偏导数概念与几何意义

y
f (x, y )
(x0 , y0 ) fx (x, y )
x (x0 , y0 ) y (x0 , y0 ) .
fx (x0 , y0 ) ; fy (x0 , y0 ) .
f (x, y )
(x0 , y0 ) fx (x, y )
. u = f (x, y, z ) : fx (x, y, z ) = lim f (x + ∆x, y, z ) − f (x, y, z ) . ∆x→0 ∆x (x, y, z ) x
4.
xy , x2 + y 2 = 0 2 + y , f (x, y ) = 2 2 0, x +y =0 fx (0, 0), fy (0, 0). x2 fx (0, 0) = lim f (0 + ∆x, 0) − f (0, 0) ∆x→0 ∆x 0−0 = 0, = lim ∆x→0 ∆x , fy (x, y ) = 0.
P V = RT (R ), : ∂P ∂V ∂T · · = −1. ∂V ∂T ∂P ∂P : , ∂V dy dx . ∂y ∂x : P =
3.
dy dx .
RT ∂P RT ,⇒ =− 2, V ∂V V RT ∂V R PV ∂T V V = ,⇒ = ; T = ,⇒ = ; P ∂T P R ∂P R ∂P ∂V ∂T RT R V RT ∴ · · =− 2 · · =− = −1. ∂V ∂T ∂P V P R PV
, x f (x, y0 ) , y0 , , . y , x0 fy (x0 , y0 ) . x
f (x, y ) y = y0 .
(x0 , y0 ) ,
f (x0 , y )
1.
f (x, y ) = e−x sin(x + 2y ),
fx (0, π 4 ),
fx (0, π 4 ). : fx (x, y ) = −e−x sin(x + 2y ) + e−x cos(x + 2y ), fy (x, y ) = 2e−x cos(x + 2y ) π π π ∴ fx (0, ) = − sin + cos = −1, 4 2 2 π π fy (0, ) = 2 cos = 0. 4 2
z = f (x, y ),
∂z ∂y M
= lim
f (x0 ,y0 +∆y )−f (x0 ,y0 ) ∆y ∆y →0
x = x0 , z = f (x, y ) :L: x=x 0 : ∂z ∂y = tan β
M

x, y
, ∴ f (x, y ) : , . : ,
(x,y )→(0,0)
lim
f (x, y ) .
,
(0, 0) ,
,
f (x, y )
(x0 , y0 ) .
. z = f (x, y ),
∂z ∂x M
= lim
f (x0 +∆x,y0 )−f (x0 ,y0 ) ∆x ∆x→0
3.1
. z = f (x, y ) 3.1 z = f (x, y ) N (M0 ) ∆x , lim , y M0 (x0 , y0 ) M0 (x0 , y0 ) y0 x x0
f (x0 + ∆x, y0 ) − f (x0 , y0 ).
x
f (x0 + ∆x, y0 ) − f (x0 , y0 ) , ∆x→0 ∆x z = f (x, y ) M0 (x0 , y0 ) ∂z ∂f , , fx (x0 , y0 ) . ∂x M0 ∂x M0
, z = f (x, y )
M0 (x0 , y0 )
y
f (x0 , y0 + ∆y ) − f (x0 , y0 ) , ∆y →0 ∆y lim ∂z ∂y , fy (x0 , y0 )
M0
∂f ∂y
.
M0
.
z = f (x, y ) z = f (x, y )
D D (x, y ) x, y
y = y0 , z = f (x, y ) :L: y=y 0 : ∂z ∂x = tan α
M
z = f (x, y ),
∂z ∂y M
= lim
f (x0 ,y0 +∆y )−f (x0 ,y0 ) ∆y ∆y →0
x = x0 , z = f (x, y ) :L: x=x 0 :
2. (1) z = xy : (1) (2) ∂z = yxy−1 , ∂x (2) z = arctan ∂z = xy ln x. ∂y y x
∂z 1 y y = , y 2 (− 2 ) = − 2 ∂x 1 + ( x ) x x + y2
∂z 1 1 x = ( ) = . y 2 ∂y 1 + (x ) 1 x2 + y 2
x ,
,
z = f (x, y ) x ∂z ∂f , fx (x, y ) . , ∂x ∂x f (x + ∆x, y ) − f (x, y ) , ∴ fx (x, y ) = lim ∆x→0 ∆x (x, y ) ∈ D.
来自百度文库 , ,
z = f (x, y ) ∂z ∂f , fy (x, y ) . ∂y ∂y