テンソル積の成分計算

 a basis set of V
 m-dim vector space

 dual space of V
 dual basis vector of V^*

  i.e. 
  i.e. 

m, n-dim vector space

 canonical basis

  i.e. 
  i.e. 
  Rem. 

 tensor product of V and W

  i.e. 
       
  i.e. 

  i.e. 
  i.e. 

  i.e. 

  with 
  i.e.  suppose each vectors are col. orderd

  i.e. 
  i.e. 

  as 

 (r,s)-tensor space on V
  with 

 is algebra
 is derivative when
(i) T(V)の型を保ち，縮約と可換
(ii)  Leibniz rule
 is a vector space
and a lie-algebra with product 

 k-th exterior power, its elem. called k-form
  where skew-symmetric means : 
  with 

 is an algebra, called exterior algebra on V