[微積分] 17.7 46題

[steve1012 (steve)]

好像很多人不會算
這題不要直接做太麻煩了

以下是jacobian的算法


set x=au y=bt  z=cv


the the original ellipsoid can transfer to a sphere

 2    2    2
u  + y  + z  =1

我用p代表partail的符號

     partial(x,y,z)        | px/pu  px/pt  px/pv |
J = -------------------- = | py/pu  py/pt  py/pv |
     partial(u,t,v)        | pz/pu  pz/pt  pz/pv |


  | a  0  0 |
= | 0  b  0 |  =  abc
  | 0  0  c |


so  ∫∫∫ dzdydx  = ∫∫∫ |J|dudtdv


   =  ∫∫∫  abcdudtdv   =  abc ∫∫∫dudtdv

transfer to sphere coordinate

       2π π  1    2
  =abc ∫  ∫ ∫   r sinφdr dφdθ
        0    0  0

it's obvious that the result of the integration is the volume of a sphere

which is 4π/3  so multiply by abc we have the volume of ellipsoid
      4
  =  --- πabc
      3



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