In vector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space . [1] . 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.

A vector field is a function that assigns to every point in space a vector. It can be imagined as a collection of arrows, each one attached to a different point in space. For example, the wind (the velocity of air) can be represented by a vector field.

In science, a field is a physical quantity, represented by a scalar, vector, or tensor, that has a value for each point in space and time. [1][2][3] An example of a scalar field is a weather map, with the surface temperature described by assigning a number to each point on the map.

In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point.

A vector field (usually defined by a vector function) is a field in which all points have a vector value (having both magnitude and direction). This is different from a scalar field, where points have only a scalar value (having only magnitude).

English: In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a (locally) Euclidean space.

A term which is usually understood to mean a function of points in some space $X$ whose values are vectors (cf. Vector), defined for this space in some way. In the classical vector calculus it is a subset of a Euclidean space that plays the part of $X$, while the vector field represents directed segments applied at the points of this subset.

In vector calculus, the curl, also known as rotor, is a vector operator that describes the infinitesimal circulation of a vector field in three-dimensional Euclidean space. The curl at a point in the field is represented by a vector whose length and direction denote the magnitude and axis of the maximum circulation. [1] .

Jan 17, 2025 · Intuitively, a vector field is a map of vectors. In this section, we study vector fields in ℝ2 and ℝ3. A vector field ⇀ F in ℝ2 is an assignment of a two-dimensional vector ⇀ F(x, y) to each point (x, y) of a subset D of ℝ2. The subset D is the domain of the vector field.

1.3 Vector Fields and Flows. This section introduces vector fields on Euclidean space and the flows they determine. This topic puts together and globalizes two basic ideas learned in undergraduate mathematics: the study of vector fields on the one hand and differential equations on the other. Definition 1.3.1. Let r ≥ 0 be an integer. A ...