L
Lv 7
L asked in 科學數學 · 1 decade ago

一題 chain rule 的題目

u : |R^n -> |R ; O : 一個 n 階正交矩陣定義 △u = Σ_[1,n] u_ii (u_ii 表示 u(x) 對其第 i 個變數作偏微分兩次)若 u 滿足 △u = 0 , 證明 △u(O(x)) = 0這題我用 chain rule 硬作後面出現三個 Σ =.= 有較好的作法嗎 ? (也許可以利用正交矩陣的某種性質 ?)

2 Answers

Rating
  • Eric
    Lv 6
    1 decade ago
    Favorite Answer

    Laplacian operator: Δ =∇T∇Δ[u(Ox)] = ∇T∇[u(Ox)]= ∇T[OT∇u(Ox)] (chain rule)= OT∇T[∇u(Ox)]= OTO∇T∇u(Ox) (chain rule, (OT)T = O)= Δu(Ox) (OTO = I)= 0

    2006-10-17 07:52:04 補充:

    請參考 http://planetmath.org/encyclopedia/Gradient.html

  • L
    Lv 7
    1 decade ago

    請問一下,這些算子的公式和一些性質那些網頁有介紹 ? 我書上竟然沒附 ... 只好拿起 chain rule 把自己當苦力下去硬幹 ...

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