Vector Field · aiwiki.page

A vector field is an assignment of a vector to each point of a specified space or region. In elementary calculus, it is a vector-valued function on a region of Euclidean space, commonly represented by arrows whose directions and lengths indicate the assigned vectors. In differential geometry, a vector field on a manifold assigns each point a vector in that point’s tangent space. Vector fields provide mathematical descriptions of fluid velocity, gravitational attraction, and other quantities that vary with position. (openstax.org)

Definition and representation

On a region $U\subseteq\mathbb R^n$ , a vector field is a map

\mathbf F:U\longrightarrow\mathbb R^n, \qquad \mathbf F(\mathbf x)= \bigl(F_1(\mathbf x),\ldots,F_n(\mathbf x)\bigr).

The functions $F_i$ are its components. For differential calculus, $U$ is usually an open set, and the components are assumed to have the continuity or differentiability required by the operation under consideration. A field is continuous, continuously differentiable, or smooth when its components have the corresponding regularity. (openstax.org)

In three-dimensional Euclidean space, a field can be written

\mathbf F(x,y,z)=P(x,y,z)\mathbf e_x+ Q(x,y,z)\mathbf e_y+R(x,y,z)\mathbf e_z,

where the $\mathbf e_i$ form the standard basis. Its magnitude is

\|\mathbf F\|=\sqrt{P^2+Q^2+R^2}.

An arrow plot samples the field at selected points; it is a visualization rather than the field itself. A unit vector field has magnitude one everywhere and records direction without a varying magnitude. (openstax.org)

A scalar field assigns a number, rather than a vector, to each point. For example, temperature is ordinarily modeled as a scalar field, while fluid velocity is modeled as a vector field. A time-dependent vector field additionally depends on a parameter $t$ , so its value is written $\mathbf F(\mathbf x,t)$ . (ocw.mit.edu)

Basic examples

Several simple fields illustrate distinct geometries:

Constant field: $\mathbf F(x,y)=(a,b)$ . All arrows have the same magnitude and direction.
Radial field: $\mathbf F(x,y)=(x,y)$ . Arrows point away from the origin, with magnitude increasing with distance.
Rotational field: $\mathbf F(x,y)=(-y,x)$ . Arrows are tangent to circles centered at the origin and point counterclockwise.
Gradient field: $\mathbf F=\nabla\phi$ , obtained by differentiating a scalar function $\phi$ .

These examples distinguish uniform translation, outward motion, circular motion, and motion governed by a scalar potential. (ocw.mit.edu)

The rotational field also shows why the geometry of arrows alone does not determine whether a field has sources: $(-y,x)$ has zero divergence, despite its varying direction and magnitude. (openstax.org)

Differential operations

The main local differential operations on vector fields measure how their components change with position. The following formulas use Cartesian coordinates.

Gradient fields

The gradient of a differentiable scalar function $\phi$ is

\nabla\phi= \left( \frac{\partial\phi}{\partial x_1},\ldots, \frac{\partial\phi}{\partial x_n} \right).

With the Euclidean metric, it points in the direction of greatest local increase of $\phi$ , when the gradient is nonzero. A field expressible as $\nabla\phi$ is called a gradient field or conservative field. For physical forces derived from potential energy $V$ , the conventional relation instead uses a minus sign:

\mathbf F=-\nabla V.

(ocw.mit.edu)

Divergence

The divergence of a differentiable vector field is the scalar function

\nabla\cdot\mathbf F= \sum_{i=1}^{n}\frac{\partial F_i}{\partial x_i}.

It measures local net outward flux per unit volume. For a velocity field, positive divergence indicates local expansion and negative divergence local compression. A field satisfying $\nabla\cdot\mathbf F=0$ is called divergence-free or solenoidal. This does not require the field to be constant or nonrotating. (openstax.org)

Curl

In three dimensions, the curl is

\nabla\times\mathbf F= \left( \frac{\partial R}{\partial y}-\frac{\partial Q}{\partial z}, \frac{\partial P}{\partial z}-\frac{\partial R}{\partial x}, \frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y} \right).

It measures local circulation; for a fluid velocity field it describes local rotation. A field with zero curl is called irrotational. For a planar field $(P,Q)$ , the corresponding scalar curl is

\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}.

(ocw.mit.edu)

For sufficiently smooth functions,

\nabla\times(\nabla\phi)=0, \qquad \nabla\cdot(\nabla\times\mathbf F)=0.

These identities follow from the equality of mixed partial derivatives. The familiar vector-valued curl formula is specific to three dimensions; more general formulations use differential forms. (ocw.mit.edu)

Integration, circulation, and flux

A vector field can be integrated along a curve or across a surface, producing different geometric quantities.

For a piecewise smooth curve $\mathbf r(t)$ , $a\leq t\leq b$ , its line integral is

\int_C\mathbf F\cdot d\mathbf r = \int_a^b \mathbf F(\mathbf r(t))\cdot\mathbf r'(t)\,dt.

When $\mathbf F$ is a force field, this integral gives work. Around a closed curve, it is often called circulation. (openstax.org)

The flux across an oriented surface $S$ is

\iint_S\mathbf F\cdot\mathbf n\,dS,

where $\mathbf n$ is the chosen unit normal. Only the component normal to the surface contributes. For a fluid velocity field, the integral measures signed volume flow per unit time through the surface. (openstax.org)

Two fundamental theorems connect these integrals with local differential operations:

The divergence theorem states
$\iint_{\partial V}\mathbf F\cdot\mathbf n\,dS = \iiint_V\nabla\cdot\mathbf F\,dV,$
with outward boundary orientation.
Stokes’ theorem states
$\oint_{\partial S}\mathbf F\cdot d\mathbf r = \iint_S(\nabla\times\mathbf F)\cdot\mathbf n\,dS,$
with compatible surface and boundary orientations.

Their application requires appropriate smoothness of the field and regularity of the region or surface. Singularities inside an integration region cannot simply be ignored. (openstax.org)

Conservative fields and domain topology

If $\mathbf F=\nabla\phi$ , then

\int_C\mathbf F\cdot d\mathbf r =\phi(\mathbf r(b))-\phi(\mathbf r(a)).

Consequently, the integral depends only on the endpoints, and every closed-curve integral is zero. On a connected open domain, path independence is equivalent to the existence of a scalar potential for a continuous field. (openstax.org)

Zero curl is a necessary condition for a continuously differentiable conservative field, but it is not sufficient on every domain. On a simply connected open region in two or three dimensions, a continuously differentiable curl-free field is conservative. The domain’s topology therefore matters, not just its local derivatives. (ocw.mit.edu)

A standard counterexample on the punctured plane is

\mathbf F(x,y)= \left( -\frac{y}{x^2+y^2}, \frac{x}{x^2+y^2} \right).

Its scalar curl is zero everywhere it is defined, but its circulation around the counterclockwise unit circle is $2\pi$ . Thus, it has no globally single-valued scalar potential on that domain. The excluded origin prevents the usual curl test from establishing global conservativeness. (ocw.mit.edu)

Integral curves and flows

An integral curve of a vector field is a parametrized curve satisfying

\frac{d\mathbf x}{dt}=\mathbf F(\mathbf x(t)).

Its velocity equals the field vector at its current position. This turns a vector field into an autonomous differential equation. For a time-dependent field, the equation becomes

\frac{d\mathbf x}{dt}=\mathbf F(\mathbf x(t),t).

(math.stanford.edu)

Under suitable local Lipschitz continuity, the initial position determines a unique local solution. A smooth field satisfies this local condition. Solutions need not exist for all time: they may leave the domain or become unbounded in finite time. A field whose integral curves exist for every real time is called complete. (math.stanford.edu)

For a steady fluid velocity field, integral curves are streamlines and coincide with particle trajectories. For an unsteady field, instantaneous streamlines and particle trajectories generally differ. Zeros of the field are equilibrium points, represented by constant integral curves. (ocw.mit.edu)

Vector fields on manifolds

On a smooth manifold $M$ , the vector assigned at $p$ belongs to its tangent space $T_pM$ :

X_p\in T_pM.

Equivalently, a smooth vector field is a smooth section of the tangent bundle $TM$ . In local coordinates,

X=\sum_i X^i\frac{\partial}{\partial x^i}.

When coordinates change, the components transform with the coordinate-change Jacobian matrix, while the underlying tangent vector remains unchanged. (math.stanford.edu)

A vector field also acts on smooth scalar functions as a directional derivative:

X(f)=\sum_iX^i\frac{\partial f}{\partial x^i}.

This operation obeys the product rule. The commutator of two such operations defines their Lie bracket, another vector field. (math.stanford.edu)

Global topology can restrict the existence of vector fields with particular properties. The hairy ball theorem states that every continuous tangent vector field on the two-dimensional sphere has a zero somewhere. A field drawn on a curved surface must therefore respect both tangency and global topological constraints. (math.stanford.edu)

Applications and scope

In physics, vector fields model velocity, force per unit mass, the electric field, and the magnetic field. Their derivatives and integrals connect local behavior with observable circulation, transport, and flux. In particular, Maxwell’s equations use divergence and curl to express electromagnetic relations. (openstax.org)

Mathematically, the choice of domain, regularity, and geometric structure is essential. Cartesian formulas cannot be transferred unchanged to arbitrary curved coordinates, and local differential conditions do not automatically imply global potentials or globally defined flows. Vector fields on manifolds address these distinctions by treating vectors as point-dependent tangent objects rather than merely lists of component functions. (math.stanford.edu)

References

1 Vector Fields — Calculus Volume 3openstax.org
2 Line Integrals — Calculus Volume 3openstax.org
3 Conservative Vector Fields — Calculus Volume 3openstax.org
5 Divergence and Curl — Calculus Volume 3openstax.org
6 Surface Integrals — Calculus Volume 3openstax.org
7 Stokes’ Theorem — Calculus Volume 3openstax.org
8 The Divergence Theorem — Calculus Volume 3openstax.org
Chapter 6 Key Concepts — Calculus Volume 3openstax.org
Chapter 15: Vector Calculusocw.mit.edu
Part B: Vector Fields and Line Integralsocw.mit.edu
MIT 3.016 Lecture 13ocw.mit.edu
MIT 18.02 Lecture 24 Transcriptocw.mit.edu