Three-vector and scalar field identities and uniqueness theorems in Euclidean and Minkowski spaces

Euclidean three-space and Minkowski four-space identities and uniqueness theorems are reviewed and extended. A Helmholtz identity is used to prove two three-vector uniqueness theorems in Euclidean three-space. The first theorem specifies the divergence and curl of the vector, and the second is a Helmholtz type theorem that sums the irrotational and solenoidal parts of the vector. The second theorem is shown to be valid for three-vector fields that are time dependent. A time-dependent extension of the Helmholtz identity is also derived. However, only the three-vector and scalar components of a Minkowski space four-vector identity are shown to yield two identities that lead to a uniqueness theorem of the first or source type. Also, the field equations of this latter theorem appear to be sufficiently general such that the field equations naturally divide into two distinct classes, a four-solenoidal electromagnetic type class in a relativistic transverse gauge and a four-irrotational class in a relativistic longitudinal gauge.