÅ ÞÁÓÔÎÙÅ ÐÒÏÉÚ×ÏÄÎÙÅ ÆÕÎËÃÉÉ z = sin(xy).
§3. þÁÓÔÎÙÅ ÐÒÏÉÚ×ÏÄÎÙÅ ×ÙÓÛÉÈ ÐÏÒÑÄËÏ× 3
òÅÛÅÎÉÅ.
zx0 = (x2 + y 2 )0x exy + (x2 + y 2 )(exy )0x =
= 2xexy + (x2 + y 2 )exy y = (2x + x2y + y 3 )exy
zy0 = (x2 + y 2 )0y exy + (x2 + y 2 )(exy )0y =
= 2yexy + (x2 + y 2 )exy x = (2y + x3 + xy 2 )exy .
§3. þÁÓÔÎÙÅ ÐÒÏÉÚ×ÏÄÎÙÅ ×ÙÓÛÉÈ ÐÏÒÑÄËÏ×
ïÐÒÅÄÅÌÅÎÉÅ. þÁÓÔÎÙÍÉ ÐÒÏÉÚ×ÏÄÎÙÍÉ ×ÔÏÒÏÇÏ ÐÏÒÑÄËÁ ÎÁÚÙ×ÁÀÔÓÑ
ÞÁÓÔÎÙÅ ÐÒÏÉÚ×ÏÄÎÙÅ ÏÔ ÞÁÓÔÎÙÈ ÐÒÏÉÚ×ÏÄÎÙÈ. ïÂÏÚÎÁÞÅÎÉÑ:
∂ 2f ∂ 2z
(zx0 )0x = 00
zxx = 00
fxx (x, y) = = 2
∂x2 ∂x
2
0 0 00 00 ∂ f ∂ 2z
(zx )y = zxy = fxy (x, y) = =
∂x∂y ∂x∂y
∂ 2f ∂ 2z
(zy0 )0x = zyx
00 00
= fyx (x, y) = =
∂y∂x ∂y∂x
∂ 2f ∂ 2z
(zy0 )0y = zyy00
= fyy00
(x, y) = 2 = 2 .
∂y ∂y
òÁÓÐÏÌÏÖÅÎÉÅ ÓÉÍ×ÏÌÏ× x É y ÉÌÉ ∂x, ∂y ÓÏÏÔ×ÅÔÓÔ×ÕÅÔ ÐÏÒÑÄËÕ ÄÉÆÆÅÒÅÎ-
ÃÉÒÏ×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 1. îÁÊÔÉ ×ÓÅ ×ÔÏÒÙÅ ÞÁÓÔÎÙÅ ÐÒÏÉÚ×ÏÄÎÙÅ ÆÕÎËÃÉÉ z = sin(xy).
