Calculus/Helmholtz Decomposition Theorem

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Helmholtz Decomposition Theorem

The Helmhortz Decomposition Theorem, regarded as the fundamental theorem of vector calculus, dictates that any vector field can be expressed as the sum of a conservative vector field and a divergence free vector field : .


Approach #1Edit

Given a vector field  , the vector field   has the same divergence as  , and is also conservative:   and  . The vector field   is divergence free.

Therefore   where   and  . Vector field   is conservative and   is divergence free.


Approach #2Edit

Given a vector field  , the vector field   has the same curl as  , and is also divergence free:   and  . The vector field   is conservative.

Therefore   where   and  . Vector field   is conservative and   is divergence free.


Approach #3Edit

The Helmholtz decomposition can be derived as follows:

Given an arbitrary point  , the divergence of the vector field   is   where   is the Dirac delta function centered on   (The subscript   clarifies that   as opposed to   is the parameter that the differential operator is being applied to). Since  , it is the case that  

Alongside the identities  , and  , and most importantly  , the following can be derived:

     

 

 

 

  is the gradient of a scalar field, and so is conservative.

  is the curl of a vector field, and so is divergence free.

In summary,   where   is conservative and   is divergence free.