L
Lv 7
L asked in 科學數學 · 1 decade ago

問一個集合看不懂的表示法 ?

S^1 = xy 平面上的單位圓

但是我書上又寫 S^1 = [0,1] mod 1

這是什麼意思 @@?

4 Answers

Rating
  • Eric
    Lv 6
    1 decade ago
    Favorite Answer

    這與代數有關。The unit circle (單位圓) in R2S1 = {(x,y) ∈ R2 : x2+y2 = 1}can be identified with the unit circle in C,T = {z ∈ C: |z| = 1},which forms a group (群) when endowed with the group operation (群運算) of complex multiplication.The author is further identifying, through an isomorphism (同構), the unit circle with the group[0,1] mod 1,that is, the set [0,1) (or R/Z) with addition modulo 1 as its group operation. This can be established by observing the isomorphismt ↦ e2πitbetween [0,1] mod 1 and T.Hence,S1 ≅ T ≅ [0,1] mod 1where ≅ reads "is isomoprhic to".

  • L
    Lv 7
    1 decade ago

    我真想哭 ...

    誰可以告訴我為什麼動態系統會把實變,線代,機率論,微分幾何,代數,和程式給通通攪在一起嗎

  • 1 decade ago

    R/Z 是較一般的寫法。

    R/Z 不但有群的結構,同時也是光滑流形,且加減為可微運算,故本身是一李群。同樣地,S^1 也是李群。

    t ---> exp(2πit)

    這個函數不但是 group isomorphism,且本身及其反函數為兩李群間的可微函數,因此,題目中的等號可視為兩者在李群的結構上是一樣的。

  • 1 decade ago

    問個笨問題?

    [0,1] mod 1

    是啥?? 閉區間? 還是純粹記號?

    沒有看過~~

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