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Helmholtz Theorem

Statement
A vector field is uniquely described within a region by its divergence and curl.
 
Explanation If the divergence and curl of any vector Description: 74215.png are given as,
 
Description: 74206.png and Description: 74197.png
 
where Description: 74188.png is the source density of Description: 74179.png and Description: 74172.png is the circulation density of Description: 74165.png, both vanishing at infinity, then, according to the Helmholtz theorem, we can write a vector field Description: 74156.png as a sum of a component Description: 74148.png whose divergence is zero, Description: 74141.png (solenoidal), and a component Description: 74133.png whose curl is zero, Description: 74125.png (irrotational).
 
Description: 74116.png
Also, we know that the divergence of curl of a vector is zero and the curl of the gradient of any scalar is zero. Using these two null identities, we can write Description: 74106.png and Description: 74097.png as follows.
 
Description: 74088.png and Description: 74080.png
 
where Description: 74072.png and Description: 74063.png are vector and scalar quantities respectively.
 
Thus, any vector can be represented by the Helmholtz theorem as,
Description: 74056.png
 




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