2. Gi其ng vi捉n: Phan 則淡c
Tun
畉i S畛 Tuy畉n T鱈nh
則7: D畉ng Ton ph動董ng
2 '' 2 2
0 0 0 0 0 0 0 0
2 2
( , ) ( , ) 2 "( , ) "( , )
2
xx xy yyd f x y f x y dx f x y dxdy f x y dy
Adx Bdxdy Cdy
= + +
= + +
Khi t狸m c畛c tr畛 c畛a hm 2 bi畉n bi to叩n s畉
d畉n 畉n vi畛c x叩c 畛nh d畉u c畛a vi ph但n c畉p 2
c畛a hm f, ngh挑a l ta c畉n x叩c 畛nh d畉u c畛a:
Khi x辿t hm 3 bi畉n th狸 ta c畉n x叩c 畛nh
d畉u c畛a vi ph但n c畉p 2:
2 2 2 2
11 12 13 22 23 332 2 2d f a dx a dxdy a dxdz a dy a dydz a dz= + + + + +
3. Gi其ng vi捉n: Phan 則淡c
Tun
畉i S畛 Tuy畉n T鱈nh
則7: D畉ng Ton ph動董ng
T畛ng qu叩t cho hm nhi畛u bi畉n th狸 vi畛c t狸m
d畉u c畛a vi ph但n c畉p 2 kh担ng 董n gi畉n, do
v畉y D畉ng ton ph動董ng l m畛t l箪 thuy畉t
h畛 tr畛 cho vi畛c t狸m d畉u c畛a vi ph但n c畉p 2
c畛a hm nhi畛u bi畉n.
4. Gi其ng vi捉n: Phan 則淡c
Tun
畉i S畛 Tuy畉n T鱈nh
則7: D畉ng Ton ph動董ng
畛nh ngh挑a: Cho V l kh担ng gian vector n
chi畛u tr棚n R, hm
x叩c 畛nh nh動 sau: v畛i m畛i
:V R
1 2( , ,..., )nx x x x V=
5. Gi其ng vi捉n: Phan 則淡c
Tun
畉i S畛 Tuy畉n T鱈nh
則7: D畉ng Ton ph動董ng
2
11 1 12 1 2 13 1 3 1 1
2
22 2 23 2 3 2 2
2
33 3 3 3
2
( ) 2 2 ... 2
2 ... 2
... 2
....................
n n
n n
n n
nn n
x a x a x x a x x a x x
a x a x x a x x
a x a x x
a x
= + + + +
+ + + +
+ + +
+
動畛c g畛i l 糸畉n乙 ton ph動董ng tr棚n V.
6. Gi其ng vi捉n: Phan 則淡c
Tun
畉i S畛 Tuy畉n T鱈nh
則7: D畉ng Ton ph動董ng
V鱈 d畛: Cho 糸畉n乙 ton ph動董ng:
3
1 2 3
2
1 1 2 1 3
2
2 2 3
2
3
2 2 2
1 1 2 1 3 2 2 3 3
: , ( , , )
( ) 2 4 6
2
8
2 4 6 2 8
R R x x x x
x x x x x x
x x x
x
x x x x x x x x x
=
= +
+
+
= + + +
11a 122a 132a 22a 232a 33a
7. Gi其ng vi捉n: Phan 則淡c
Tun
畉i S畛 Tuy畉n T鱈nh
則7: D畉ng Ton ph動董ng
畛nh ngh挑a: Cho 糸畉n乙 ton ph動董ng
2
11 1 12 1 2 13 1 3 1 1
2
22 2 23 2 3 2 2
2
33 3 3 3
2
( ) 2 2 ... 2
2 ... 2
... 2
....................
n n
n n
n n
nn n
x a x a x x a x x a x x
a x a x x a x x
a x a x x
a x
= + + + +
+ + + +
+ + +
+
khi 坦, ma tr畉n sau:
8. Gi其ng vi捉n: Phan 則淡c
Tun
畉i S畛 Tuy畉n T鱈nh
則7: D畉ng Ton ph動董ng
G畛i l ma tr畉n c畛a 糸畉n乙 ton ph動董ng
11 12 1
12 22 2
1 2
...
...
... ... ... ...
...
n
n
n n nn
a a a
a a a
A
a a a
錚 錚
錚 錚
錚 錚=
錚 錚
錚 錚
錚 錚削０ 錚
9. Gi其ng vi捉n: Phan 則淡c
Tun
畉i S畛 Tuy畉n T鱈nh
則7: D畉ng Ton ph動董ng
V鱈 d畛: Cho 糸畉n乙 ton ph動董ng
3
1 2 3
2 2 2
1 1 2 1 3 2 2 3 3
: , ( , , )
( ) 2 4 6 2 8
R R x x x x
x x x x x x x x x x
=
= + + +
Khi 坦, ma tr畉n c畛a 糸畉n乙 ton ph動董ng l:
2 2 3
2 1 1
3 1 8
A
錚 錚
錚 錚= 錚 錚
錚 錚財錚 錚
10. Gi其ng vi捉n: Phan 則淡c
Tun
畉i S畛 Tuy畉n T鱈nh
則7: D畉ng Ton ph動董ng
Bi t畉p: T狸m ma tr畉n c畛a 糸畉n乙 ton ph動董ng sau:
2 2 2
1 2 3 1 1 2 2 2 3 3( , , ) 6 3 4 5x x x x x x x x x x = + +
1 3 0
3 3 2
0 2 5
A
錚 錚
錚 錚= 錚 錚
錚 錚財錚 錚
11. Gi其ng vi捉n: Phan 則淡c
Tun
畉i S畛 Tuy畉n T鱈nh
則7: D畉ng Ton ph動董ng
Bi t畉p: T狸m ma tr畉n c畛a 糸畉n乙 ton ph動董ng sau:
2 2 2
1 2 3 1 2 1 3 2 3( ) 3 7 3 8 10 8x x x x x x x x x x = + +
3 4 5
4 7 4
5 4 3
A
錚 錚
錚 錚= 錚 錚
錚 錚財 錚 錚
12. Gi其ng vi捉n: Phan 則淡c
Tun
畉i S畛 Tuy畉n T鱈nh
則7: D畉ng Ton ph動董ng
V鱈 d畛: Cho 糸畉n乙 ton ph動董ng c坦 ma tr畉n:
2 2 2
1 2 3 1 2 1 3 2 3( ) 4 5 4 6 2x x x x x x x x x x = + + +
1 2 3
2 4 1
3 1 5
A
錚 錚
錚 錚= 錚 錚
錚 錚財錚 錚
Khi 坦, 糸畉n乙 ton ph動董ng t動董ng 畛ng l:
14. Gi其ng vi捉n: Phan 則淡c
Tun
畉i S畛 Tuy畉n T鱈nh
則7: D畉ng Ton ph動董ng
D畉ng ch鱈nh t畉c c畛a 糸畉n乙 ton ph動董ng
Khi ma tr畉n c畛a 糸畉n乙 ton ph動董ng l ma tr畉n
ch辿o
錚
錚
錚
錚
錚
錚
錚
錚
錚
錚
錚
錚
錚
錚
nna
a
a
000
............
0...0
0...0
22
11
15. Gi其ng vi捉n: Phan 則淡c
Tun
畉i S畛 Tuy畉n T鱈nh
則7: D畉ng Ton ph動董ng
....)( 22
222
2
111 nnn xaxaxax +++= Hay
Th狸 ta g畛i 坦 l 糸畉n乙 ch鱈nh t畉c c畛a 糸畉n乙
ton ph動董ng.
16. Gi其ng vi捉n: Phan 則淡c
Tun
畉i S畛 Tuy畉n T鱈nh
則7: D畉ng Ton ph動董ng
動a 糸畉n乙 ton ph動董ng v畛 糸畉n乙 ch鱈nh t畉c
Ph動董ng ph叩p Lagrange (xem ti li畛u)
V鱈 d畛: 動a 糸畉n乙 ton ph動董ng sau v畛 糸畉n乙
ch鱈nh t畉c.
2 2 2
1 2 3 1 2 1 3 2 3( ) 2 10 2 4 8x x x x x x x x x x = + + +
17. Gi其ng vi捉n: Phan 則淡c
Tun
畉i S畛 Tuy畉n T鱈nh
2 2 2 2 2
2 2 2 2 2
( ) 2 2
( ) 2 2
a b a ab b a b ab
a b a ab b a b ab
+ = + + = + +
= + = +
2 2 2 2
2 2 2 2
( ) 2 2 2
( ) 2 2 2
a b c a b c ab ac bc
a b c a b c ab ac bc
+ = + + +
= + + +
18. Gi其ng vi捉n: Phan 則淡c
Tun
畉i S畛 Tuy畉n T鱈nh
則7: D畉ng Ton ph動董ng
2 2 2
1 2 3 1 2 1 3 2 3
2 2 2
1 2 3 2 3 2 3
2 2 2
1 2 3 2 3 3
( ) 2 10 2 4 8
( 2 ) 6 4
( 2 ) ( 2 ) 2
x x x x x x x x x x
x x x x x x x
x x x x x x
= + + +
= + + +
= + + +
1 1 2 3
2 2 3
3 3
2
2
y x x x
y x x
y x
= +
=
=
2 2 2
1 2 3( ) 2y y y y = + +
畉t
19. Gi其ng vi捉n: Phan 則淡c
Tun
畉i S畛 Tuy畉n T鱈nh
則7: D畉ng Ton ph動董ng
V鱈 d畛: 動a DT ph動董ng sau v畛 糸畉n乙 ch鱈nh t畉c:
2 2 2
1 2 3 1 2 1 3 2 3( ) 6 13 4 6 2x x x x x x x x x x = + + +
2
1( )x= 22x+ 33x 2
22x+ 2
34x+ 2 310x x+
2 2 2
1 2 3 2 3 2 3( 2 3 ) 2[ 2 5 ]x x x x x x x= + + + +
2 2
1 2 3 2( 2 3 ) 2[( ) ]x x x x= + + 3
5
2
x+ 2
3
17
4
x
2 2 2
1 2 3 2 3 3
5 17
( 2 3 ) 2( )
2 2
x x x x x x= + + +
2 2 2
1 2 3
17
2
2
y y y= +
20. Gi其ng vi捉n: Phan 則淡c
Tun
畉i S畛 Tuy畉n T鱈nh
則7: D畉ng Ton ph動董ng
V鱈 d畛: 動a DT ph動董ng sau v畛 糸畉n乙 ch鱈nh t畉c:
2 2 2
1 2 3 1 2 1 3 2 3( ) 4 4 2x x x x x x x x x x = + + + +
2
1( )x= 22x+ 3x 2 35x x+
1 1 2 3
2 3 3 2
2 3
2
,
2 2
y x x x
x x x x
y y
= +
+
= =
2 2 2
1 2 3( ) 5 5y y y y = +
畉t
21. Gi其ng vi捉n: Phan 則淡c
Tun
畉i S畛 Tuy畉n T鱈nh
則7: D畉ng Ton ph動董ng
Bi t畉p: 動a 糸畉n乙 ton ph動董ng sau v畛 糸畉n乙
ch鱈nh t畉c b畉ng ph動董ng ph叩p Lagrange:
2 2 2
1 2 3 1 2 1 3 2 3( ) 5 10 4 8 2x x x x x x x x x x = + + +
2
1 )(x=
22. Gi其ng vi捉n: Phan 則淡c
Tun
畉i S畛 Tuy畉n T鱈nh
則7: D畉ng Ton ph動董ng
動a 糸畉n乙 ton ph動董ng v畛 糸畉n乙 ch鱈nh t畉c
Ph動董ng ph叩p Jacobi (xem ti li畛u)
V鱈 d畛: 動a 糸畉n乙 ton ph動董ng sau v畛 糸畉n乙
ch鱈nh t畉c.
2 2 2
1 2 3 1 2 1 3 2 3( ) 2 3 2 4 6x x x x x x x x x x = + +
錚
錚
錚
錚
錚
錚
錚
錚
錚
錚
=
132
331
212
A
23. Gi其ng vi捉n: Phan 則淡c
Tun
畉i S畛 Tuy畉n T鱈nh
則7: D畉ng Ton ph動董ng
錚
錚
錚
錚
錚
錚
錚
錚
錚
錚
=
132
331
212
A
,2111 == aD0 1,D = 11 12
2
21 22
2 1
5,
1 3
a a
D
a a
= = =
畉t
11 12 13
3 21 22 23
31 32 33
2 1 2
1 3 3 35,
2 3 1
a a a
D a a a
a a a
= = =
24. Gi其ng vi捉n: Phan 則淡c
Tun
畉i S畛 Tuy畉n T鱈nh
則7: D畉ng Ton ph動董ng
N畉u th狸 糸畉n乙 ton ph動董ng
c坦 糸畉n乙 ch鱈nh t畉c l:
2 2 20 1 2
1 2 3
1 2 3
( )
D D D
y y y y
D D D
= + +
2 2 2
1 2 3
2 5 35
( )
1 2 5
y y y y
= + +
,...2,1,0 = iDi
25. Gi其ng vi捉n: Phan 則淡c
Tun
畉i S畛 Tuy畉n T鱈nh
則7: D畉ng Ton ph動董ng
Bi t畉p: 動a 糸畉n乙 ton ph動董ng sau v畛 糸畉n乙
ch鱈nh t畉c b畉ng ph動董ng ph叩p Jacobi:
2 2 2
1 2 3 1 2 1 3 2 3( ) 2 3 4 2 8x x x x x x x x x x = + +
錚
錚
錚
錚
錚
錚
錚
錚
錚
錚
=
341
422
121
A
![Gi其ng vi捉n: Phan 則淡c
Tun
畉i S畛 Tuy畉n T鱈nh
則7: D畉ng Ton ph動董ng
V鱈 d畛: 動a DT ph動董ng sau v畛 糸畉n乙 ch鱈nh t畉c:
2 2 2
1 2 3 1 2 1 3 2 3( ) 6 13 4 6 2x x x x x x x x x x = + + +
2
1( )x= 22x+ 33x 2
22x+ 2
34x+ 2 310x x+
2 2 2
1 2 3 2 3 2 3( 2 3 ) 2[ 2 5 ]x x x x x x x= + + + +
2 2
1 2 3 2( 2 3 ) 2[( ) ]x x x x= + + 3
5
2
x+ 2
3
17
4
x
2 2 2
1 2 3 2 3 3
5 17
( 2 3 ) 2( )
2 2
x x x x x x= + + +
2 2 2
1 2 3
17
2
2
y y y= +](https://image.slidesharecdn.com/07dangtoanphuong-160420161652/85/07-dang-toan-phuong-19-320.jpg)
