※ 引述《subtropical (風大雨大)》之銘言：
: 1.
: f(x,y) = 0 , if (x,y)=(0,0)
: x^3
: --------------- , other
: (x^2+y^2)^(1/2)
: 求: f在x in R 可全微分
fx(0,0)=lim 1/h * {f(h,0)-f(0,0)}= lim (1/h) * (h^3/h) = 0
h->0
fy(0,0)=lim 1/k * {f(0,k)-f(0,0)}= lim (1/k) * (0/k) = 0
k->0
=> epsilon((h,k)) = f(h,k) - f(0,0) - (1,0)*(h,k)
= (h^3 / sqrt(h^2+k^2)) -h
let h = rcost k = rsint
=> r(r(cost)^3 - cost )
然後就卡住了....
: 2.
: f(x,y) = 1, if exist t in R, t /= 0 with (x,y) = (t,t^2)
: 0, other
: 求: f在(0,0)不可全微分 但Directional derivative皆存在
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