# 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  are given as,

and

where  is the source density of  and  is the circulation density of , both vanishing at infinity, then, according to the Helmholtz theorem, we can write a vector field  as a sum of a component  whose divergence is zero,  (solenoidal), and a component  whose curl is zero,  (irrotational).

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  and  as follows.

and

where  and  are vector and scalar quantities respectively.

Thus, any vector can be represented by the Helmholtz theorem as,