A vector field is a map that assigns each a vector . Several vector fields are illustrated above. A vector field is uniquely specified by giving its divergence and curl within a region and its normal component over the boundary, a result known as Helmholtz's theorem (Arfken 1985, p. 79).
Vector fields can be plotted in the Wolfram Language using VectorPlot[f, x, xmin, xmax, y, ymin, ymax].
Flows are generated by vector fields and vice versa. A vector field is a tangent bundle section of its tangent bundle.