In physics and mathematics, in the area of vector calculus, Helmholtz's theorem, also known as the fundamental theorem of vector calculus, states that any sufficiently smooth, rapidly decaying vector field in three dimensions can be resolved into the sum of an irrotational (curl-free) vector field and a solenoidal (divergence-free) vector field; this is known as the Helmholtz decomposition or Helmholtz representation. It is named after Hermann von Helmholtz. As an irrotational vector field has a scalar potential and a solenoidal vector field has a vector potential, the Helmholtz decomposition states that a vector field (satisfying appropriate smoothness and decay conditions) can be decomposed as the sum of the form , where is a scalar field called "scalar potential", and is a vector field, called a vector potential. Statement of the theorem Let be a vector field on a bounded domain , which is twice continuously differentiable inside , and let be the surface that encloses the domain . Then can be decomposed into a curl-free component and a divergence-free component: where and is the nabla operator with respect to , not . If and is therefore unbounded, and vanishes at least as fast as as , then one has This holds in particular if is twice continuously differentiable in and of bounded support. Derivation Suppose we have a vector function of which we know the curl, , and the divergence, , in the domain and the fields on the boundary. Writing the function using delta function in the form where is the Laplace operator, we have where we have used the definition of the vector Laplacian: differentiation/integration with respect to by and in the last line, linearity of function arguments: Then using the vectorial identities we get Thanks to the divergence theorem the equation can be rewritten as with outward surface normal . Defining we finally obtain Generalization to higher dimensions In a -dimensional vector space with , should be replaced by the appropriate Green's function for the Laplacian, defined by where Einstein summation convention is used for the index . For example, in 2D. Following the same steps as above, we can write where is the Kronecker delta (and the summation convention is again used). In place of the definition of the vector Laplacian used above, we now make use of an identity for the Levi-Civita symbol , which is valid in dimensions, where is a -component multi-index. This