A vector field is a map 
that assigns each 
a vector 
.
Several vector fields are illustrated above. On the closure
of a bounded simply
connected region in 
whose boundary is a smooth
surface, a continuously differentiable
vector field is uniquely specified by its divergence
and curl within the region and
its dot product with the outward normal
vector on the boundary (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.
See also
Flow,
Hilbert Transform Along a Vector Field,
Newtonian
Vector Field,
Pólya Plot,
Scalar
Field,
Seifert Conjecture,
Slope
Field,
Tangent Bundle,
Vector,
Wilson Plug Explore this topic in the MathWorld classroom
Explore with Wolfram|Alpha
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.Referenced
on Wolfram|Alpha
Vector Field
Cite this as:
Weisstein, Eric W. "Vector Field." From
MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/VectorField.html
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