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 #1[edit]

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 #2[edit]

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 #3[edit]

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.