Re: [問題] 有關梯度向量運算的問題

[JohnMash (John)]

※ 引述《zero97442 (liner)》之銘言：
: 之前看流力書上看到一個式子
  (V˙▽)V=(1/2)▽(V˙V)-V╳(▽╳V)

  This is my 4th time to do such problems.


  index repetition implies summation

  d_i means d/dx_i

  d(ij) means delta function

  e(ijk) is full antisymmetric function and e(123)=1


  [(V˙▽)V]_k=(V_i d_i)V_k=V_i (d_i V_k)                 (1)

  (1/2)[▽(V˙V)]_k=(1/2)d_k (V_i V_i)=V_i (d_k V_i)      (2)

  [V╳(▽╳V)]_k=e_{kij} V_i (▽╳V)_j

                =e_{kij} V_i e_{jmn} (d_m V_n)

                =[d(km) d(in)-d(kn) d(im)] V_i (d_m V_n)

                =V_i (d_k V_i) - V_i (d_i V_k)            (3)


  (2)-(3)= V_i (d_i V_k)=(1)

  Done.

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推 jimiras:推  簡潔
→ zero97442:謝謝
推 amozartea:用epsilon是簡潔　但初次證明還是把分量寫出來算過較好
推 DDMO:這個問題怎麼好像很眼熟@@?
→ JohnMash:if you are not used to epsilon,
→ JohnMash:you can not derive any complicate formulas.