to be compact. Since this is not true of R3, the Hodge decomposition theorem is not strictly a generalization of the Helmholtz theorem. However, the compactness restriction in the usual formulation of the Hodge decomposition can be replaced by suitable decay assumptions at infinity on the differential forms involved, giving a proper generalization of the Helmholtz theorem. Weak formulation The Helmholtz decomposition can also be generalized by reducing the regularity assumptions (the need for the existence of strong derivatives). Suppose is a bounded, simply-connected, Lipschitz domain. Every square-integrable vector field has an orthogonal decomposition: where is in the Sobolev space of square-integrable functions on whose partial derivatives defined in the distribution sense are square integrable, and , the Sobolev space of vector fields consisting of square integrable vector fields with square integrable curl. For a slightly smoother vector field , a similar decomposition holds: where . Longitudinal and transverse fields A terminology often used in physics refers to the curl-free component of a vector field as the longitudinal component and the divergence-free component as the transverse component. This terminology comes from the following construction: Compute the three-dimensional Fourier transform of the vector field . Then decompose this field, at each point k, into two components, one of which points longitudinally, i.e. parallel to k, the other of which points in the transverse direction, i.e. perpendicular to k. So far, we have Now we apply an inverse Fourier transform to each of these components. Using properties of Fourier transforms, we derive: Since and , we can get so this is indeed the Helmholtz decomposition. See also Clebsch representation for a related decomposition of vector fields Darwin Lagrangian for an application Poloidal–toroidal decomposition for a further decomposition of the divergence-free component . Scalar–vector–tensor decomposition Hodge theory generalizing Helmholtz decomposition Polar factorization theorem Helmholtz–Leray decomposition used for defining the Leray projection Notes References General references George B. Arfken and Hans J. Weber, Mathematical Methods for Physicists, 4th edition, Academic Press: San Diego (1995) pp. 92–93 George B. Arfken and Hans J. Weber, Mathematical Methods for Physicists – International Edition, 6th edition, Academic Press: San Diego (2005) pp. 95–101 Rutherford Aris, Vectors, tensors, and the basic equations of fluid mechanics, Prentice-Hall (1962), , pp. 70–72 References for the weak formulation R. Dautray and J.-L. Lions. Spectral Theory and Applications, volume 3 of Mathematical Analysis and Numerical Methods for Science and Technology. Springer-Verlag, 1990. V.