Re: [微分]偏導數
※ 引述《rebe212296 (綠豆冰)》之銘言：
: 1.
: Let f:R^2→R be defined by f(x,y)=√|xy| for all (x.y)屬於R^2
: Show that f_x and f_y both exist on R^2 and continuous on R^2\{(0,0)}
: but f is not differentable at (0,0).
: 2.
: Let f:R^2→R be defined by f(x,y)=[xy(x^2-y^2)]/(x^2+y^2) if(x,y)≠(0,0)
: and f(x,y) = 0 if(x,y)=(0,0)
: Show that f_xy(0,0) and f_yx(0,0) both exist but are not equal.
: 請問以上兩題怎麼解? 謝謝!
f(x,0) - f(0,0)
2. f (0,0) = lim -----------------
x x→0 x - 0
0 - 0
= lim ------- = lim 0 = 0
x→0 x x→0
f(0,y) - f(0,0)
f (0,0) = lim -----------------
y y→0 y - 0
0 - 0
= lim ------- = lim 0 = 0
y→0 y y→0
x≠0
f(x,y) - f(x,0)
f (x,0) = lim -----------------
y y→0 y - 0
xy(x^2 - y^2)
--------------- - 0
x^2 + y^2
= lim ---------------------
y→0 y - 0
(x)(x^2 - y^2) (x)(x^2 - 0)
= lim ---------------- = -------------- = x
y→0 x^2 + y^2 x^2
y≠0
f(x,y) - f(0,y)
f (0,y) = lim -----------------
x x→0 x - 0
xy(x^2 - y^2)
--------------- - 0
x^2 + y^2
= lim ---------------------
x→0 x - 0
y(x^2 - y^2) y(0 - y^2)
= lim -------------- = ------------ = -y
x→0 x^2 + y^2 y^2
f (0,y) - f (0,0)
x x
f (0,0) = lim --------------------
xy y→0 y - 0
-y - 0
= lim -------- = lim -1 = -1
y→0 y - 0 y→0
f (x,0) - f (0,0)
y y
f (0,0) = lim -------------------
yx x→0 x - 0
x - 0
= lim ------- = lim 1 = 1
x→0 x - 0 x→0
∴ f (0,0) and f (0,0) both exist but are not equal
xy yx
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