"
"
"
"
 p
p (a1, a2, . . . , an) ai " R i = 1, 2, . . . n
ai
p
p
Rp A = (a1, a2, . . . , ap)
R2
A = (a1, a2)
a1 a2 A
R3
a1, a2, a3 A =
(a1, a2, a3)
A
R2 f
P = (x, y) A z
z = f(x, y).

1
z = , z = x2 - y2
x - y
z = f(x, y)
(x0, y0) x
f(x0 + h, y0) - f(x0, y0)
lim .
h0
h
z = f(x, y) (x0, y0)
y
f(x0, y0 + h) - f(x0, y0)
lim .
h0
h
z = f(x, y) x
"z "f(x, y)

, zx, fx(x, y),
"x "x
 y
"z "f(x, y)

, zy, fy(x, y).
"y "y
x
x y
y x
C1
C1
z = 3x2y2 - x cos y
"
y
z = x
z = ln (ex + ey)
"z "z
+ = 1
"x "y
x
y2
z = e
"z "z
2x + y = 0
"x "y
z = f(x, y) C1 (x0, y0)
df x = x - x0 y = y - y0

df( x, y) := fx(x0, y0) x + fy(x0, y0) y.
z = f(x, y) C1
f(x0 + x, y0 + y) H" f(x0, y0) + df( x, y).
(1.02)3.01

(6.2)2 + (8.1)2
8.04
2.02
 z = f(x, y)

fx(x, y) fy(x, y)
z = f(x, y)

"2f " "f "2f " "f

fxx = = , fxy = = ,
"x2 "x "x "x"y "y "x

"2f " "f "2f " "f

fyx = = , fyy = = ,
"y"x "x "y "y2 "y "y
"x2 "x"x

fxx fyy

fxy fyx
C2
C2
f C2
z = ln(x2 +
y)
z = f(x, y) C1 (x0, y0)

fx(x0, y0) = 0 fy(x0, y0) = 0.
z = f(x, y) C2

fx(x0, y0) = 0 fy(x0, y0) = 0,



fxx(x0, y0) fxy(x0, y0)

W (x0, y0) = > 0,


fyx(x0, y0) fyy(x0, y0)
z = f(x, y) (x0, y0)

fxx(x0, y0) > 0,

fxx(x0, y0) < 0,
 W (x0, y0) < 0 z = f(x, y)
(x0, y0) W (x0, y0) = 0
(x0, y0) z = x4 + y4
(0, 0) z = x3 + y2 (0, 0)
f(x, y) = 3x2y - x3 - y4.
32m3