Vector field

Description

In vector calculus and physics, a vector field is an assignment of a Vector to each point in a subset of space. For instance, a vector field in the plane can be visualised as a collection of arrows with a given magnitude and direction, each attached to a point in the plane. Vector fields are often used to model, for example, the speed and direction of a moving fluid throughout space, 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).

In coordinates, a vector field on a domain in n-dimensional Euclidean space 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 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 $S$ in $R^{n}$ , a vector field is represented by a vector-valued function $V : S \to R^{n}$ in standard Cartesian coordinates $(x_{1}, \dots, x_{n})$ . If each component of $V$ is continuous, then $V$ is a continuous vector field, and more generally $V$ is a $C^{k}$ vector field if each component of $V$ is k times continuously differentiable.

A vector field can be visualized as assigning a vector to individual points within an n-dimensional space.

Given two $C^{k}$ -vector fields $V$ , $W$ defined on $S$ and a real-valued $C^{k}$ -function $f$ defined on $S$ , the two operations scalar multiplication and vector addition

$(f V) (p) := f (p) V (p)$

$(V + W) (p) := V (p) + W (p)$

define the module of C^k-vector fields over the ring of C^k-functions where the multiplication of the functions is defined pointwise (therefore, it is commutative with the multiplicative identity being $f_{i d} (p) := 1$ ).

Coordinate transformation law

In physics, a vector is additionally distinguished by how its coordinates change when one measures the same vector with respect to a different background coordinate system. The transformation properties of vectors distinguish a vector as a geometrically distinct entity from a simple list of scalars, or from a covector.

Thus, suppose that (x1, ..., xn) is a choice of Cartesian coordinates, in terms of which the components of the vector V are

$V_{x} = (V_{1, x}, . . . V_{n, x})$

and suppose that $(y_{1}, . . ., y_{n})$ are n functions of the $x_{i}$ defining a different coordinate system. Then the components of the vector $V$ in the new coordinates are required to satisfy the transformation law.

$V_{i, y} = \sum_{j = 1}^{n} \frac{d y_{i}}{d x_{j}} V_{j, x}$

Such a transformation law is called contravariant. A similar transformation law characterizes vector fields in physics: specifically, a vector field is a specification of n functions in each coordinate system subject to the transformation law relating the different coordinate systems.

Vector fields are thus contrasted with scalar fields, which associate a number or scalar to every point in space, and are also contrasted with simple lists of scalar fields, which do not transform under coordinate changes.

Vector fields on manifolds

Given a differentiable manifold $M$ , a vector field on $M$ is an assignment of a tangent vector to each point in $M$ . More precisely, a vector field $F$ is a mapping from $M$ into the tangent bundle $T M$ so that $p \circ F$ is the identity mapping where $p$ denotes the projection from $T M$ to $M$ . In other words, a vector field is a section of the tangent bundle.

An alternative definition: A smooth vector field $X$ on a manifold $M$ is a linear map: $X : C^{\infty} (M) \to C^{\infty} (M)$ such that $X$ is a derivation: $X (f g) = f X (g) + X (f) g$ for all $f, g \in C^{\infty} (M)$ .

If the manifold $M$ is smooth or analytic—that is, the change of coordinates is smooth (analytic)—then one can make sense of the notion of smooth (analytic) vector fields. The collection of all smooth vector fields on a smooth manifold $M$ is often denoted by $Γ (T M)$ or $C^{\infty} (M, T M)$ (especially when thinking of vector fields as sections); the collection of all smooth vector fields is also denoted by $X (M)$ (a fraktur "X").

Gradient field in Euclidean spaces

Wiki

Vector fields can be constructed out of scalar fields using the gradient operator (denoted by the del: ∇).

A vector field $V$ defined on an open set $S$ is called a gradient field or a conservative field if there exists a real-valued function (a scalar field)$ f$ on $S$ such that

$V = \nabla f = (\frac{\partial f}{\partial x_{1}}, \frac{\partial f}{\partial x_{2}}, \frac{\partial f}{\partial x_{3}}, \dots, \frac{\partial f}{\partial x_{n}})$

The associated flow is called the gradient flow, and is used in the method of gradient descent.

The path integral along any closed curve γ (γ(0) = γ(1)) in a conservative field is zero:

$\oint_{γ} V (x) \cdot d x = \oint_{γ} \nabla f (x) \cdot d x = f (γ (1)) - f (γ (0))$ .

Central field in Euclidean spaces

A $C^{\infty}$ vector field over $R n$ \{0} is called a central field if

$V (T (p)) = T (V (p)) (T \in O (n, R))$

where $O (n, R)$ is the orthogonal group. We say central fields are invariant under orthogonal transformations around 0.
The point 0 is called the center of the field.

Since orthogonal transformations are actually rotations and reflections, the invariance conditions mean that vectors of a central field are always directed towards, or away from, 0; this is an alternate (and simpler) definition. A central field is always a gradient field, since defining it on one semiaxis and integrating gives an antigradient.

Operations on vector fields

Index of a vector field

The index of a vector field is an integer that helps to describe the behaviour of a vector field around an isolated zero (i.e., an isolated singularity of the field). In the plane, the index takes the value $- 1$ at a saddle singularity but $+ 1$ at a source or sink singularity.

Let the dimension of the manifold on which the vector field is defined be $n$ . Take a small sphere $S$ around the zero so that no other zeros lie in the interior of $S$ . A map from this sphere to a unit sphere of dimensions $n - 1$ can be constructed by dividing each vector on this sphere by its length to form a unit length vector, which is a point on the unit sphere $S^{n - 1}$ . This defines a continuous map from $S$ to $S^{n - 1}$ . The index of the vector field at the point is the degree of this map. It can be shown that this integer does not depend on the choice of $S$ , and therefore depends only on the vector field itself.

The index of the vector field as a whole is defined when it has just a finite number of zeroes. In this case, all zeroes are isolated, and the index of the vector field is defined to be the sum of the indices at all zeroes.

The index is not defined at any non-singular point (i.e., a point where the vector is non-zero). It is equal to $+ 1$ around a source, and more generally equal to $(- 1)^{k}$ around a saddle that has k contracting dimensions and $n - k$ expanding dimensions. For an ordinary (2-dimensional) sphere in three-dimensional space, it can be shown that the index of any vector field on the sphere must be 2. This shows that every such vector field must have a zero. This implies the hairy ball theorem, which states that if a vector in $R^{3}$ is assigned to each point of the unit sphere $S^{2}$ in a continuous manner, then it is impossible to "comb the hairs flat", i.e., to choose the vectors in a continuous way such that they are all non-zero and tangent to $S^{2}$ .

For a vector field on a compact manifold with a finite number of zeroes, the Poincaré-Hopf theorem states that the index of the vector field is equal to the Euler characteristic of the manifold.