A vector field is a function \(\bm{F}\colon \mathbf{R}^n \to \mathbf{R}^n,\) a multivariable, vector-valued function, that assigns a vector to each point in your space. \(\bm{F}\colon \mathbf{R}^n \to \mathrm{R}^n\) E.g. in three-dimensional space \(\mathbf{R}^3\) both the domain and the codomain of a
For a function \(f\) the gradient operator \(\nabla\) gives us a vector field \(\nabla f\) called the gradient vector field. A vector field is called conservative if it is the gradient vector field of some scalar function \(f.\) In this situation that \(\bm{F} = \nabla f\) is conservative, we call \(f\) a potential function for \(\bm{F}.\)
TK conservative intuition
solenoidal