Phép tính vi phân hàm nhiều biến

Chu
.
o
.
ng 9
Ph´ep t´ınh vi phˆan h`am
nhiˆe
`
ubiˆe
´
n
9.1 D
-
a
.
oh`amriˆeng .................110
9.1.1 D
-
a
.
o h`am riˆeng cˆa
´
p1.............110
9.1.2 D
-
a
.
o h`am cu
’
a h`am ho
.

.
p............111
9.1.3 H`am kha
’
vi..................111
9.1.4 D
-
a
.
o h`am theo hu
.
´o
.
ng.............112
9.1.5 D
-
a
.
o h`am riˆeng cˆa
´
pcao............113
9.2 Vi phˆan cu
’
a h`am nhiˆe
`
ubiˆe
´
n.........125
9.2.1 Vi phˆan cˆa
´

p1.................126
9.2.2
´
Ap du
.
ng vi phˆan d
ˆe
’
t´ınh gˆa
`
nd´ung . . . . . 126
9.2.3 C´ac t´ınh chˆa
´
tcu
’
aviphˆan..........127
9.2.4 Vi phˆan cˆa
´
pcao ...............127
9.2.5 Cˆong th´u
.
cTaylor...............129
9.2.6 Vi phˆan cu
’
ah`amˆa
’
n .............130
9.3 Cu
.
.

c tri
.
cu
’
a h`am nhiˆe
`
ubiˆe
´
n .........145
110 Chu
.
o
.
ng 9. Ph´ep t´ınh vi phˆan h`am nhiˆe
`
ubiˆe
´
n
9.3.1 Cu
.
.
c tri
.
....................145
9.3.2 Cu
.
.
c tri
.
c´o d

iˆe
`
ukiˆe
.
n .............146
9.3.3 Gi´a tri
.
l´o
.
n nhˆa
´
t v`a b´e nhˆa
´
tcu
’
a h`am . . . . 147
9.1 D
-
a
.
oh`am riˆeng
9.1.1 D
-
a
.
o h`am riˆeng cˆa
´
p1
Gia
’

su
.
’
w = f(M), M =(x, y) x´ac d
i
.
nh trong lˆan cˆa
.
n n`ao d´o c u
’
adiˆe
’
m
M(x, y). Ta
.
id
iˆe
’
m M ta cho biˆe
´
n x sˆo
´
gia t`uy´y∆x trong khi vˆa
˜
ngi˜u
.
gi´a tri
.
cu
’

abiˆe
´
n y khˆong d
ˆo
’
i. Khi d´o h`am f(x, y) nhˆa
.
nsˆo
´
gia tu
.
o
.
ng
´u
.
ng l`a
∆
x
w = f(x +∆x, y)− f(x, y)
go
.
il`asˆo
´
gia riˆeng cu
’
a h`am f(x, y) theo biˆe
´
n x ta
.

id
iˆe
’
m M(x, y).
Tu
.
o
.
ng tu
.
.
d
a
.
ilu
.
o
.
.
ng
∆
y
w = f(x, y +∆y) − f(x, y)
go
.
il`asˆo
´
gia riˆeng cu
’
a h`am f(x, y) theo biˆe

´
n y ta
.
id
iˆe
’
m M(x, y).
D
-
i
.
nh ngh˜ıa 9.1.1
1. Nˆe
´
utˆo
`
nta
.
i gi´o
.
iha
.
nh˜u
.
uha
.
n
lim
∆x→0
∆

x
w
∆x
= lim
∆x→0
f(x +∆x, y)− f(x, y)
∆x
th`ı gi´o
.
iha
.
nd
´odu
.
o
.
.
cgo
.
il`ad
a
.
o h`am riˆeng cu
’
a h`am f(x, y) theo biˆe
´
n
x ta
.
id

iˆe
’
m(x, y)v`adu
.
o
.
.
cchı
’
bo
.
’
imˆo
.
t trong c´ac k´yhiˆe
.
u
∂w
∂x
,
∂f(x, y)
∂x
,f

x
(x, y),w

x
.
9.1. D

-
a
.
o h`am riˆeng 111
2. Tu
.
o
.
ng tu
.
.
:nˆe
´
utˆo
`
nta
.
i gi´o
.
iha
.
n
lim
∆y→0
∆
y
w
∆y
= lim
∆y→0

f(x, y +∆y)− f(x, y)
∆y
th`ı gi´o
.
iha
.
nd
´odu
.
o
.
.
cgo
.
il`ad
a
.
o h`am riˆeng cu
’
a h`am f(x, y) theo biˆe
´
n
y ta
.
id
iˆe
’
m M(x, y)v`adu
.
o

.
.
cchı
’
bo
.
’
imˆo
.
t trong c´ac k´yhiˆe
.
u
∂w
∂y
,
∂f(x, y)
∂y
,f

y
(x, y),w

y
.
T`u
.
d
i
.
nh ngh˜ıa suy r˘a

`
ng da
.
o h`am riˆeng cu
’
a h`am hai biˆe
´
n theo biˆe
´
n
x l`a d
a
.
o h`am thˆong thu
.
`o
.
ng cu
’
a h`am mˆo
.
tbiˆe
´
n x khi cˆo
´
d
i
.
nh gi´a tri
.

cu
’
abiˆe
´
n y.Dod
´o c ´a c da
.
o h`am riˆeng du
.
o
.
.
c t´ınh theo c´ac quy t˘a
´
cv`a
cˆong th´u
.
c t´ınh d
a
.
o h`am thˆong thu
.
`o
.
ng cu
’
a h`am mˆo
.
tbiˆe
´

n.
Nhˆa
.
nx´et. Ho`an to`an tu
.
o
.
ng tu
.
.
ta c´o thˆe
’
d
i
.
nh ngh˜ıa da
.
o h`am riˆeng
cu
’
a h`am ba (ho˘a
.
c nhiˆe
`
uho
.
n ba) biˆe
´
nsˆo
´

.
9.1.2 D
-
a
.
o h`am cu
’
a h`am ho
.
.
p
Nˆe
´
u h`am w = f(x, y), x = x(t), y = y(t)th`ıbiˆe
’
uth´u
.
c w =
f[x(t),y(t)] l`a h`am ho
.
.
pcu
’
a t. Khi d
´o
dw
dt
=
∂w
∂x

·
dx
dt
+
∂w
∂y
·
dy
dt
· (9.1)
Nˆe
´
u w = f(x, y), trong d
´o x = x(u, v), y = y(u, v)th`ı







∂w
∂u
=
∂w
∂x
∂x
∂u
+
∂w

∂y
∂y
∂u
,
∂w
∂v
=
∂w
∂x
∂x
∂v
+
∂w
∂y
∂y
∂v
·
(9.2)
9.1.3 H`am kha
’
vi
Gia
’
su
.
’
h`am w = f(M) x´ac d
i
.
nh trong mˆo

.
t lˆan cˆa
.
n n`ao d´ocu
’
adiˆe
’
m
M(x, y). H`am f d
u
.
o
.
.
cgo
.
i l`a h`am kha
’
vi ta
.
id
iˆe
’
m M(x, y)nˆe
´
usˆo
´
gia
112 Chu
.

o
.
ng 9. Ph´ep t´ınh vi phˆan h`am nhiˆe
`
ubiˆe
´
n
∆f(M)=f(x +∆,y +∆y) − f(x, y)cu
’
a h`am khi chuyˆe
’
nt`u
.
d
iˆe
’
m
M(x, y)d
ˆe
´
ndiˆe
’
N(x +∆,y+∆y) c´o thˆe
’
biˆe
’
udiˆe
˜
ndu
.

´o
.
ida
.
ng
∆f(M)=D
1
∆x + D
2
∆y + o(ρ),ρ→ 0
trong d
´o ρ =

∆x
2
+∆y
2
.
Nˆe
´
u h`am f(x, y) kha
’
vi ta
.
id
iˆe
’
m M(x, y)th`ı
∂f
∂x

(M)=D
1
,
∂f
∂y
(M)=D
2
v`a khi d´o
∆f(M)=
∂f
∂x
(M)∆x +
∂f
∂y
∆y + o(ρ),ρ→ 0. (9.3)
9.1.4 D
-
a
.
o h`am theo hu
.
´o
.
ng
Gia
’
su
.
’
:

(1) w = f(M) l`a h`am x´ac d
i
.
nh trong lˆan cˆa
.
n n`ao d´o c u
’
adiˆe
’
m
M(x, y);
(2) e = (cos α, cos β) l`a vecto
.
d
o
.
nvi
.
trˆen d
u
.
`o
.
ng th˘a
’
ng c´o hu
.
´o
.
ng

L qua d
iˆe
’
m M(x, y);
(3) N = N(x +∆x, y +∆y)l`ad
iˆe
’
m thuˆo
.
c L v`a ∆e l`a dˆo
.
d`ai cu
’
a
d
oa
.
n th˘a
’
ng MN.
Nˆe
´
utˆo
`
nta
.
i gi´o
.
iha
.

nh˜u
.
uha
.
n
lim
∆→0
(N →M)
∆w
∆
th`ı gi´o
.
iha
.
nd
´odu
.
o
.
.
cgo
.
il`ad
a
.
o h`am ta
.
idiˆe
’
m M(x, y) theo hu

.
´o
.
ng cu
’
a
vecto
.
e v`a d
u
.
o
.
.
ck´yhiˆe
.
ul`a
∂w
∂e
,t´u
.
cl`a
∂w
∂e
= lim
∆→0
∆w
∆
·
9.1. D

-
a
.
o h`am riˆeng 113
Da
.
o h`am theo hu
.
´o
.
ng cu
’
a vecto
.
e = (cos α, cos β)d
u
.
o
.
.
c t´ınh theo
cˆong th´u
.
c
∂f
∂e
=
∂f
∂x
(M) cos α +

∂f
∂y
(M) cos β. (9.4)
trong d
´o cos α v`a cos β l`a c´ac cosin chı
’
phu
.
o
.
ng cu
’
a vecto
.
e .
Vecto
.
v´o
.
i c´ac to
.
ad
ˆo
.
∂f
∂x
v`a
∂F
∂y
(t´u

.
c l`a vecto
.

∂f
∂x
,
∂f
∂y

)d
u
.
o
.
.
cgo
.
i
l`a vecto
.
gradiˆen cu
’
a h`am f(M)ta
.
id
iˆe
’
m M(x, y)v`adu
.

o
.
.
ck´yhiˆe
.
ul`a
gradf(M).
T`u
.
d
´o d a
.
o h`am theo hu
.
´o
.
ng
∂f
∂e
c´o biˆe
’
uth´u
.
cl`a
∂f
∂e
=

gradf,e


.
Ta lu
.
u´yr˘a
`
ng: 1) Nˆe
´
u h`am w = f(x, y) kha
’
vi ta
.
id
iˆe
’
m M(x, y)
th`ı n´o liˆen tu
.
cta
.
i M v`a c´o c´ac d
a
.
o h`am riˆeng cˆa
´
p1ta
.
id´o ;
2) N´eu h`am w = f(x, y) c´o c´ac d
a
.

o h`am riˆeng cˆa
´
p 1 theo mo
.
ibiˆe
´
n
trong lˆan cˆa
.
nn`aod
´ocu
’
adiˆe
’
m M(x, y) v`a c´ac da
.
o h`am riˆeng n`ay liˆen
tu
.
cta
.
id
iˆe
’
m M(x, y) th`ı n´o kha
’
vi ta
.
idiˆe
’

m M.
Nˆe
´
u h`am f(x, y) kha
’
vi ta
.
id
iˆe
’
m M(x, y) th`ı n´o c´o da
.
o h`am theo
mo
.
ihu
.
´o
.
ng ta
.
id
iˆe
’
md´o .
Ch´u´y.Nˆe
´
u h`am f(x, y)c´od
a
.

o h`am theo mo
.
ihu
.
´o
.
ng ta
.
id
iˆe
’
m M
0
th`ı khˆong c´o g`ıda
’
mba
’
o l`a h`am f(x, y) kha
’
vi ta
.
idiˆe
’
m M
0
(xem v´ı
du
.
4).
9.1.5 D

-
a
.
o h`am riˆeng cˆa
´
p cao
Gia
’
su
.
’
miˆe
`
n D ⊂ R
2
v`a
f : D → R
114 Chu
.
o
.
ng 9. Ph´ep t´ınh vi phˆan h`am nhiˆe
`
ubiˆe
´
n
l`a h`am hai biˆe
´
n f(x, y)du
.

o
.
.
c cho trˆen D.Tad
˘a
.
t
D
x
=

(x, y) ∈ D : ∃
∂f
∂x
= ±∞

,
D
y
=

(x, y) ∈ D : ∃
∂f
∂y
= ±∞

.
D
∗
= D

x
∩ D
y
D
-
i
.
nh ngh˜ıa. 1) C´ac da
.
o h`am riˆeng
∂f
∂x
v`a
∂f
∂y
d
u
.
o
.
.
cgo
.
i l`a c´ac d
a
.
o
h`am riˆeng cˆa
´
p1.

2) Nˆe
´
u h`am
∂f
∂x
: D
x
→ R v`a
∂f
∂y
: D
y
→ R c´o c´ac da
.
o h`am riˆeng
∂
∂x

∂f
∂x

=
∂
2
f
∂x∂x
=
∂
2
f

∂x
2
,
∂
∂y

∂f
∂x

=
∂
2
f
∂x∂y
,
∂
∂x

∂f
∂y

=
∂
2
f
∂y∂x
,
∂
∂y


∂f
∂y

=
∂
2
f
∂y∂y
=
∂
2
f
∂y
2
th`ı ch´ung du
.
o
.
.
cgo
.
i l`a c´ac d
a
.
o h`am riˆeng cˆa
´
p2theo x v`a theo y.
C´ac d
a
.

o h`am riˆeng cˆa
´
p3du
.
o
.
.
cd
i
.
nh ngh˜ıa nhu
.
l`a c´ac d
a
.
o h`am riˆeng
cu
’
ad
a
.
o h`am riˆeng cˆa
´
p 2, v.v...
Ta lu
.
u´yr˘a
`
ng nˆe
´

u h`am f(x, y) c´o c´ac d
a
.
o h`am hˆo
˜
nho
.
.
p
∂
2
f
∂x∂y
v`a
∂
2
f
∂y∂x
liˆen tu
.
cta
.
id
iˆe
’
m(x, y) th`ı ta
.
idiˆe
’
md´o c´ac da

.
o h`am hˆo
˜
nho
.
.
p n`ay
b˘a
`
ng nhau:
∂
2
f
∂x∂y
=
∂
2
f
∂y∂x
·
C
´
AC V
´
IDU
.
9.1. D
-
a
.

o h`am riˆeng 115
V´ı d u
.
1. T´ınh da
.
o h`am riˆeng cˆa
´
p1cu
’
a c´ac h`am
1) 4w = x
2
− 2xy
2
+ y
3
.2)w = x
y
.
Gia
’
i. 1) D
a
.
o h`am riˆeng
∂w
∂x
d
u
.

o
.
.
c t´ınh nhu
.
l`a d
a
.
o h`am cu
’
a h`am w
theo biˆe
´
n x v´o
.
i gia
’
thiˆe
´
t y = const. Do d
´o
∂w
∂x
=(x
2
− 2xy
2
+ y
3
)


x
=2x − 2y
2
+0=2(x − y
2
).
Tu
.
o
.
ng tu
.
.
, ta c´o
∂w
∂y
=(x
2
− 2xy
2
+ y
3
)

y
=0− 4xy +3y
2
= y(3y − 4x).
2) Nhu

.
trong 1), xem y = const ta c´o
∂w
∂x
=

x
y


x
= yx
y−1
.
Tu
.
o
.
ng tu
.
.
, khi xem x l`a h˘a
`
ng sˆo
´
ta thu d
u
.
o
.

.
c
∂w
∂y
= x
y
lnx.
(v`ı w = x
y
l`a h`am m˜udˆo
´
iv´o
.
ibiˆe
´
n y khi x = const. 
V´ı d u
.
2. Cho w = f(x, y)v`ax = ρ cos ϕ, y = ρ sin ϕ. H˜ay t´ınh
∂w
∂ρ
v`a
∂w
∂ϕ
.
Gia
’
i. D
ˆe
’

´ap du
.
ng cˆong th´u
.
c (9.2), ta lu
.
u´yr˘a
`
ng
w = f(x, y)=f(ρ cos ϕ, ρ sin ϕ)=F(ρ, ϕ).
Do d
´o theo (9.2) v`a biˆe
’
uth´u
.
cd
ˆo
´
iv´o
.
i x v`a y ta c´o
∂w
∂ρ
=
∂w
∂x
∂x
∂ρ
+
∂w

∂y
∂y
∂ρ
=
∂w
∂x
cos ϕ +
∂w
∂y
sin ϕ
∂w
∂ϕ
=
∂w
∂x
∂x
∂ϕ
+
∂w
∂y
∂y
∂ϕ
=
∂w
∂x
(−ρ sin ϕ)+
∂w
∂y
(ρ cos ϕ)
= ρ


−
∂w
∂x
sin ϕ +
∂w
∂y
cos ϕ

. 
116 Chu
.
o
.
ng 9. Ph´ep t´ınh vi phˆan h`am nhiˆe
`
ubiˆe
´
n
V´ı d u
.
3. T´ınh da
.
o h`am cu
’
a h`am w = x
2
+ y
2
x ta

.
idiˆe
’
m M
0
(1, 2) theo
hu
.
´o
.
ng cu
’
a vecto
.
−→
M
0
M
1
, trong d´o M
1
l`a diˆe
’
mv´o
.
ito
.
ad
ˆo
.

(3, 0).
Gia
’
i. D
ˆa
`
u tiˆen ta t`ım vecto
.
d
o
.
nvi
.
e c´o hu
.
´o
.
ng l`a hu
.
´o
.
ng d
˜a cho.
Ta c´o
−→
M
0
M
1
=(2,−2)=2e

1
− 2e
2
,
⇒|
−→
M
0
M
1
| =2
√
2 ⇒ e =
M
0
M
1
|M
0
M
1
|
=
2e
1
− 2e
2
2
√
2

=
1
√
2
e
1
−
1
√
2
e
2
.
trong d
´o e
1
, e
2
l`a vecto
.
d
o
.
nvi
.
cu
’
a c´ac tru
.
cto

.
ad
ˆo
.
.T`u
.
d
´o suy r˘a
`
ng
cos α =
1
√
2
, cos β = −
1
√
2
·
Tiˆe
´
p theo ta t´ınh c´ac d
a
.
o h`am riˆeng ta
.
idiˆe
’
m M
0

(1, 2). Ta c´o
f

x
=2x + y
2
⇒ f

x
(M
0
)=f

x
(1, 2)=6,
f

y
=2xy ⇒ f

y
(M
0
)=f

y
(1, 2)=4.
Do d
´o theo cˆong th´u
.

c (9.4) ta thu d
u
.
o
.
.
c
∂f
∂e
=6·
1
√
2
− 4 ·
1
√
2
=
√
2. 
V´ı d u
.
4. H`am f(x, y)=x + y +

|xy| c´o da
.
o h`am theo mo
.
ihu
.

´o
.
ng
ta
.
id
iˆe
’
m O(0, 0) nhu
.
ng khˆong kha
’
vi ta
.
id
´o.
Gia
’
i. 1. Su
.
.
tˆo
`
nta
.
id
a
.
o h`am theo mo
.

ihu
.
´o
.
ng.
Ta x´et hu
.
´o
.
ng cu
’
a vecto
.
e d
irat`u
.
Ov`alˆa
.
pv´o
.
i tru
.
c Ox g´oc α.Ta
c´o
∆
e
f(0, 0) = ∆x +∆y +

|∆x∆y|
=


cos α + sin α +

| cos α sin α|

ρ,
9.1. D
-
a
.
o h`am riˆeng 117
trong d´o ρ =

∆x
2
+∆y
2
,∆x = ρ cos α,∆y = ρ sin α.
T`u
.
d
´o suy ra
∂f
∂e
(0, 0) = lim
ρ→0
∆
e
f(0, 0)
ρ

= cos α + sin α +

| sin α cos α|
t´u
.
cl`ad
a
.
o h`am theo hu
.
´o
.
ng tˆo
`
nta
.
i theo mo
.
ihu
.
´o
.
ng.
2. Tuy nhiˆen h`am d
˜a cho khˆong kha
’
vi ta
.
i O. Thˆa
.

tvˆa
.
y, ta c´o
∆f(0, 0) = f(∆x, ∆y)− f(0, 0)=∆x +∆y +

|∆x||∆y|−0.
V`ı f

x
=1v`af

y
= 1 (ta
.
i sao ? ) nˆen nˆe
´
u f kha
’
vi ta
.
i O(0, 0) th`ı
∆f(0, 0) = ∆x +∆y +

|∆x∆y| =1· ∆x +1· ∆y + ε(ρ)ρ
ε(ρ) → 0(ρ → 0),ρ=

∆x
2
+∆y
2

hay l`a lu
.
u´y∆x = ρ cos α,∆y = ρ sin α ta c´o
ε(ρ)=

| cos α sin α|.
Vˆe
´
pha
’
id
˘a
’
ng th´u
.
c n`ay khˆong pha
’
i l`a vˆo c`ung b´e khi ρ → 0 (v`ı n´o
ho`an to`an khˆong phu
.
thuˆo
.
c v`ao ρ). Do d
´o theo di
.
nh ngh˜ıa h`am f(x, y)
d
˜a cho khˆong kha
’
vi ta

.
idiˆe
’
mO.
V´ı d u
.
5. T´ınh c´ac d
a
.
o h`am riˆeng cˆa
´
p2cu
’
a c´ac h`am:
1) w = x
y
,2)w = arctg
x
y
·
Gia
’
i. 1) D
ˆa
`
u tiˆen t´ınh c´ac da
.
o h`am riˆeng cˆa
´
p1.Tac´o

∂w
∂x
= yx
y−1
,
∂w
∂y
= x
y
lnx.
Tiˆe
´
p theo ta c´o
∂
2
w
∂x
2
= y(y − 1)x
y−2
,
∂
2
w
∂y∂x
= x
y−1
+ yx
y−1
lnx = x

y−1
(1 + ylnx),
∂
2
w
∂x∂y
= yx
y−1
lnx + x
y
·
1
x
= x
y−1
(1 + ylnx),
∂
2
f
∂y
2
= x
y
(lnx)
2
.
118 Chu
.
o
.

ng 9. Ph´ep t´ınh vi phˆan h`am nhiˆe
`
ubiˆe
´
n
2) Ta c´o
∂w
∂x
=
y
x
2
+ y
2
,
∂w
∂y
= −
x
x
2
+ y
2
·
T`u
.
d
´o
∂
2

w
∂x
2
=
∂
∂x

y
x
2
+ y
2

= −
2xy
(x
2
+ y
2
)
2
,
∂
2
w
∂y
2
=
∂
∂y


−x
x
2
+ y
2

=
2xy
x
2
+ y
2
,
∂
2
w
∂x∂y
=
∂
∂y

y
x
2
+ y
2

=
x

2
− y
2
(x
2
+ y
2
)
2
,
∂
2
w
∂y∂x
=
∂
∂x

−
x
x
2
+ y
2

=
x
2
− y
2

(x
2
+ y
2
)
2
·
Nhˆa
.
nx´et. Trong ca
’
1) lˆa
˜
n2)tad
ˆe
`
uc´o
∂
2
w
∂x∂y
=
∂
2
w
∂y∂x
. 
V´ı d u
.
6. T´ınh c´ac d

a
.
o h`am riˆeng cˆa
´
p1cu
’
a h`am w = f(x+y
2
,y+x
2
)
ta
.
id
iˆe
’
m M
0
(−1, 1), trong d´o x v`a y l`a biˆe
´
ndˆo
.
clˆa
.
p.
Gia
’
i. D
˘a
.

t t = x + y
2
, v = y + x
2
. Khi d´o
w = f(x + y
2
,y+ x
2
)=f(t, v).
Nhu
.
vˆa
.
y w = f(t, v) l`a h`am ho
.
.
pcu
’
a hai biˆe
´
nd
ˆo
.
clˆa
.
p x v`a y. N´o phu
.
thuˆo
.

c c´ac biˆe
´
nd
ˆo
.
clˆa
.
p thˆong qua hai biˆe
´
n trung gian t, v. Theo cˆong
th´u
.
c (9.2) ta c´o:
∂w
∂x
=
∂f
∂t
·
∂t
∂x
+
∂f
∂v
·
∂v
∂x
= f

t

(x + y
2
,y+ x
2
) · 1+f

v
(x + y
2
,y+ x
2
) · 2x
= f

t
+2xf

v
.
9.1. D
-
a
.
o h`am riˆeng 119
∂w
∂x
(−1, 1) =
∂f
∂x
(0, 2) = f


t
(0, 2) − 2f

v
(0, 2)
∂w
∂y
=
∂f
∂t
·
∂t
∂y
+
∂f
∂v
·
∂v
∂y
= f

t
(·)2y + f

v
(·)1
=2yf

t

+ f

v
∂w
∂y
(−1, 1) =
∂f
∂y
(0, 2)=2f

t
(0, 2) + f

v
(0, 2). 
B
`
AI T
ˆ
A
.
P
T´ınh d
a
.
o h`am riˆeng cu
’
a c´ac h`am sau dˆay
1. f(x, y)=x
2

+ y
3
+3x
2
y
3
.
(D
S. f

x
=2x +6xy
3
, f

y
=3y
2
+9x
2
y
2
)
2. f(x, y, z)=xyz +
x
yz
.
(D
S. f


x
= yz +
1
yz
, f

y
= xz −
x
y
2
z
, f

z
= xy −
x
yz
2
)
3. f(x, y, z) = sin(xy + yz). (D
S. f

x
= y cos(xy + yz),
f

y
=(x + z) cos(xy + yz), f


z
= y cos(xy + yz))
4. f(x, y) = tg(x + y)e
x/y
.
(D
S. f

x
=
e
x/y
cos
2
(x + y)
+ tg(x + y)e
x/y
1
y
,
f

y
=
e
x/y
cos
2
(x + y)
+ tg(x + y)e

x/y

−
x
y
2

.)
5. f = arc sin
x

x
2
+ y
2
.(DS. f

x
=
|y|
x
2
+ y
2
, f

y
=
−xsigny
x

2
+ y
2
)
6. f(x, y)=xyln(xy). (D
S. f

x
= yln(xy)+y, f

y
= xln(xy)+x)
120 Chu
.
o
.
ng 9. Ph´ep t´ınh vi phˆan h`am nhiˆe
`
ubiˆe
´
n
7. f(x, y, z)=

y
x

z
.
(D
S. f


x
= z

y
x

z−1

−
y
x
2

= −
z
x

y
x

z
,
f

y
=
z
y


y
x

z
,f

z
=

y
x

z
ln
y
x
)
8. f(x, y, z)=z
x/y
.
(D
S. f

x
= x
x/y
lnz ·

1
y


, f

y
= z
x/y
lnz ·

−x
y
2

, f

z
=

x
y

z
x/y−1
)
9. f(x, y, z)=x
y
z
.
(D
S. f


x
= y
z
x
y
z
−1
, f

y
= x
y
z
zy
z−1
lnx, f

z
= x
y
z
ln(x)
z
lny)
10. f(x, y, z)=x
y
y
z
z
x

.
(D
S. f

x
= x
y−1
y
z+1
z
x
+ x
y
y
z
z
x
lnz, f

y
= x
y
lnxy
z
z
x
+ x
y
y
z−1

z
x+1
,
f

z
= x
y
y
z
lny · z
x
+ x
y+1
y
z
z
x−1
)
11. f(x, y) = ln sin
x + a
√
y
.
(D
S. f

x
=
1

√
y
cotg
x + a
√
y
, f

y
= −
x + a
y
cotg
x + a
√
y
)
12. f(x, y)=
x
y
− e
x
arctgy.
(D
S. f

x
=
1
y

− e
x
arctgy, f

y
= −
x
y
2
−
e
x
1+y
2
)
13. f(x, y)=ln

x +

x
2
+ y
2

.
(D
S. f

x
=

1

x
2
+ y
2
, f

y
=
1
x +

x
2
+ y
2
·
y

x
2
+ y
2
).
T`ım d
a
.
o h`am riˆeng cu
’

a h`am ho
.
.
p sau d
ˆay (gia
’
thiˆe
´
t h`am f(x, y)
kha
’
vi)
14. f(x, y)=f(x + y,x
2
+ y
2
).
(D
S. f

x
= f

t
+ f

v
2x, f

y

= f

t
+ f

v
2y, t = x + y, v = x
2
+ y
2
)
15. f(x, y)=f

x
y
,
y
x

.
9.1. D
-
a
.
o h`am riˆeng 121
(DS. f

x
=
1

y
f

t
−
y
x
2
f

v
, f

y
=
−x
y
2
f

t
+
1
x
f

v
, t =
x
y

, v =
y
x
)
16. f(x, y)=f(x − y,xy).
(D
S. f

x
= f

t
+ yf

v
, f

y
= −f

t
+ xf

v
, t = x − y, v = xy)
17. f(x, y)=f(x − y
2
,y− x
2
,xy).

(D
S. f

x
= f

t
− 2xf

v
+ yf

w
, f

y
= −2yf

t
+ f

v
+ xf

w
,
t = x− y
2
, v = y − x
2

, w = xy)
18. f(x, y, z)=f(

x
2
+ y
2
,

y
2
+ z
2
,
√
z
2
+ x
2
).
(D
S. f

x
=
xf

t

x

2
+ y
2
+
xf

w
√
z
2
+ x
2
,f

y
=
yf

t

x
2
+ y
2
+
yf

v
√
x

2
+ z
2
,
f

z
=
zf

v

x
2
+ y
2
+
zf

w
√
z
2
+ x
2
,t=

x
2
+ y

2
,
v =

y
2
+ z
2
,w=
√
z
2
+ x
2
)
19. w = f(x, xy, xyz).
(D
S. f

x
= f

t
+ yf

u
+ yzf

v
,

f

y
= xf

u
+ xzf

v
,
f

z
= xyf

v
t = x, u = xy, v = xyz).
Trong c´ac b`ai to´an sau d
ˆay h˜ay ch´u
.
ng to
’
r˘a
`
ng h`am f(x, y) tho
’
a
m˜an phu
.
o

.
ng tr`ınh d
˜a cho tu
.
o
.
ng ´u
.
ng (f(x, y)-kha
’
vi).
20. f = f(x
2
+ y
2
), y
∂f
∂x
− x
∂f
∂y
=0.
21. f = x
n
f

y
x
2


, x
∂f
∂y
+2y
∂f
∂y
= nf.
22. f = yf(x
2
− y
2
), y
2
∂f
∂x
+ xy
∂f
∂y
= xyf.
23. f =
y
2
3x
+ f(x, y), x
2
∂f
∂x
− xy
∂f
∂y

+ y
2
=0.
122 Chu
.
o
.
ng 9. Ph´ep t´ınh vi phˆan h`am nhiˆe
`
ubiˆe
´
n
24. f = x
n
f

y
x
α
,
z
x
β

, x
∂f
∂x
+ αy
∂f
∂y

+ βz
∂f
∂z
= nf.
25. f =
xy
z
lnx + xf

y
x
,
z
x

, x
∂f
∂x
+ y
∂f
∂y
+ z
∂f
∂z
= f +
xy
z
.
26. T´ınh
∂

2
f
∂x
2
,
∂
2
f
∂x∂y
,
∂
2
f
∂y
2
nˆe
´
u f = cos(xy)
(D
S. f

xx
= −y
2
cos xy, f

xy
= − sin xy − xy cos xy, f

yy

=
−x
2
cos xy)
27. T´ınh c´ac d
a
.
o h`am riˆeng cˆa
´
p hai cu
’
a h`am f = sin(x + yz).
(D
S. f

xx
= − sin t, f

xy
= −z sin t, f

xz
= −y sin t, f

yy
= −z
2
sin t,
f


yz
= −yz sin t, f

zz
= −y
2
sin t, t = x + yz)
28. T´ınh
∂
2
f
∂x∂y
nˆe
´
u f =

x
2
+ y
2
e
x+y
.
(D
S.
e
x+y
(x
2
+ y

2
)
3/2

− xy +(x + y)(x
2
+ y
2
)+(x
2
+ y
2
)
2

)
29. T´ınh
∂
2
f
∂x∂y
,
∂
2
f
∂y∂z
,
∂
2
f

∂x∂z
nˆe
´
u f = x
yz
.
(D
S. f

xy
= x
yz−1
z(1 + yzlnx), f

xz
= x
yz−1
y(1 + yzlnx),
f

yz
=lnx · x
yz
(1 + yzlnx))
30. T´ınh
∂
2
f
∂x∂y
nˆe

´
u f = arctg
x + y
1 − xy
.(D
S.
∂
2
f
∂x∂y
=0)
31. T´ınh f

xx
(0, 0), f

xy
(0, 0), f

yy
(0, 0) nˆe
´
u
f(x, y)=(1+x)
m
(1 + y)
n
.
(D
S. f


xx
(0, 0) = m(m− 1), f

xy
(0, 0) = mn, f

yy
(0, 0) = n(n − 1))
32. T´ınh
∂
2
r
∂x
2
nˆe
´
u r =

x
2
+ y
2
+ z
2
.(DS.
r
2
− x
2

r
3
)
33. T´ınh f

xy
, f

yz
, f

xz
nˆe
´
u f =

x
y

z
.
(D
S. f

xy
= −z
2
y
−2


xy
−1

z−1
, f

xz
=

1
y

x
y

z−1

1+zln
x
y

,
9.1. D
-
a
.
o h`am riˆeng 123
f

yz

= −
1
y

x
y

z
·

1+zln
x
y

)
34. Ch´u
.
ng minh r˘a
`
ng
∂
2
f
∂x∂y
=
∂
2
f
∂y∂x
nˆe

´
u f = arc sin

x − y
x
.
T´ınh c´ac d
a
.
o h`am cˆa
´
p hai cu
’
a c´ac h`am (gia
’
thiˆe
´
t hai lˆa
`
n kha
’
vi)
35. u = f(x + y,x
2
+ y
2
).
(D
S. u


xx
= f

tt
+4xf

tv
+4x
2
f

vv
+2f

v
,
u

xy
= f

tt
+2(x + y)f

tv
+4xyf

vv
,
u


yy
= f

tt
+4yf

tv
+4y
2
f

vv
+2f

v
,
t = x + y, v = x
2
+ y
2
.)
36. u = f

xy,
x
y

.
(D

S. u

xx
= y
2
f

tt
+2f

tv
+
1
y
2
f

vv
,
u

xy
= xyf

tt
−
x
y
3
f


vv
+ f

t
−
1
y
2
f

v
,
u

yy
= x
2
f

tt
− 2
x
2
y
2
f

tv
+

x
2
y
4
f

vv
+
2x
y
3
f

v
,
t = xy, v =
x
y
)
37. u = f(sin x + cos y).
(D
S. u

xx
= cos
2
x · f

− sin x · f


, u

xy
= − sin y cos x· f

,
u

yy
= sin
2
y · f

− cos y · f

)
38. Ch´u
.
ng minh r˘a
`
ng h`am
f =
1
2a
√
πt
e
−
(x−x
0

)
2
4a
2
t
(trong d´o a, x
0
l`a c´ac sˆo
´
) tho
’
a m˜an phu
.
o
.
ng tr`ınh truyˆe
`
n nhiˆe
.
t
∂f
∂t
= a
2
∂
2
f
∂x
2
·

124 Chu
.
o
.
ng 9. Ph´ep t´ınh vi phˆan h`am nhiˆe
`
ubiˆe
´
n
39. Ch´u
.
ng minh r˘a
`
ng h`am f =
1
r
trong d
´o
r =

(x − x
0
)
2
+(y − y
0
)
2
+(z − z
0

)
2
tho
’
a m˜an phu
.
o
.
ng tr`ınh Laplace:
∆f ≡
∂
2
f
∂x
2
+
∂
2
f
∂y
2
+
∂
2
f
∂z
2
=0,r=0.
Trong c´ac b`ai to´an 40 - 44 ch´u
.

ng minh r˘a
`
ng c´ac h`am d
˜a cho tho
’
a
m˜an phu
.
o
.
ng tr`ınh tu
.
o
.
ng ´u
.
ng (gia
’
thiˆe
´
t f v`a g l`a nh˜u
.
ng h`am hai lˆa
`
n
kha
’
vi)
40. u = f(x − at)+g(x + at),
∂

2
u
∂t
2
= a
2
∂
2
u
∂x
2
41. u = xf(x + y)+yg(x + y),
∂
2
u
∂x
2
− 2
∂
2
u
∂x∂y
+
∂
2
u
∂y
2
=0.
42. u = f


y
x

+ xg

y
x

, x
2
∂
2
u
∂x
2
+2xy
∂
2
u
∂x∂y
+ y
2
∂
2
u
∂y
2
=0.
43. u = x

n
f

y
x

+ x
1−n
g

y
x

,
x
2
∂
2
u
∂x
2
+2xy
∂
2
u
∂x∂y
+ y
2
∂
2

u
∂y
2
= n(n − 1)u.
44. u = f(x + g(y)),
∂u
∂x
·
∂
2
u
∂x∂y
=
∂u
∂y
·
∂
2
u
∂x
2
·
45. T`ım d
a
.
o h`am theo hu
.
´o
.
ng ϕ = 135

◦
cu
’
a h`am sˆo
´
f(x, y)=3x
4
+ xy + y
3
ta
.
idiˆe
’
m M(1, 2). (DS. −
√
2
2
)
46. T`ım d
a
.
o h`am cu
’
a h`am f(x, y)=x
3
− 3x
2
y +3xy
2
+1 ta

.
idiˆe
’
m
M(3, 1) theo hu
.
´o
.
ng t`u
.
d
iˆe
’
m n`ay dˆe
´
ndiˆe
’
m(6, 5). (DS. 0)
47. T`ım d
a
.
o h`am cu
’
a h`am f(x, y)=ln

x
2
+ y
2
ta

.
idiˆe
’
m M(1, 1)
theo hu
.
´o
.
ng phˆan gi´ac cu
’
a g´oc phˆa
`
ntu
.
th´u
.
nhˆa
´
t. (D
S.
√
2
2
)
9.2. Vi phˆan cu
’
a h`am nhiˆe
`
ubiˆe
´

n 125
48. T`ım da
.
o h`am cu
’
a h`am f(x, y, z)=z
2
− 3xy +5 ta
.
idiˆe
’
m
M(1, 2,−1) theo hu
.
´o
.
ng lˆa
.
pv´o
.
i c´ac tru
.
cto
.
ad
ˆo
.
nh˜u
.
ng g´oc b˘a

`
ng nhau.
(D
S. −
√
3
3
)
49. T`ım d
a
.
o h`am cu
’
a h`am f(x, y, z)=ln(e
x
+ e
y
+ e
z
)ta
.
igˆo
´
cto
.
adˆo
.
v`a hu
.
´o

.
ng lˆa
.
pv´o
.
i c´ac tru
.
cto
.
ad
ˆo
.
x, y, z c´ac g´oc tu
.
o
.
ng ´u
.
ng l`a α, β, γ.
(D
S.
cos α + cos β + cos γ
3
)
50. T´ınh d
a
.
o h`am cu
’
a h`am f(x, y)=2x

2
− 3y
2
ta
.
idiˆe
’
m M(1, 0) theo
hu
.
´o
.
ng lˆa
.
pv´o
.
i tru
.
c ho`anh g´oc b˘a
`
ng 120
◦
.(DS. −2)
51. T`ım d
a
.
o h`am cu
’
a h`am z = x
2

− y
2
ta
.
idiˆe
’
m M
0
(1, 1) theo hu
.
´o
.
ng
vecto
.
e lˆa
.
pv´o
.
ihu
.
´o
.
ng du
.
o
.
ng tru
.
c ho`anh g´oc α =60

◦
.(DS. 1 −
√
3)
52. T`ım d
a
.
o h`am cu
’
a h`am z = ln(x
2
+ y
2
)ta
.
idiˆe
’
m M
0
(3, 4) theo
hu
.
´o
.
ng gradien cu
’
a h`am d
´o. (DS.
2
5

)
53. T`ım gi´a tri
.
v`a hu
.
´o
.
ng cu
’
a vecto
.
gradien cu
’
a h`am
w =tgx − x + 3 sin y − sin
3
y + z + cotgz
ta
.
id
iˆe
’
m M
0

π
4
,
π
3

,
π
2

.
(D
S. (gradw)
M
=

i +
3
8

j, cos α =
8
√
73
, cos β =
3
√
73
)
54. T`ım d
a
.
o h`am cu
’
a h`am w = arc sin
z


x
2
+ y
2
ta
.
idiˆe
’
m M
0
(1, 1, 1)
theo hu
.
´o
.
ng vecto
.
−→
M
0
M, trong d´o M =(3, 2, 3). (DS.
1
6
)
9.2 Vi phˆan cu
’
ah`am nhiˆe
`
ubiˆe

´
n
Trong mu
.
c n`ay ta x´et vi phˆan cu
’
a h`am nhiˆe
`
ubiˆe
´
nm`adˆe
’
cho go
.
nta
chı
’
cˆa
`
n tr`ınh b`ay cho h`am hai biˆe
´
nl`ad
u
’
.Tru
.
`o
.
ng ho
.

.
psˆo
´
biˆe
´
nl´o
.
n
ho
.
nhaid
u
.
o
.
.
c tr`ınh b`ay ho`an to`an tu
.
o
.
ng tu
.
.
.
126 Chu
.
o
.
ng 9. Ph´ep t´ınh vi phˆan h`am nhiˆe
`

ubiˆe
´
n
9.2.1 Vi phˆan cˆa
´
p1
Gia
’
su
.
’
h`am w = f(x, y) kha
’
vi ta
.
id
iˆe
’
m M(x, y), t´u
.
cl`ata
.
id
´o s ˆo
´
gia
to`an phˆa
`
ncu
’

a h`am c´o thˆe
’
biˆe
’
udiˆe
˜
ndu
.
´o
.
ida
.
ng
∆f(M)=f(x +∆x, y +∆y) − f(x, y)
= D
1
∆x + D
2
∆y + o(ρ) (9.5)
trong d
´o ρ =

∆x
2
+∆y
2
, D
1
v`a D
2

khˆong phu
.
thuˆo
.
cv`ao∆x v`a
∆y. Khi d
´obiˆe
’
uth´u
.
c (go
.
il`aphˆa
`
nch´ınh tuyˆe
´
n t´ınh d
ˆo
´
iv´o
.
i ∆x v`a ∆y
cu
’
asˆo
´
gia ∆f)
D
1
∆x + D

2
∆y
d
u
.
o
.
.
cgo
.
il`avi phˆan (hay vi phˆan to`an phˆa
`
n ≡ hay vi phˆan th´u
.
nhˆa
´
t)
cu
’
a h`am w = f(x, y)v`ad
u
.
o
.
.
ck´yhiˆe
.
ul`adf :
df = D
1

∆x + D
2
∆y.
V`ı∆x = dx,∆y = dy v`a v`ı f(x, y) kha
’
vi ta
.
i M nˆen D
1
=
∂f
∂y
,
D
2
=
∂f
∂y
v`a
df =
∂f
∂x
dx +
∂f
∂y
dy (9.6)
Nhu
.
vˆa
.

y, nˆe
´
u w = f(x, y) kha
’
vi ta
.
i M(x, y)th`ıt`u
.
(9.5) v`a (9.6)
ta c´o
∆f(M)=df (M)+o(ρ)hay∆f(M)=df (M)+ε(ρ)ρ (9.7)
trong d
´o ε(ρ) → 0 khi ρ → 0.
9.2.2
´
Ap du
.
ng vi phˆan dˆe
’
t´ınh gˆa
`
nd´ung
Dˆo
´
iv´o
.
i∆x v`a ∆y d
u
’
b´e ta c´o thˆe

’
thay xˆa
´
pxı
’
sˆo
´
gia ∆f(M)bo
.
’
ivi
phˆan df (M), t´u
.
cl`a
∆f(M) ≈ df (M)
9.2. Vi phˆan cu
’
a h`am nhiˆe
`
ubiˆe
´
n 127
hay l`a
f(x +∆x, y +∆y) ≈ f(x, y)+
∂f
∂x
(M)∆x +
∂f
∂y
(M)∆y

(9.8)
Cˆong th´u
.
c (9.8) l`a co
.
so
.
’
d
ˆe
’
´ap du
.
ng vi phˆan t´ınh gˆa
`
nd´ung. Dˆo
´
i
v´o
.
i h`am c´o sˆo
´
biˆe
´
n nhiˆe
`
uho
.
n2tac˜ung c´o cˆong th´u
.

ctu
.
o
.
ng tu
.
.
.
9.2.3 C´ac t´ınh chˆa
´
tcu
’
a vi phˆan
Dˆo
´
iv´o
.
i c´ac h`am kha
’
vi f v`a g ta c´o:
(i) d(f ± g)=df ± dg;
(ii) d(fg)=fdg + gdf, d(αf)=αdf, α ∈ R;
(iii) d

f
g

=
gdf − fdg
g

2
, g =0;
(iv) Vi phˆan cˆa
´
p1cu
’
a h`am hai biˆe
´
n f(x, y)bˆa
´
tbiˆe
´
nvˆe
`
da
.
ng bˆa
´
t
luˆa
.
n x v`a y l`a biˆe
´
nd
ˆo
.
clˆa
.
p hay l`a h`am cu
’

a c´ac biˆe
´
ndˆo
.
clˆa
.
p kh´ac.
9.2.4 Vi phˆan cˆa
´
p cao
Gia
’
su
.
’
h`am w = f(x, y) kha
’
vi trong miˆe
`
n D. Khi d
´o vi phˆan cˆa
´
p1
cu
’
a n´o ta
.
id
iˆe
’

m(x, y) ∈ D tu
.
o
.
ng ´u
.
ng v´o
.
i c´ac sˆo
´
gia dx v`a dy cu
’
a c´ac
biˆe
´
nd
ˆo
.
clˆa
.
pdu
.
o
.
.
cbiˆe
’
udiˆe
˜
nbo

.
’
i cˆong th´u
.
c
df =
∂f
∂x
dx +
∂f
∂y
dy. (9.9)
O
.
’
d
ˆa y , dx =∆x, dy =∆y l`a nh˜u
.
ng sˆo
´
gia t`uy ´y cu
’
abiˆe
´
nd
ˆo
.
clˆa
.
p, d´o

l`a nh˜u
.
ng sˆo
´
khˆong phu
.
thuˆo
.
c v`ao x v`a y.Nhu
.
vˆa
.
y, khi cˆo
´
d
i
.
nh dx v`a
dy vi phˆan df l`a h`am cu
’
a x v`a y.
Theo d
i
.
nh ngh˜ıa: Vi phˆan th´u
.
hai d
2
f (hay vi phˆan cˆa
´

p 2) cu
’
a
h`am f(x, y)ta
.
id
iˆe
’
m M(x, y)du
.
o
.
.
cd
i
.
nh ngh˜ıa nhu
.
l`a vi phˆan cu
’
avi
phˆan th´u
.
nhˆa
´
tta
.
id
iˆe
’

m M v´o
.
i c´ac d
iˆe
`
ukiˆe
.
n sau dˆay:
(1) Vi phˆan df l`a h`am chı
’
cu
’
a c´ac biˆe
´
nd
ˆo
.
clˆa
.
p x v`a y.
128 Chu
.
o
.
ng 9. Ph´ep t´ınh vi phˆan h`am nhiˆe
`
ubiˆe
´
n
(2) Sˆo

´
gia cu
’
a c´ac biˆe
´
ndˆo
.
clˆa
.
p x v`a y xuˆa
´
thiˆe
.
n khi t´ınh vi phˆan
cu
’
a f

x
v`a f

y
du
.
o
.
.
c xem l`a b˘a
`
ng sˆo

´
gia d
ˆa
`
u tiˆen, t´u
.
cl`ab˘a
`
ng dx v`a dy.
T`u
.
d
´o
d
2
f(M)=
∂
2
f(M)
∂x
2
dx
2
+2
∂
2
f
∂x∂y
(M)dxdy +
∂

2
f
∂y
2
(M)dy
2
(9.10)
trong d
´o dx
2
=(dx)
2
, dy
2
=(dy)
2
v`a ta xem c´ac da
.
o h`am riˆeng hˆo
˜
n
ho
.
.
pb˘a
`
ng nhau.
Mˆo
.
t c´ach h`ınh th´u

.
c d
˘a
’
ng th´u
.
c (9.10) c´o thˆe
’
viˆe
´
tdu
.
´o
.
ida
.
ng
d
2
f =

∂
∂x
dx +
∂
∂y
dy

2
f(x, y)

t´u
.
c l`a sau khi thu
.
.
chiˆe
.
n ph´ep “b`ınh phu
.
o
.
ng” ta cˆa
`
nd
iˆe
`
n f(x, y) v`ao
“ˆo trˆo
´
ng”.
Tu
.
o
.
ng tu
.
.
d
3
f =


∂
∂x
dx +
∂
∂y
dy

3
f(x, y)
=
∂
3
f
∂x
3
dx
3
+3
∂
3
f
∂x
2
∂y
dx
2
dy +3
∂
3

f
∂x∂y
2
dxdy
2
+
∂
3
f
∂y
3
dy
3
,
v.v... Mˆo
.
t c´ach quy na
.
p ta c´o
d
n
f(x, y)=
n

k=0
C
k
n
∂
n

f
∂x
n−k
∂y
k
dx
n−k
dy
k
. (9.11)
Trong tru
.
`o
.
ng ho
.
.
pnˆe
´
u
w = f(t, v),t= ϕ(x, y),v= ψ(x, y)
th`ı
dw =
∂f
∂t
dt +
∂f
∂v
dx (t´ınh bˆa
´

tbiˆe
´
nvˆe
`
da
.
ng !)
d
2
w =
∂
2
f
∂t
2
dt
2
+2
∂
2
f
∂t∂v
dtdy +
∂
2
f
∂v
2
dv
2

+
∂f
∂t
d
2
t +
∂f
∂v
d
2
v. (9.12)