[微積] 台大95碩士班考試高微
4.
Let u and v be two real-valued C^1 functions on R^2 such that the gradient
▽u is never zero, and such that, at each point, ▽u and ▽v are linearly
dependent vectors. Given p = (x_0,y_0) belonging to R^2. Must there exist
a C^1 function F of one variable such that v(x,y) = F(u(x,y))?
5.
Given h : R → R a nonzero smooth function with compact support i.e. the
closure of {x : h(x)≠0} is compact. For ε > 0, let
∞ -K(x,y)/ε
∫ (x - y) e dy
-∞
u_ε(x) = ————————————— , for any x belonging to R,
∞ -K(x,y)/ε
∫ e dy
-∞
where K(x,y) = ((x-y)^2)/4 + h(y)/2 for x,y belonging to R.
(1) Can each u_ε be a smooth function with compact support?
(2) Can the limit lim u_ε(x) always exist? in what sense?
ε→0+
這兩題要怎麼著手? 有任何想法可以提供嗎? 感謝
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