Conservative Vector Fields

Fig 1

The above field $\mathbf{v}(x,y,z) = (\frac{−y}{x^2+y^2}, \frac{x}{x^2+y^2}, 0)$ includes a vortex at its center, meaning it is non-irrotational; it is neither conservative, nor does it have path independence. However, any simply connected subset that excludes the vortex line $(0,0,z)$ will have zero curl, $\nabla \mathbf{v}=0$. Such vortex-free regions are examples of irrotational vector fields.

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Conservative Vector Fields

Fig 1

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