i biˆe´n trong lˆan cˆa.. o h`am riˆeng n`ay liˆen tu... Dˆ` u tiˆen ta t`ım vecto... Tuy nhiˆen h`am d˜a cho khˆong kha’ vi ta.i O... thuˆo.c c´ac biˆe´n dˆo.c lˆa.p thˆong qua hai biˆe
Trang 12 Tu.o.ng tu : nˆe´u tˆo`n ta.i gi´o.i ha.n
lim
∆y→0
∆yw
∆y = lim∆y→0
f (x, y + ∆y) − f (x, y)
∆y
th`ı gi´o.i ha.n d´o du.o c go.i l`a da.o h`am riˆeng cu’a h`am f(x, y) theo biˆe´n
y ta.i diˆe’m M(x, y) v`a du.o c chı’ bo.’i mˆo.t trong c´ac k´y hiˆe.u
∂w
∂y ,
∂f (x, y)
0
y (x, y), w0y
T`u di.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 da.o h`am thˆong thu.`o.ng cu’a h`am mˆo.t biˆe´n x khi cˆo´ di.nh gi´a tri.
cu’a biˆe´n y Do d´o c´ac da o h`am riˆeng du.o..c t´ınh theo c´ac quy t˘a´c v`a
cˆong th´u.c t´ınh da o h`am thˆong thu.`o.ng cu’a h`am mˆo t biˆe´n
Nhˆa n x´et Ho`an to`an tu.o.ng tu ta c´o thˆe’ di.nh ngh˜ıa da.o h`am riˆeng
cu’a h`am ba (ho˘a.c nhiˆe`u ho.n ba) biˆe´n sˆo´
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`ı biˆe’u th´u.c w =
f [x(t), y(t)] l`a h`am ho..p cu’a t Khi d´o
dw
∂w
∂x ·
dx
dt +
∂w
∂y ·
dy
Nˆe´u w = f (x, y), trong d´ o x = x(u, v), y = y(u, v) th`ı
∂w
∂w
∂x
∂x
∂w
∂y
∂y
∂u ,
∂w
∂w
∂x
∂x
∂w
∂y
∂y
∂v·
(9.2)
9.1.3 H` am kha ’ vi
Gia’ su.’ h`am w = f (M ) x´ac di.nh trong mˆo.t lˆan cˆa.n n`ao d´o cu’a diˆe’m
M (x, y) H` am f du.o c go.i l`a h`am kha’ vi ta.i diˆe’m M(x, y) nˆe´u sˆo´ gia
Trang 2∆f (M ) = f (x + ∆, y + ∆y) − f (x, y) cu’a h`am khi chuyˆe’n t`u diˆe’m
M (x, y) dˆe´n diˆe’N (x + ∆, y + ∆y) c´o thˆe’ biˆe’u diˆe˜n du.´o.i da.ng
∆f (M ) = D1∆x + D2∆y + o(ρ), ρ → 0
trong d´o ρ =p
∆x2+ ∆y2
Nˆe´u h`am f (x, y) kha’ vi ta.i diˆe’m M(x, y) th`ı
∂f
∂x (M ) = D1 ,
∂f
∂y (M ) = D2
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 di.nh trong lˆan cˆa.n n`ao d´o cu’a diˆe’m
M (x, y);
(2) ~ e = (cos α, cos β) l`a vecto do.n vi trˆen du.`o.ng th˘a’ng c´o hu.´o.ng
L qua diˆe’m M (x, y);
(3) N = N (x + ∆x, y + ∆y) l`a diˆe’m thuˆo.c L v`a ∆e l`a dˆo d`ai cu’a doa.n th˘a’ng MN.
Nˆe´u tˆ` n ta.i gi´o.i ha.n h˜u.u ha.no
lim
∆`→0 (N →M )
∆w
∆`
th`ı gi´o.i ha.n d´o du.o c go.i l`a da.o h`am ta.i diˆe’m M(x, y) theo hu.´o.ng cu’a vecto ~ e v`a du.o c k´y hiˆe.u l`a ∂w
∂~ e, t´u.c l`a
∂w
∂~ e = lim∆`→0
∆w
∆` ·
Trang 3Da.o h`am theo hu.´o.ng cu’a vecto ~e = (cos α, cos β) du.o c t´ınh theo
cˆong th´u.c
∂f
∂~ e =
∂f
∂x (M ) cos α +
∂f
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.a dˆo ∂f
∂x v`a
∂F
∂y (t´u.c l`a vecto.
∂f
∂x ,
∂f
∂y
) du.o c go.i l`a vecto gradiˆen cu’a h`am f (M ) ta.i diˆe’m M(x, y) v`a du.o c k´y hiˆe.u l`a
gradf (M ).
T`u d´o da.o h`am theo hu.´o.ng ∂f
∂~ e c´o biˆe’u th´u.c l`a
∂f
∂~ e = gradf, ~ e
.
Ta lu.u ´y r˘a`ng: 1) Nˆe´u h`am w = f (x, y) kha’ vi ta i diˆe’m M (x, y)
th`ı n´o liˆen tu c ta i M v`a c´o c´ac da o h`am riˆeng cˆa´p 1 ta i d´o;
2) N´eu h`am w = f (x, y) c´o c´ac da o h`am riˆeng cˆa´p 1 theo mo i biˆe´n
trong lˆan cˆa n n`ao d´o cu’a diˆe’m M (x, y) v`a c´ac da o h`am riˆeng n`ay liˆen
tu c ta i diˆe’m M (x, y) th`ı n´o kha’ vi ta i diˆe’m M
Nˆe´u h`am f (x, y) kha’ vi ta.i diˆe’m M(x, y) th`ı n´o c´o da.o h`am theo
mo.i hu.´o.ng ta.i diˆe’m d´o
Ch´u ´y Nˆe´u h`am f (x, y) c´ o da.o h`am theo mo.i hu.´o.ng ta.i diˆe’m M0
th`ı khˆong c´o g`ı da’m ba’o l`a h`am f (x, y) kha’ vi ta.i diˆe’m M0 (xem v´ı
du 4)
9.1.5 D - a.o h`am riˆeng cˆa´p cao
Gia’ su.’ miˆe`n D ⊂ R2
v`a
f : D → R
Trang 4l`a h`am hai biˆe´n f (x, y) du.o .c cho trˆen D Ta d˘a.t
D x =
n
(x, y) ∈ D : ∃ ∂f
∂x 6= ±∞
o
,
D y =
n
(x, y) ∈ D : ∃ ∂f
∂y 6= ±∞
o
.
D∗ = Dx ∩ Dy
D - i.nh ngh˜ıa 1) C´ac da.o h`am riˆeng ∂f
∂x v`a
∂f
∂y du.o c go.i l`a c´ac da.o h`am riˆeng cˆa´p 1
2) Nˆe´u h`am ∂f
∂x : Dx →R v`a
∂f
∂y : Dy →R c´o c´ac da.o h`am riˆeng
∂
∂x
∂f
∂x
= ∂
2
f
∂2f
∂x2 ,
∂
∂y
∂f
∂x
= ∂
2f
∂x∂y ,
∂
∂x
∂f
∂y
= ∂
2f
∂y∂x ,
∂
∂y
∂f
∂y
= ∂
2
f
∂2f
∂y2 th`ı ch´ung du.o c go.i l`a c´ac da.o h`am riˆeng cˆa´p 2 theo x v`a theo y.
C´ac da.o h`am riˆeng cˆa´p 3 du.o c di.nh ngh˜ıa nhu l`a c´ac da.o h`am riˆeng cu’a da.o h`am riˆeng cˆa´p 2, v.v
Ta lu.u ´y r˘a`ng nˆe´u h`am f (x, y) c´o c´ac da.o h`am hˆo˜n ho p ∂
2
f
∂x∂y v`a
∂2f
∂y∂x liˆen tu.c ta.i diˆe’m (x, y) th`ı ta.i diˆe’m d´o c´ac da.o h`am hˆo˜n ho p n`ay
b˘a`ng nhau:
∂2f
∂2f
∂y∂x·
C ´ AC V´ I DU .
Trang 5V´ ı du 1 T´ınh da.o h`am riˆeng cˆa´p 1 cu’a c´ac h`am
1) 4w = x2− 2xy2+ y3 2) w = x y
Gia’i 1) Da.o h`am riˆeng ∂w
∂x du.o..c t´ınh nhu l`a da.o h`am cu’a h`am w
theo biˆe´n x v´o.i gia’ thiˆe´t y = const Do d´o
∂w
2
− 2xy2+ y3)0x = 2x − 2y2+ 0 = 2(x − y2).
Tu.o.ng tu , ta c´o
∂w
2
− 2xy2+ y3)0y = 0 − 4xy + 3y2 = y(3y − 4x).
2) Nhu trong 1), xem y = const ta c´o
∂w
y0
x = yx y−1
Tu.o.ng tu.., khi xem x l`a h˘a`ng sˆo´ ta thu du.o c
∂w
y lnx.
(v`ı w = x y l`a h`am m˜u dˆo´i v´o.i biˆe´n y khi x = const N
V´ ı du 2 Cho w = f (x, y) v` a x = ρ cos ϕ, y = ρ sin ϕ H˜ay t´ınh ∂w
∂ρ
v`a ∂w
∂ϕ.
Gia’i Dˆe’ ´ap du.ng cˆong th´u.c (9.2), ta lu.u ´y r˘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’u th´u.c dˆo´i v´o.i x v` a y ta c´o
∂w
∂w
∂x
∂x
∂w
∂y
∂y
∂w
∂x cos ϕ +
∂w
∂y sin ϕ
∂w
∂w
∂x
∂x
∂w
∂y
∂y
∂w
∂x (−ρ sin ϕ) +
∂w
∂y (ρ cos ϕ)
= ρ
− ∂w
∂x sin ϕ +
∂w
∂y cos ϕ
Trang 6
V´ ı du 3 T´ınh da.o h`am cu’a h`am w = x2+ y2
x ta.i diˆe’m M0(1, 2) theo hu.´o.ng cu’a vecto
−→
M0M1, trong d´o M1 l`a diˆe’m v´o.i to.a dˆo (3, 0).
Gia’i Dˆ` u tiˆen ta t`ım vecto do.n vi ~e c´o hu.´o.ng l`a hu.´o.ng d˜a cho.a
Ta c´o
−→
M0M1= (2, −2) = 2e1 − 2e2 ,
⇒ |
−→
M0M1| = 2
√
2 ⇒ ~ e = M0M1
|M0 M1| = 2e1 − 2e2
2
√ 2
= √1
2~ e1
− 1
√
2~ e2. trong d´o ~ e1, ~e2 l`a vecto do.n vi cu’a c´ac tru.c to.a dˆo T`u d´o suy r˘a`ng
cos α = √1
2, cos β = −
1
√
2· Tiˆe´p theo ta t´ınh c´ac da.o h`am riˆeng ta.i diˆe’m M0(1, 2) Ta c´o
f x0 = 2x + y2⇒ f x0(M0) = f x0(1, 2) = 6,
f y0 = 2xy ⇒ f y0(M0) = f y0(1, 2) = 4.
Do d´o theo cˆong th´u.c (9.4) ta thu du.o c
∂f
∂~ e = 6 ·
1
√
2− 4 ·
1
√
2 =
√
2. N
V´ ı du 4 H`am f (x, y) = x + y +p
|xy| c´o da.o h`am theo mo.i hu.´o.ng
ta.i diˆe’m O(0, 0) nhu.ng khˆong kha’ vi ta.i d´o.
Gia’i 1 Su tˆo`n ta.i da.o h`am theo mo.i hu.´o.ng
Ta x´et hu.´o.ng cu’a vecto ~ e di ra t`u O v`a lˆa.p v´o.i tru.c Ox g´oc α Ta
c´o
∆ef (0, 0) = ∆x + ∆y +p
|∆x∆y|
= cos α + sin α +p
| cos α sin α|
ρ,
Trang 7trong d´o ρ = p
∆x2+ ∆y2, ∆x = ρ cos α, ∆y = ρ sin α.
T`u d´o suy ra
∂f
∂~ e (0, 0) = lim ρ→0
∆ef (0, 0)
ρ = cos α + sin α +
p
| sin α cos α|
t´u.c l`a da.o h`am theo hu.´o.ng tˆo`n ta.i theo mo.i hu.´o.ng
2 Tuy nhiˆen h`am d˜a cho khˆong kha’ vi ta.i O Thˆa.t vˆa.y, ta c´o
∆f (0, 0) = f (∆x, ∆y) − f (0, 0) = ∆x + ∆y +p
|∆x| |∆y| − 0.
V`ı f x0 = 1 v`a f y0 = 1 (ta.i sao ? ) nˆen nˆe´u f kha’ vi ta.i O(0, 0) th`ı
∆f (0, 0) = ∆x + ∆y +p
|∆x∆y| = 1 · ∆x + 1 · ∆y + ε(ρ)ρ
∆x2+ ∆y2 hay l`a lu.u ´y ∆x = ρ cos α, ∆y = ρ sin α ta c´o
| cos α sin α|.
Vˆe´ pha’i d˘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.i diˆe’m O N
V´ ı du 5 T´ınh c´ac da.o h`am riˆeng cˆa´p 2 cu’a c´ac h`am:
1) w = x y, 2) w = arctg x
y·
Gia’i 1) Dˆ` u tiˆen t´ınh c´ac da.o h`am riˆeng cˆa´p 1 Ta c´oa
∂w
y−1
y lnx.
Tiˆe´p theo ta c´o
∂2w
∂x2 = y(y − 1)x y−2 ,
∂2w
y−1 + yx y−1 lnx = x y−1 (1 + ylnx),
∂2w
y−1 lnx + x y· 1
y−1 (1 + ylnx),
∂2f
∂y2 = x y (lnx)2.
Trang 82) Ta c´o
∂w
y
x
x2+ y2· T`u d´o
∂2w
∂x
x2+ y2
(x2+ y2)2 ,
∂2w
∂y2 = ∂
∂y
−x
x2+ y2
x2+ y2 ,
∂2w
∂
∂y
x2+ y2
2− y2
(x2+ y2)2,
∂2w
∂
∂x
x2+ y2
2− y2
(x2+ y2)2 ·
Nhˆa n x´et Trong ca’ 1) lˆa˜n 2) ta dˆe`u c´o ∂
2
w
∂2w
∂y∂x N
V´ ı du 6 T´ınh c´ac da.o h`am riˆeng cˆa´p 1 cu’a h`am w = f(x+y2, y + x2)
ta.i diˆe’m M0(−1, 1), trong d´ o x v` a y l`a biˆe´n dˆo.c lˆa.p
Gia’i D˘a.t t = x + y2, v = y + x2 Khi d´o
w = f (x + y2, y + x2) = f (t, v).
Nhu vˆa.y w = f(t, v) l`a h`am ho p cu’a hai biˆe´n dˆo.c lˆa.p x v`a y N´o phu.
thuˆo.c c´ac biˆe´n dˆo.c lˆa.p thˆong qua hai biˆe´n trung gian t, v Theo cˆong
th´u.c (9.2) ta c´o:
∂w
∂f
∂t ·
∂t
∂f
∂v ·
∂v
∂x
= f t0(x + y2, y + x2) · 1 + f v0(x + y2, y + x2) · 2x
= f t0+ 2xf v0.
Trang 9∂x (−1, 1) =
∂f
∂x (0, 2) = f
0
t (0, 2) − 2f v0(0, 2)
∂w
∂f
∂t ·
∂t
∂f
∂v ·
∂v
∂y = f
0
t (·)2y + f v0(·)1
= 2yf t0+ f v0
∂w
∂y (−1, 1) =
∂f
∂y (0, 2) = 2f
0
t (0, 2) + f v0(0, 2) N
B ` AI T ˆ A P
T´ınh da.o h`am riˆeng cu’a c´ac h`am sau dˆay
1 f (x, y) = x2+ y3+ 3x2y3
(DS f0
x = 2x + 6xy3, f0
y = 3y2 + 9x2y2)
2 f (x, y, z) = xyz + x
yz.
(DS f x0 = yz + 1
yz , f
0
y = xz − x
y2z , f
0
z = xy − x
yz2)
3 f (x, y, z) = sin(xy + yz). (DS f0
x = y cos(xy + yz),
f y0 = (x + z) cos(xy + yz), f z0 = y cos(xy + yz))
4 f (x, y) = tg(x + y)e x/y
(DS f x0 = e
x/y
cos2(x + y) + tg(x + y)e
x/y1
y ,
x/y
cos2(x + y) + tg(x + y)e
x/y
− x
y2
.)
5 f = arc sinp x
x2+ y2 (DS f0
x = |y|
x2+ y2, f0
y = −xsigny
x2+ y2 )
6 f (x, y) = xyln(xy). (DS f0 = yln(xy) + y, f0 = xln(xy) + x)
Trang 107 f (x, y, z) = y
x
z
(DS f x0 = z y
x
z−1
− y
x2
= −z
x
y
x
z
,
f y0 = z
y
y
x
z
, f z0 =y
x
z
lny
x)
8 f (x, y, z) = z x/y
(DS f x0 = x x/y lnz ·1
y
, f y0 = z x/y lnz · −x
y2
, f z0 =x
y
z x/y−1)
9 f (x, y, z) = x yz
(DS f0
x = y z x yz −1, f0
y = x yz
zy z−1 lnx, f0
z = x yz
ln(x) z lny)
10 f (x, y, z) = x y y z z x
(DS f0
x = x y−1 y z+1 z x + x y y z z x lnz, f0
y = x y lnxy z z x + x y y z−1 z x+1,
f0
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 .
(DS f0
x= √1
ycotg
x + a
√
y , f
0
y = −x + a
x + a
√
12 f (x, y) = x
y − e
x arctgy.
(DS f0
x= 1
y − e
x arctgy, f0
y = −x
y2 − e x
1 + y2)
13 f (x, y) = ln x +p
x2+ y2
(DS f0
x= p 1
x2+ y2, f0
x2+ y2 · y
p
x2+ y2)
T`ım da.o h`am riˆeng cu’a h`am ho..p sau dˆay (gia’ thiˆe´t h`am f(x, y)
kha’ vi)
14 f (x, y) = f (x + y, x2+ y2)
(DS f0
x = f0
t + f0
v 2x, f0
y = f0
t + f0
v 2y, t = x + y, v = x2+ y2)
15 f (x, y) = f x
y ,
y x
Trang 11
(DS f0
x = 1
y f
0
t− y
x2f0
v , f0
y = −x
y2 f0
t+ 1
x f
0
v , t = x
y , v =
y
x)
16 f (x, y) = f (x − y, xy).
(DS f x0 = f t0+ yf v0, f y0 = −f t0+ xf v0, t = x − y, v = xy)
17 f (x, y) = f (x − y2, y − x2, xy).
(DS f x0 = f t0− 2xf v0+ yf w0, f y0 = −2yf t0+ f v0 + xf w0,
t = x − y2, v = y − x2, w = xy)
18 f (x, y, z) = f (p
x2+ y2,p
y2+ z2,
√
z2+ x2)
(DS f x0 = xf
0
t
p
0
w
√
z2+ x2, f y0 = yf
0
t
p
0
v
√
x2+ z2,
f z0 = zf
0
v
p
0
w
√
z2+ x2, t =p
x2+ y2,
y2+ z2, w =
√
z2+ x2)
19 w = f (x, xy, xyz).
(DS f x0 = f t0+ yf u0 + yzf v0,
f y0 = xf u0 + xzf v0,
t = x, u = xy, v = xyz).
Trong c´ac b`ai to´an sau dˆay 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 (x2+ y2), y ∂f
∂f
21 f = x n f y
x2
, x ∂f
∂f
∂y = nf
22 f = yf (x2− y2), y2∂f
∂f
∂y = xyf
23 f = y
2
3x + f (x, y), x
2∂f
∂f
∂y + y
2
= 0
Trang 1224 f = x n f y
x α , z
x β
, x ∂f
∂f
∂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
∂x2, ∂
2
f
∂x∂y,
∂2f
∂y2 nˆe´u f = cos(xy)
(DS f00
xx = −y2cos xy, f00
xy = − sin xy − xy cos xy, f00
yy =
−x2cos xy)
27 T´ınh c´ac da.o h`am riˆeng cˆa´p hai cu’a h`am f = sin(x + yz).
(DS f xx00 = − sin t, f xy00 = −z sin t, f xz00 = −y sin t, f yy00 = −z2sin t,
f00
yz = −yz sin t, f00
zz = −y2sin t, t = x + yz)
28 T´ınh ∂
2
f
∂x∂y nˆe´u f =
p
x2+ y2e x+y
x+y
(x2+ y2)3/2
h
− xy + (x + y)(x2+ y2) + (x2+ y2)2
i )
29 T´ınh ∂
2
f
∂x∂y,
∂2f
∂y∂z,
∂2f
∂x∂z nˆe´u f = x
yz
(DS f xy00 = x yz−1 z(1 + yzlnx), f xz00 = x yz−1 y(1 + yzlnx),
f yz00 = lnx · x yz (1 + yzlnx))
30 T´ınh ∂
2
f
∂x∂y nˆe´u f = arctg
x + y
1 − xy. (DS.
∂2f
31 T´ınh f xx00 (0, 0), f xy00 (0, 0), f yy00(0, 0) nˆe´u
f (x, y) = (1 + x) m (1 + y) n
(DS f00
xx (0, 0) = m(m − 1), f00
xy (0, 0) = mn, f00
yy (0, 0) = n(n − 1))
32 T´ınh ∂
2r
∂x2 nˆe´u r =p
x2+ y2+ z2 (DS r
2− x2
33 T´ınh f00
xy , f yz00 , f xz00 nˆe´u f = x
y
z
(DS f xy00 = −z2y−2 xy−1z−1
, f xz00 = 1
y
x
y
z−1
1 + zln x
y
,
Trang 13yz = −1
y
x
y
z
·
1 + zln x
y
)
34 Ch´u.ng minh r˘a`ng ∂
2
f
∂2f
∂y∂x nˆe´u f = arc sin
r
x − y
T´ınh c´ac da.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, x2+ y2)
(DS u00xx = f tt00+ 4xf tv00 + 4x2f vv00 + 2f v0,
u00xy = f tt00+ 2(x + y)f tv00 + 4xyf vv00,
u00yy = f tt00+ 4yf tv00 + 4y2f vv00 + 2f v0,
t = x + y, v = x2 + y2.)
36 u = f
xy, x
y
(DS u00xx = y2f tt00+ 2f tv00 + 1
y2f vv00 ,
u00xy = xyf tt00− x
y3f vv00 + f t0− 1
y2f v0,
u00yy = x2f tt00− 2x
2
y2f tv00 + x
2
y4f vv00 + 2x
y3f v0,
t = xy, v = x
y )
37 u = f (sin x + cos y).
(DS u00xx = cos2x · f00− sin x · f0, u00xy = − sin y cos x · f00,
u00yy = sin2y · f00− cos y · f0)
38 Ch´u.ng minh r˘a`ng h`am
2a
√
πt e
−(x−x0 )2
4a2t
(trong d´o a, x0 l`a c´ac sˆo´) tho’a m˜an phu.o.ng tr`ınh truyˆ`n nhiˆe.te
∂f
2∂2f
∂x2 ·
Trang 1439 Ch´u.ng minh r˘a`ng h`am f = 1
r trong d´o
(x − x0)2+ (y − y0)2 + (z − z0)2 tho’a m˜an phu.o.ng tr`ınh Laplace:
∆f ≡ ∂
2f
∂x2 +∂
2f
∂y2 +∂
2f
∂z2 = 0, r 6= 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 lˆ` na kha’ vi)
40 u = f (x − at) + g(x + at), ∂
2u
∂t2 = a2∂2u
∂x2
41 u = xf (x + y) + yg(x + y), ∂
2
u
∂x2 − 2 ∂
2
u
∂2u
∂y2 = 0
42 u = f y
x
+ xg y
x
, x2∂
2
u
∂x2 + 2xy ∂
2
u
2∂2u
∂y2 = 0
43 u = x n f y
x
+ x 1−n g y
x
,
x2∂
2u
∂x2 + 2xy ∂
2u
2∂2u
∂y2 = n(n − 1)u.
44 u = f (x + g(y)), ∂u
∂x·
∂2u
∂u
∂y ·
∂2u
∂x2·
45 T`ım da.o h`am theo hu.´o.ng ϕ = 135◦
cu’a h`am sˆo´
f (x, y) = 3x4+ xy + y3
ta.i diˆe’m M(1, 2). (DS −
√ 2
2 )
46 T`ım da.o h`am cu’a h`am f(x, y) = x3 − 3x2y + 3xy2 + 1 ta.i diˆe’m
M (3, 1) theo hu.´o.ng t`u diˆe’m n`ay dˆe´n diˆe’m (6, 5). (DS 0)
47 T`ım da.o h`am cu’a h`am f(x, y) = lnp
x2+ y2 ta.i diˆe’m M(1, 1)
theo hu.´o.ng phˆan gi´ac cu’a g´oc phˆ` n tu th´a u nhˆa´t (DS
√ 2
2 )
Trang 1548. T`ım da.o h`am cu’a h`am f(x, y, z) = z2 − 3xy + 5 ta.i diˆe’m
M (1, 2, −1) theo hu.´o.ng lˆa.p v´o.i c´ac tru.c to.a dˆo nh˜u.ng g´oc b˘a`ng nhau
(DS −
√
3
3 )
49 T`ım da.o h`am cu’a h`am f(x, y, z) = ln(e x + e y + e z) ta.i gˆo´c to.a dˆo
v`a hu.´o.ng lˆa.p v´o.i c´ac tru.c to.a dˆo x, y, z c´ac g´oc tu.o.ng ´u.ng l`a α, β, γ.
(DS cos α + cos β + cos γ
50 T´ınh da.o h`am cu’a h`am f(x, y) = 2x2− 3y2 ta.i diˆe’m M(1, 0) theo
hu.´o.ng lˆa.p v´o.i tru.c ho`anh g´oc b˘a`ng 120◦ (DS −2)
51 T`ım da.o h`am cu’a h`am z = x2− y2 ta.i diˆe’m M0(1, 1) theo hu.´o.ng
vecto ~ e lˆ a.p v´o.i hu.´o.ng du.o.ng tru.c ho`anh g´oc α = 60◦ (DS 1 −
√ 3)
52. T`ım da.o h`am cu’a h`am z = ln(x2+ y2) ta.i diˆe’m M0(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 − sin3y + z + cotgz
ta.i diˆe’m M0
π
4,
π
3,
π
2
(DS (gradw)M = ~i + 3
8~j, cos α = √8
73, cos β =
3
√
73)
54 T`ım da.o h`am cu’a h`am w = arc sinp z
x2+ y2 ta.i diˆe’m M0(1, 1, 1) theo hu.´o.ng vecto
−→
M0M , trong d´ o M = (3, 2, 3) (DS. 1
6)
Trong mu.c n`ay ta x´et vi phˆan cu’a h`am nhiˆe`u biˆe´n m`a dˆe’ cho go.n ta
chı’ cˆ` n tr`ınh b`ay cho h`am hai biˆe´n l`a du’ Tru.`o.ng ho p sˆo´ biˆe´n l´o.na
ho.n hai du.o c tr`ınh b`ay ho`an to`an tu.o.ng tu