Line Integral

The line integral of a vector field F(x) on a curve sigma is defined by

where a·b denotes a dot product. In Cartesian coordinates, the line integral can be written

where

For z complex and gamma:z=z(t) a path in the complex plane parameterized by t in [a,b],

Poincaré's theorem states that if del xF=0 in a simply connected neighborhood U(x) of a point x, then in this neighborhood, F is the gradient of a scalar field phi(x),

for x in U(x), where del is the gradient operator. Consequently, the gradient theorem gives

for any path sigma located completely within U(x), starting at x_1 and ending at x_2.

This means that if del xF=0 (i.e., F(x) is an irrotational field in some region), then the line integral is path-independent in this region. If desired, a Cartesian path can therefore be chosen between starting and ending point to give

If del ·F=0 (i.e., F(x) is a divergenceless field, a.k.a. solenoidal field), then there exists a vector field A such that

where A is uniquely determined up to a gradient field (and which can be chosen so that del ·A=0).

See also

Conservative Field, Contour Integral, Gradient Theorem, Irrotational Field, Path Integral, Poincaré's Theorem

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References

Krantz, S. G. "The Complex Line Integral." §2.1.6 in Handbook of Complex Variables. Boston, MA: Birkhäuser, p. 22, 1999.

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Line Integral

Cite this as:

Weisstein, Eric W. "Line Integral." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/LineIntegral.html

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