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Prove-that-div-curlA-0-Help-




Question Number 186074 by Mastermind last updated on 31/Jan/23
Prove that div(curlA^− )=0      Help!
Provethatdiv(curlA)=0Help!
Answered by aleks041103 last updated on 30/Apr/23
Using einstein convention:  div(curl(v))=  =∂_i (ε_(ijk) ∂_j v_k )=  =ε_(ijk) ∂_i ∂_j v_k =  =−ε_(jik) ∂_i ∂_j v_k   Schwartz rule : ∂_i ∂_j =∂_j ∂_i   ⇒ε_(jik) ∂_i ∂_j v_k =ε_(jik) ∂_j ∂_i v_k   renaming i→j and j→i  ⇒ε_(jik) ∂_i ∂_j v_k =ε_(ijk) ∂_i ∂_j v_k =div(curl(v))  ⇒div(curl(v))=−div(curl(v))  ⇒div(curl(v))=0    Note:  ε_(ijk)  is the Levi−Civita symbol  where ε_(ijk) = { ((+1, (ijk) is an even perm. of (123))),((−1, (ijk) is an odd perm. of (123))),((0, if any two of i,j or k are equal)) :}  ⇒ε_(ijk)  is completely antisymmetric, i.e.  ε_(ijk) =−ε_(jik)   Also, it is true that:  a^→ =a_i e_i ^(→) , b^(→) =b_j e_j ^(→)   a^→ ×b^(→)  = ε_(ijk) a_i b_j  e_k ^(→)
Usingeinsteinconvention:div(curl(v))==i(εijkjvk)==εijkijvk==εjikijvkSchwartzrule:ij=jiεjikijvk=εjikjivkrenamingijandjiεjikijvk=εijkijvk=div(curl(v))div(curl(v))=div(curl(v))div(curl(v))=0Note:εijkistheLeviCivitasymbolwhereεijk={+1,(ijk)isanevenperm.of(123)1,(ijk)isanoddperm.of(123)0,ifanytwoofi,jorkareequalεijkiscompletelyantisymmetric,i.e.εijk=εjikAlso,itistruethat:a=aiei,b=bjeja×b=εijkaibjek

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