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.
See also
Flow, Newtonian Vector Field, Pólya Plot, Scalar Field, Seifert Conjecture, Slope Field, Tangent Bundle, Vector, Wilson Plug Explore this topic in the MathWorld classroom
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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
Cite this as:
Weisstein, Eric W. "Vector Field." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/VectorField.html