Vector field

On an open set $U\subset {\mathbb{R}}^n$, a vector field assigns a vector $X(x)\in {\mathbb{R}}^n$ to each point $x\in U$. Its components are scalar functions. For a smooth manifold $M$, the corresponding definition requires $X(p)\in T_{p}M$: the vector belongs to the tangent space at its base point. A smooth vector field is a smooth section of the tangent bundle.^[ref]

For example, $X(x,y)=(-y,x)$ is tangent to circles centred at the origin. Its integral curves solve $\dot{x}=-y$, $\dot{y}=x$, so $x(t{)}^2+y(t{)}^2$ is constant. This distinguishes the field, which assigns a direction and magnitude at every point, from any one trajectory following it. A fluid velocity is a physical vector field; its divergence measures local expansion, while its curl measures local rotation in three dimensions.

A vector field is a mathematical construct that assigns a vector to every point in a space. It can be thought of as a way to represent spatial variations in a quantity that has both magnitude and direction. Vector fields are widely used in physics and engineering to model phenomena such as fluid flow, electromagnetic fields, and gravitational fields, among others.