Consider the vector field ʊ(r) = (x² + y²)êx + (x² + y²)êy + z²êz. Decompose the vector field (r) into the sum of two other vector fields, a (r) and 5(r), such that a(r) has no divergence (it is solenoidal) and 5 (r) has no curl (it is irrotational). The answer is not unique. This is the Helmholtz decomposition.

Consider the vector field ʊ(r) = (x² + y²)êx + (x² + y²)êy + z²êz. Decompose the vector field (r) into the sum of two other vector fields, a (r) and 5(r), such that a(r) has no divergence (it is solenoidal) and 5 (r) has no curl (it is irrotational). The answer is not unique. This is the Helmholtz decomposition.