A conservative vector field is a vector field which is equal to the gradient of a scalar function. By the fundamental theorem of line integrals, a vector field being conservative is equivalent to a closed line integral over it being equal to zero. Vector fields which are conservative are also irrotational (the curl is equal to zero), although the converse is only true if the domain is simply connected. As a corollary of Green's theorem, a two-dimensional vector field f is conservative if

\vec{f} (x,y)= P(x,y) \mathbf{\hat{i}} + Q(x,y) \mathbf{\hat{j}} , \ \ \frac{ \partial Q}{\partial x} = \frac{ \partial P}{\partial y}

Examples of conservative vector fields in the real word would be gravitational or static electrical fields.

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