Vector field

A vector field is a function $\mathbf{F} : S \subseteq \mathbb{R}^N → \mathbb{R}^N$ defined in a Vector space $S$ that assigns a vector to each point in space.

Properties#

Vector fields find many applications in vector calculus through their derivatives. Being multivariate functions, vector fields admit three kinds of derivatives: the Gradient, the Divergence and the Curl. If the curl is zero, the field is said to be irrotational. If the divergence is zero, the field is said to be solenoidal.

A vector field is said to be conservative if it admits a Potential $V$ that states that $\mathbf{F}=-\nabla V$. If $\mathbf{F}$ is continuous, differentiable and conservative then it is also irrotational. The converse also applies: if a field is irrotational, then it admits a potential.

Alternatively, it can admit a Vector potential $\mathbf{A}$ such that $\mathbf{F}=\nabla\times \mathbf{A}$. If $\mathbf{F}$ is continuous, differentiable and admits a vector potential then it is also solenoidal. The converse also applies: if a field is solenoidal, then it admits a vector potential.