A vector field is a function that assigns a vector to each point in a subset of space. In mathematical terms, if $\mathbf{F}$ is a vector field, then for each point $(x, y, z)$ in space, $\mathbf{F}(x, y, z)$ is a vector.

Examples and Applications

1. Physics: Vector fields are used to represent various physical quantities such as gravitational fields, electric fields, and velocity fields in fluid dynamics.

2. Mathematics: In differential geometry and calculus, vector fields are used to study the behavior of functions and differential equations on manifolds.

Formal Definition

For a vector field in three-dimensional space, it can be written as: $$ \mathbf{F}(x, y, z) = P(x, y, z) \mathbf{i} + Q(x, y, z) \mathbf{j} + R(x, y, z) \mathbf{k} $$ where $P, Q,$ and $R$ are functions that describe the components of the vector field in the $x, y,$ and $z$ directions, respectively, and $\mathbf{i}, \mathbf{j},$ and $\mathbf{k}$ are the unit vectors in the $x, y,$ and $z$ directions.

Visualization

A vector field can be visualized by drawing arrows at various points in the space, where each arrow represents the vector assigned to that point. The length and direction of each arrow correspond to the magnitude and direction of the vector at that point.

Example

Consider the vector field $\mathbf{F}(x, y) = -y \mathbf{i} + x \mathbf{j}$ in two dimensions. At each point $(x, y)$, the vector is rotated 90 degrees counterclockwise. This particular vector field represents a rotational or swirling pattern around the origin.