The line integral of a vector field on a curve is defined by

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

where

For z complex and a path in the complex plane parameterized by ,

Poincaré's theorem states that if in a simply connected neighborhood of a point x, then in this neighborhood, F is the gradient of a scalar field ,

for , where is the gradient operator. Consequently, the gradient theorem gives

for any path located completely within , starting at and ending at .

This means that if (i.e., 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

	(7)

If (i.e., 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 ).

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

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