[分析] divergence與solenoidal這樣定義

[nomorethings (不可測之五度空間)]

 1                                                  2
H (Q)= the Soblev space formed by all functions in L (Q)
                                    2   m
       ,whose gradients belong to (L(Q))
 1                            ∞       1
H (Q)=the closure of the set C (Q) in H(Q)
 0                            0
 -1                        1
H  (Q)= the dual space of H(Q)
                           0
                              2   m                   -1
Then the DIVERGENCE of p in (L(Q))  is an element of H (Q)
                                                       1
and defined by :(div p,ψ)= -∫ p˙▽ψdx  for ψ in  H (Q)
                              Q                        0

The CURL of p is defined by :
                           partial ψ         partial ψ
(curl p, ψ)   = - ∫ (p  -----------   - p  ------------)
            ij      Q   i  partial x       j  partial x
                                    j                  i

                                                      1
and divp=o if :   ∫ p˙▽ψdx =0  for every ψ in  H (Q)
                   Q                                  0


also curl p = o if :
                           partial ψ         partial ψ
                   ∫ (p  -----------   - p  ------------) = 0
                    Q   i  partial x       j  partial x       ij
                                    j                  i
                                       1
                     for every ψ in  H (Q)
                                       0
-----
想請教這樣的定義在哪些領域或者在哪些書上可以找到如此定義的緣由呢？
或者願意直接告訴我原因的也很拜託>_<

我自己的理解是他類似 weak-derivative的想法去定義，把後面的ψ當成是
test function的功能。
然後覺得這樣想，很沒有感覺，所以還是來請教版上臥虎藏龍的各位m(_ _)m.

先謝謝了^^

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