In vector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space R n {\displaystyle \mathbb {R} ^{n}} . A vector field on a plane can be visualized as a collection of arrows with given magnitudes and directions, each attached to a point on the plane. Vector fields often have unit of measurement (for example, metres or kilometres per hour), forming a vector physical quantity. They may be used to model, for example, the speed and direction of a moving fluid throughout three dimensional space, such as the wind, or the strength and direction of some force, such as the magnetic or gravitational force, as it changes from one point to another point. The elements of differential and integral calculus extend naturally to vector fields. When a vector field represents force, the line integral of a vector field represents the work done by a force moving along a path, and under this interpretation conservation of energy is exhibited as a special case of the fundamental theorem of calculus. Vector fields can usefully be thought of as representing the velocity of a moving flow in space, and this physical intuition leads to notions such as the divergence (which represents the rate of change of volume of a flow) and curl (which represents the rotation of a flow). A vector field is a special case of a vector-valued function, whose domain's dimension has no relation to the dimension of its range; for example, the position vector of a space curve is defined only for smaller subset of the ambient space. Likewise, n coordinates, a vector field on a domain in n-dimensional Euclidean space R n {\displaystyle \mathbb {R} ^{n}} can be represented as a vector-valued function that associates an n-tuple of real numbers to each point of the domain. This representation of a vector field depends on the coordinate system, and there is a well-defined transformation law (covariance and contravariance of vectors) in passing from one coordinate system to the other. Vector fields are often discussed on open subsets of Euclidean space, but also make sense on other subsets such as surfaces, where they associate an arrow tangent to the surface at each point (a tangent vector). More generally, vector fields are defined on differentiable manifolds, which are spaces that look like Euclidean space on small scales, but may have more complicated structure on larger scales. In this setting, a vector field gives a tangent vector at each point of the manifold (that is, a section of the tangent bundle to the manifold). Vector fields are one kind of tensor field.
Definition
Vector fields on subsets of Euclidean space
Given a subset S of Rn, a vector field is represented by a vector-valued function V: S → Rn in standard Cartesian coordinates (x1, …, xn). If each component of V is continuous, then V is a continuous vector field. It is common to focus on smooth vector fields, meaning that each component is a smooth function (differentiable any number of times). A vector field can be visualized as assigning a vector to individual points within an n-dimensional space. One standard notation is to write ∂ ∂ x 1 , … , ∂ ∂ x n {\displaystyle {\frac {\partial }{\partial x_{1}}},\ldots ,{\frac {\partial }{\partial x_{n}}}} for the unit vectors in the coordinate directions. In these terms, every smooth vector field V {\displaystyle V} on an open subset S {\displaystyle S} of R n {\displaystyle {\mathbf {R} }^{n}} can be written as
∑ i = 1 n V i ( x 1 , … , x n ) ∂ ∂ x i {\displaystyle \sum _{i=1}^{n}V_{i}(x_{1},\ldots ,x_{n}){\frac {\partial }{\partial x_{i}}}}
… excerpt ends here. Continue reading the full article.
