Abstract
This paper describes the variational construction of potentials for 3-dimensional vector fields on nice domains in space. Variational principles for the scalar and vector potentials in the Hodge-Weyl decomposition of fields in are first described. Then a description of orthonormal bases of the associated subspaces is developed. These bases are defined using various eigen-problems involving the scalar and vector Laplacian on the domain. The null eigenspaces of two of these problems define the deRham cohomology groups of the domain. The usual Helmholtz decomposition is also developed, including a description of its spectral expansion, for comparison purposes.
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