Based on a generalized Helmholtz's identity, definitions of an irrotational vector and a solenoidal vector are reviewed, and new definitions are presented. It is pointed out that the well-known uniqueness theorem of a vector function is incomplete. Although the divergence and curl are specified, for problems with finite boundary surfaces, normal components are not sufficient for uniquely determininga vector function. A complete uniqueness theorem and its two corollaries are then presented. It is proven that a vector function can be uniquely determined by specifyingits divergence and curl in the problem region, its value (both normal and tangential components) on the boundary.
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