In [9] Weyl discusses the problem of determining when a vector field i
s the gradient of some function. He introduces a method of orthogonal
projections to solve this problem for all square integrable (but not n
ecessarily differentiable) vector fields. In the construction of Sobol
ev gradients for problems in nonlinear partial differential equations
[6] a family of problems of a somewhat similar nature arises. Using an
idea of Beurling and Deny [2],[3] we give a generalization of the Lax
-Milgram Theorem [5] which unifies a wide class of Sobolev gradient co
nstructions.