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,
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.
where and are vector and scalar quantities respectively.
Thus, any vector can be represented by the Helmholtz theorem as,