Vector Field 

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.

REFERENCES:

Arfken, G. "Vector Analysis." Ch. 1 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 1-84, 1985.

Gray, A. "Vector Fields on " and "Derivatives of Vector Fields on ." §11.4 and 11.5 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 255-258, 1997.

Morse, P. M. and Feshbach, H. "Vector Fields." §1.2 in Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 8-21, 1953.