215 Vector Field
A mapping that assigns a vector to each point of a domain that is used to represent force-like quantities that vary with position.
definition [d] (Vector Field = Vector Function of Position) A function \(\mathbf{F}\) that assigns to each point \((x,y,z)\) in its domain a unique vector \(\mathbf{F}(x,y,z)\). In \(\mathbb{R}^{3}\):
- \(\mathbf{F}(x,y,z) = P(x,y,z)\, \mathbf{i} + Q(x,y,z)\, \mathbf{j} + R(x,y,z)\, \mathbf{k}\) .
where
- \(\mathbf{F}\) is the vector field.
- \(P, Q, R\) are scalar-valued component functions of position.
- \(\mathbf{i}, \mathbf{j}, \mathbf{k}\) are the standard unit basis vectors.
- \((x,y,z)\) is a point in the domain.
- \(\mathbb{R}^{3}\) is three-dimensional Euclidean space.
Note:
- the vector is visualized as attached at the point \((x,y,z)\).