[微積] 台大95碩士班考試高微

[tasukuchiyan (Tasuku)]

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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推 znmkhxrw    :第一題給p幹嘛??哪裡用到了??
推 herstein    :問題應該是F的存在與p有關係
→ herstein    :題目大概沒PO完整
→ tasukuchiyan:題目照著試卷打的，給p點我也不懂是幹嘛
→ keroro321   :1.給定p點,應用反函數(x,y)<->(u,v) 就知道答案了
→ tasukuchiyan:但是▽u和▽v線性相依
→ keroro321   :喔喔 看錯 XDD