Poincaré's Theorem

If del xF=0 (i.e., F(x) is an irrotational field) 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),

 F(x)=-del 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, the line integral of F is path-independent.

See also

Conservative Field, Gradient Theorem, Irrotational Field, Line Integral

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Cite this as:

Weisstein, Eric W. "Poincaré's Theorem." From MathWorld--A Wolfram Web Resource.

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