Vector field

A portion of the vector field (sin y, sin x)
A vector field on a sphere
The flow field around an airplane is a vector field in R3, here visualized by bubbles that follow the streamlines showing a wingtip vortex.
Vector fields are commonly used to create patterns in computer graphics. Here: abstract composition of curves following a vector field generated with OpenSimplex noise.
A vector field that has circulation about a point cannot be written as the gradient of a function.
Magnetic field lines of an iron bar (magnetic dipole)

Assignment of a vector to each point in a subset of space.

- Vector field

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Differentiable manifold

Type of manifold that is locally similar enough to a vector space to allow one to apply calculus.

A nondifferentiable atlas of charts for the globe. The results of calculus may not be compatible between charts if the atlas is not differentiable. In the center and right charts, the Tropic of Cancer is a smooth curve, whereas in the left chart it has a sharp corner. The notion of a differentiable manifold refines that of a manifold by requiring the functions that transform between charts to be differentiable.

A locally differential structure allows one to define the globally differentiable tangent space, differentiable functions, and differentiable tensor and vector fields.

Curl (mathematics)

Depiction of a two-dimensional vector field with a uniform curl.

In vector calculus, the curl is a vector operator that describes the infinitesimal circulation of a vector field in three-dimensional Euclidean space.

Vector calculus

The graph of a function, drawn in black, and a tangent line to that graph, drawn in red. The slope of the tangent line is equal to the derivative of the function at the marked point.

Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space The term "vector calculus" is sometimes used as a synonym for the broader subject of multivariable calculus, which spans vector calculus as well as partial differentiation and multiple integration.

Line integral

Integral where the function to be integrated is evaluated along a curve.

The trajectory of a particle (in red) along a curve inside a vector field. Starting from a, the particle traces the path C along the vector field F. The dot product (green line) of its tangent vector (red arrow) and the field vector (blue arrow) defines an area under a curve, which is equivalent to the path's line integral. (Click on image for a detailed description.)

The function to be integrated may be a scalar field or a vector field.

Tangent bundle


Informally, the tangent bundle of a manifold (which in this case is a circle) is obtained by considering all the tangent spaces (top), and joining them together in a smooth and non-overlapping manner (bottom).

A smooth assignment of a tangent vector to each point of a manifold is called a vector field.

Magnetic field

The shape of the magnetic field produced by a horseshoe magnet is revealed by the orientation of iron filings sprinkled on a piece of paper above the magnet.
Right hand grip rule: a current flowing in the direction of the white arrow produces a magnetic field shown by the red arrows.
A Solenoid with electric current running through it behaves like a magnet.
A sketch of Earth's magnetic field representing the source of the field as a magnet. The south pole of the magnetic field is near the geographic north pole of the Earth.
One of the first drawings of a magnetic field, by René Descartes, 1644, showing the Earth attracting lodestones. It illustrated his theory that magnetism was caused by the circulation of tiny helical particles, "threaded parts", through threaded pores in magnets.
Hans Christian Ørsted, Der Geist in der Natur, 1854

A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials.


In vector calculus, the gradient of a scalar-valued differentiable function

The gradient, represented by the blue arrows, denotes the direction of greatest change of a scalar function. The values of the function are represented in greyscale and increase in value from white (low) to dark (high).

of several variables is the vector field (or vector-valued function).

Gravitational field

Model used to explain the influences that a massive body extends into the space around itself, producing a force on another massive body.

Various examples of physical phenomena

is a vector field consisting at every point of a vector pointing directly towards the particle.

Scalar field

In mathematics and physics, a scalar field or scalar-valued function associates a scalar value to every point in a space – possibly physical space.

A scalar field such as temperature or pressure, where intensity of the field is represented by different hues of colors.

Scalar fields are contrasted with other physical quantities such as vector fields, which associate a vector to every point of a region, as well as tensor fields and spinor fields.

Tensor field

In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold).

3rd century BC Greek mathematician Euclid (holding calipers), as imagined by Raphael in this detail from The School of Athens (1509–1511)

As a tensor is a generalization of a scalar (a pure number representing a value, for example speed) and a vector (a pure number plus a direction, like velocity), a tensor field is a generalization of a scalar field or vector field that assigns, respectively, a scalar or vector to each point of space.