Helmholtz's Theorem

Theorem
Let $R$ be a region of ordinary space.

Let $\mathbf V$ be a vector field over $R$.

Let $\mathbf V$ be both non-conservative and non-solenoidal.

Then $\mathbf V$ can be decomposed into the sum of $2$ vector fields:
 * one being conservative, with scalar potential $S$, but not solenoidal
 * one being solenoidal, with vector potential $\mathbf A$, but not conservative.

Thus $\mathbf V$ satifies the partial differential equations:

where:


 * $\operatorname {div}$ denotes the divergence operator
 * $\grad$ denotes the gradient operator
 * $\curl$ denotes the curl operator
 * $\nabla^2$ denotes the Laplacian.

Proof
Let us write:
 * $\mathbf V = \grad S + \curl \mathbf A$

where:
 * $S$ is a scalar field
 * $\mathbf A$ is a vector field chosen to be solenoidal

Then:

and: