#todo # Definitions > [!NOTE] Definition (Conservative Vector Field) > Let $\underline{F}:U\subset \mathbb{R}^{3}\to \mathbb{R}$ be a [[Vector Field on Subset of Real n-Space|vector field]]. $\underline{F}$ is conservative iff there exists a (continuously differentiable) [[Real-Valued Function on Real n-Space (Multivariable Function)|scalar field]] $f:\mathbb{R}^{3}\to \mathbb{R}$ so that $\underline{F}=\nabla f,$ where $\nabla f$ denotes the [[Fréchet Differentiation|gradient]] of $f.$ # Properties By [[Curl of Conservative Vector Field on Subset of Real 3-Space is Real Zero Vector]], $\text{curl } F= (0,0,0).$ By [[Line Integral of Conservative Vector Field of Real 3-Space is Path Independent]], ...