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# Vector field

Vector field given by vectors of the form (-y, x)

In mathematics a vector field is a construction in vector calculus which associates a vector to every point in Euclidean space.

Vector fields are often used in physics 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 point to point.

In the rigorous mathematical treatment, vector fields are defined on manifolds as a section of the manifold's tangent bundle.

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## Definition

Given an open and connected subset X in Rn a vector field is a vector valued function

$\mathbf{F}: X \rightarrow \mathbb{R}^n$

We say F is a Ck vector field if F is k times continuously differentiable in X.

A point x in X is called stationary if

$\mathbf{F}(\mathbf{x}) = \mathbf{0}$

A vector field can be visualized as a n-dimensional space with a n-dimensional vector attached to each point in X.

Given two Ck-vector fields F,G defined over X and a real valued Ck-function f defined over X

$(f \mathbf{F})(\mathbf{x}) = f(\mathbf{x}) \mathbf{F}(\mathbf{x})$
$\mathbf{(F+G)}(\mathbf{x}) = \mathbf{F}(\mathbf{x}) + \mathbf{G}(\mathbf{x})$

defines the module of Ck-vector fields over the ring of Ck-functions.

## Notes

Vector fields should be compared to scalar fields, which associate a number or scalar to every point in space (or every point of some manifold).

The derivatives of a vector field, resulting in a scalar field or another vector field, are called the divergence and curl respectively.

## Examples

• A vector field for the movement of air on Earth will associate for every point on the surface of the Earth a vector with the wind speed and direction for that point. This can be drawn using arrows to represent the wind; the length (magnitude) of the arrow will be an indication of the wind speed. A "high" on the usual barometric pressure map would then act as a source (arrows pointing away), and a "low" would be a sink (arrows pointing towards), since air tends to move from high pressure areas to low pressure areas.
• Velocity field of a moving fluid. In this case, a velocity vector is associated to each point in the fluid. In wind tunnels, the fieldlines can be revealed using smoke.
• Magnetic fields. The fieldlines can be revealed using small iron filings.
• Maxwell's equations allow us to use a given set of initial conditions to deduce, for every point in Euclidean space, a magnitude and direction for the force experienced by a charged test particle at that point; the resulting vector field is the electromagnetic field.

Vector fields can be constructed out of scalar fields using the vector operator gradient which gives rise to the following definition.

A Ck-vector field F over X is called a gradient field or conservative field if there exists a real valued Ck+1-function f : X → R (a scalar field) so that

$\mathbf{F}(\mathbf{x}) = \nabla f(\mathbf{x}) \qquad (\mathbf{x} \in X)$

The curve integral along any closed curve (e.g. γ(a) = γ(b)) in a gradient field is always zero.

$\oint_\gamma \langle \mathbf{F}( \mathbf{x} ), d\mathbf{x} \rangle = \int_a^b \langle \nabla f( \mathbf{\gamma} (t)), \mathbf{\gamma}'(t) \rangle \, dt = \int_a^b \frac{d}{dt} f \circ \mathbf{\gamma}(t) \, dt = f(\mathbf{\gamma}(b)) - f(\mathbf{\gamma}(a)) = 0$

### Central field

A C-vector field over Rn \ {0} is called central field if

$\mathbf{F}(\mathbf{O}(\mathbf{x})) = \mathbf{O}(\mathbf{F}(\mathbf{x})) \qquad (\mathbf{O} \in O(n, \mathbf{R}) \mbox{ , } \mathbf{x} \in R^n \setminus \lbrace 0 \rbrace )$

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.

A central field is always a gradient field.

## Curve integral

A common technique in physics is to integrate a vector field along a curve. Given a particle in a gravitational vector field, where each vector represents the force acting on the particle at this point in space, the curve integral is the work done on the particle when it travels along a certain path.

The curve integral is constructed analogously to the Riemann integral and it exists if the curve is rectifiable (has finite length) and the vector field is continuous.

Given a vector field F(x) and a curve γ(t) from a to b the curve integral is defined as

$\int_\gamma \langle \mathbf{F}( \mathbf{x} ), d\mathbf{x} \rangle = \int_a^b \langle \mathbf{F}( \mathbf{\gamma}(t) ), \mathbf{\gamma}'(t) \rangle dt$

A few simple rules for calculation of curve integrals are

$\int_\gamma \langle (\mathbf{F} + \mathbf{G})( \mathbf{x} ), d\mathbf{x} \rangle = \int_\gamma \langle \mathbf{F}( \mathbf{x} ), d\mathbf{x} \rangle + \int_\gamma \langle \mathbf{G}( \mathbf{x} ), d\mathbf{x} \rangle$
$\int_\gamma \langle \alpha \cdot \mathbf{F}( \mathbf{x} ), d\mathbf{x} \rangle = \alpha \cdot \int_\gamma \langle \mathbf{F}( \mathbf{x} ), d\mathbf{x} \rangle$
$\int_{-\gamma} \langle \mathbf{F}( \mathbf{x} ), d\mathbf{x} \rangle = -\int_\gamma \langle \mathbf{F}( \mathbf{x} ), d\mathbf{x} \rangle$
$\int_{\gamma_1 + \gamma_2} \langle \mathbf{F}( \mathbf{x} ), d\mathbf{x} \rangle = \int_{\gamma_1} \langle \mathbf{F}( \mathbf{x} ), d\mathbf{x} \rangle + \int_{\gamma_2} \langle \mathbf{F}( \mathbf{x} ), d\mathbf{x} \rangle$

## Integral curves

Vector fields have a nice interpretation in terms of autonomous, first order ordinary differential equations.

Given a C0 vector field F defined over X.

$\mathbf{y} = \mathbf{F}(\mathbf{x}) \qquad (\mathbf{x} \in X)$

we can try to define curves γ(t) over X so that for a each t in an interval I

$\mathbf{\gamma}(t) = \mathbf{x} \qquad (t \in I)$

and

$\mathbf{\gamma}'(t) = \mathbf{y} \qquad (t \in I)$

Put in our vector field equation we get

$\mathbf{\gamma}'(t) = \mathbf{F}(\mathbf{\gamma}(t)) \qquad (t \in I)$

which is the definition of an explicit first order ordinary differential equation with the curves γ(t) as solutions.

If F is Lipschitz continuous we can find a unique C1-curve γx for each point x in X so that

$\mathbf{\gamma}_x(0) = \mathbf(x)$
$\mathbf{\gamma}'_x(t) = \mathbf{F}(\mathbf{\gamma}_x(t)) \qquad ( t \in (-\epsilon, +\epsilon) \subset \mathbb{R})$

The curves γx are called integral curves of the vector field F and partition X into equivalence classes. It is not always possible to extend the interval (-ε, +ε) to the whole real number line. The flow may for example reach the edge of X in a finite time.

Integrating the vector field along any integral curve γ yields

$\int_\gamma \langle \mathbf{F}( \mathbf{x} ), d\mathbf{x} \rangle = \int_a^b \langle \mathbf{F}( \mathbf{\gamma}(t) ), \mathbf{\gamma}'(t) \rangle dt = \int_a^b dt = \mbox{constant}.$

In 2 or three dimension one can visualize the vector field as given rise to a flow on X. If we drop a particle into this flow at point x it will move along the a curve γx in the flow depending on the inital point x. If x is a stationary point in F then the particle will remain stationary.

Typical applications are streamline in fluid, geodesic flow, and one-parameter subgroups and the exponential map in Lie groups.