D Rn =
(x1, x2, . . . , xn) " D f( ) = y " R
n- Rn n f : D R
f( ) f(x1, x2, . . . , xn) = (x1, x2, . . . , xn)
D
(x, f(x)) R2
n- (x1, x2, . . . , xn, f(x1, x2, . . . , xn))
Rn+1 R3
(x1, x2, . . . , xn) f f( ) = c
OXY
(x1, x2, . . . , xn)
R
 f : D R " D f
0 0
lim 0 f( ) = f( )
0
f D
f [ , ] ‚" D
1 2
f( ) < c < f( ) " [ , ] f( ) = c
1 2 1 2
f( ) =
c0 + c1x1 + c2x2 + . . . + cnxn
n n
f : Rn R n-1
xk xk h = "xk
f : Rn R y =
f(x1, x2, . . . , xn) xk (x10, x20, . . . , xk0, . . . , xn0)
f(x10, x20, . . . , xk0 + h, . . . , xn0) - f(x10, x20, . . . , xk0, . . . , xn0)
lim
h0 h
"f( )
=
0
"xk
 xk
x0
0
xk
f : Rn R " D
0
"f( ) "f( ) "f( )
0 0 0
. . .
"x1 "x2 "xn
"f( ) gradf( )
0 0 0
0
"f( ) =
0 0
"f
f : Rn R
0
"xi
"f
"( )
"xi
xj
"xj
"2f
f xi xj
0
"xi"xj
"2f
n ( )
0
"xi"xj 0
"2f "2f "2f
i = j ( ) = ( ) ( )
"xi"xj 0 "xi"xi 0 "x2 0
i
f : Rn R
îÅ‚ Å‚Å‚
"2f "2f "2f
( ) ( ) . . . ( )
ïÅ‚
"x2 0 "x1"x2 0 "x1"xn 0 śł
ïÅ‚ śł
1
ïÅ‚ śł
"2f "2f "2f
ïÅ‚
( ) ( ) . . . ( )śł
ïÅ‚
"x2"x1 0 "x2 0 "x2"xn 0 śł
Hf( ) =
ïÅ‚ śł
0
2
ïÅ‚ śł
ïÅ‚ śł
. . . . . . . . . . . .
ïÅ‚ śł
ðÅ‚ ûÅ‚
"2f "2f "2f
( ) ( ) . . . ( )
"xn"x1 0 "xn"x2 0 "x2 0
n
0
"2f "2f "2f "2f
=
0
"xi"xj "xj"xi "xi"xj "xj"xi
X ‚" Rn f : X R " X
0
"
"r>0"K( 0,r)‚"X" "K( 0,r) f( ) d" f( )
0
"
"r>0"K( 0,r)‚"X" "K( 0,r)\ 0 f( ) < f( )
0
"
"r>0"K( 0,r)‚"X" "K( 0,r) f( ) e" f( )
0
"
"r>0"K( 0,r)‚"X" "K( 0,r)\ 0 f( ) > f( )
0
 X ‚" R2
f : X R
" X "f( ) =
0 0
X ‚"
R2 f : X R " X "f( ) =
0 0
"2f "2f "2f
" f > 0 -
0
"x1"x1 "x1"x1 "x2"x2
2
"2f
> 0
"x1"x2
"2f
" f < 0
0
"x1"x1
2
"2f "2f "2f
- > 0
"x1"x1 "x2"x2 "x1"x2
2
"2f "2f "2f "2f
= 0 - = 0
"x1"x1 "x1"x1 "x2"x2 "x1"x2
f
"xi
"y "xi
÷
y xi
y = f(x)
"y "x x "y x "y x
lim ÷ = lim = lim = f (x)
"x0 y x "x0 y "x y "x0 "x y
"x 0
f "f
x x
y = f(x) x Ex = f (x)f(x) =
"x f(x)
f
Ex = -2 x 1%
f
f(x) 2% Ex = 3 x 1%
f(x) 3%
y = f(x1, x2, . . . , xn)
f "f(x1,x2,...,xn)
xi
Exi = fxi(x1, x2, . . . , xn)f(x1,xxi =
,...,xn) "xi f(x1,x2,...,xn)
2
 x1
a1 a2
Y = AX1 X2
X1
Y a1-1 a2
EX1 = a1AX1 X2 a1 a2 = a1
AX1 X2
X2
Y a1 a2-1
EX2 = a2AX1 X2 a1 a2 = a2
AX1 X2
f : D R D ‚" Rn
f " D
0
" "D f( ) d" f( ) (" "D f( ) e" f( ))
0 0
y = f( )
y = f( ) D
D
D
 y = f( ) D
D
y = f( ) D
D