Vector Fields

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.

Fields in vector calculus

A depiction of xyz Cartesian coordinates with the ijk elementary basis vectors.

Scalar fields

A scalar field is a function $f:\mathbb {R} ^{3}\to \mathbb {R}$ that assigns a real number to each point in space. Scalar fields typically denote densities or potentials at each specific point. For the sake of simplicity, all scalar fields considered by this chapter will be assumed to be defined at all points and differentiable at all points.

Vector fields

A vector field is a function $\mathbf {F} :\mathbb {R} ^{3}\to \mathbb {R} ^{3}$ that assigns a vector to each point in space. Vector fields typically denote flow densities or potential gradients at each specific point. For the sake of simplicity, all vector fields considered by this chapter will be assumed to be defined at all points and differentiable at all points.

A depiction of cylindrical coordinates and the accompanying orthonormal basis vectors.

Vector fields in cylindrical coordinates

The cylindrical coordinate system used here has the three parameters: $(\rho ,\phi ,z)$. The Cartesian coordinate equivalent to the point $(\rho ,\phi ,z)$ is

$x=\rho \cos \phi$

$y=\rho \sin \phi$

$z=z$

Any vector field in cylindrical coordinates is a linear combination of the following 3 mutually orthogonal unit length basis vectors:

${\hat {\mathbf {\rho } }}=\cos \phi \mathbf {i} +\sin \phi \mathbf {j}$

${\hat {\mathbf {\phi } }}=-\sin \phi \mathbf {i} +\cos \phi \mathbf {j}$

${\hat {\mathbf {z} }}=\mathbf {k}$

Note that these basis vectors are not constant with respect to position. The fact that the basis vectors change from position to position should always be considered. The cylindrical basis vectors change according to the following rates:

	${\frac {\partial }{\partial \rho }}$	${\frac {\partial }{\partial \phi }}$	${\frac {\partial }{\partial z}}$
${\hat {\mathbf {\rho } }}$	${\frac {\partial {\hat {\mathbf {\rho } }}}{\partial \rho }}=\mathbf {0}$	${\frac {\partial {\hat {\mathbf {\rho } }}}{\partial \phi }}={\hat {\mathbf {\phi } }}$	${\frac {\partial {\hat {\mathbf {\rho } }}}{\partial z}}=\mathbf {0}$
${\hat {\mathbf {\phi } }}$	${\frac {\partial {\hat {\mathbf {\phi } }}}{\partial \rho }}=\mathbf {0}$	${\frac {\partial {\hat {\mathbf {\phi } }}}{\partial \phi }}=-{\hat {\mathbf {r} }}$	${\frac {\partial {\hat {\mathbf {\phi } }}}{\partial z}}=\mathbf {0}$
${\hat {\mathbf {z} }}$	${\frac {\partial {\hat {\mathbf {z} }}}{\partial \rho }}=\mathbf {0}$	${\frac {\partial {\hat {\mathbf {z} }}}{\partial \phi }}=\mathbf {0}$	${\frac {\partial {\hat {\mathbf {z} }}}{\partial z}}=\mathbf {0}$

Any vector field $\mathbf {F}$ expressed in cylindrical coordinates has the form: $\mathbf {F} (\mathbf {q} )=F_{\rho }(\mathbf {q} ){\hat {\mathbf {\rho } }}+F_{\phi }(\mathbf {q} ){\hat {\mathbf {\phi } }}+F_{z}(\mathbf {q} ){\hat {\mathbf {z} }}$

Given an arbitrary position $\mathbf {q} =(\rho ,\phi ,z)$ that changes with time, the velocity of the position is:

${\frac {d\mathbf {q} }{dt}}={\frac {d\rho }{dt}}{\hat {\mathbf {\rho } }}+\rho {\frac {d\phi }{dt}}{\hat {\mathbf {\phi } }}+{\frac {dz}{dt}}{\hat {\mathbf {z} }}$

The coefficient of $\rho$ for the term $\rho {\frac {d\phi }{dt}}{\hat {\mathbf {\phi } }}$ originates from the fact that as the azimuth angle $\phi$ increases, the position $\mathbf {q}$ swings around at a speed of $\rho$.

A depiction of spherical coordinates and the accompanying orthonormal basis vectors.

Vector fields in spherical coordinates

The spherical coordinate system used here has the three parameters: $(r,\theta ,\phi )$. The Cartesian coordinate equivalent to the point $(r,\theta ,\phi )$ is

$x=r\sin \theta \cos \phi$

$y=r\sin \theta \sin \phi$

$z=r\cos \theta$

Any vector field in spherical coordinates is a linear combination of the following 3 mutually orthogonal unit length basis vectors:

${\hat {\mathbf {r} }}=\sin \theta \cos \phi \mathbf {i} +\sin \theta \sin \phi \mathbf {j} +\cos \theta \mathbf {k}$

${\hat {\mathbf {\theta } }}=\cos \theta \cos \phi \mathbf {i} +\cos \theta \sin \phi \mathbf {j} -\sin \theta \mathbf {k}$

${\hat {\mathbf {\phi } }}=-\sin \phi \mathbf {i} +\cos \phi \mathbf {j}$

Note that these basis vectors are not constant with respect to position. The fact that the basis vectors change from position to position should always be considered. The spherical basis vectors change according to the following rates:

	${\frac {\partial }{\partial r}}$	${\frac {\partial }{\partial \theta }}$	${\frac {\partial }{\partial \phi }}$
${\hat {\mathbf {r} }}$	${\frac {\partial {\hat {\mathbf {r} }}}{\partial r}}=\mathbf {0}$	${\frac {\partial {\hat {\mathbf {r} }}}{\partial \theta }}={\hat {\mathbf {\theta } }}$	${\frac {\partial {\hat {\mathbf {r} }}}{\partial \phi }}=\sin \theta {\hat {\mathbf {\phi } }}$
${\hat {\mathbf {\theta } }}$	${\frac {\partial {\hat {\mathbf {\theta } }}}{\partial r}}=\mathbf {0}$	${\frac {\partial {\hat {\mathbf {\theta } }}}{\partial \theta }}=-{\hat {\mathbf {r} }}$	${\frac {\partial {\hat {\mathbf {\theta } }}}{\partial \phi }}=\cos \theta {\hat {\mathbf {\phi } }}$
${\hat {\mathbf {\phi } }}$	${\frac {\partial {\hat {\mathbf {\phi } }}}{\partial r}}=\mathbf {0}$	${\frac {\partial {\hat {\mathbf {\phi } }}}{\partial \theta }}=\mathbf {0}$	${\frac {\partial {\hat {\mathbf {\phi } }}}{\partial \phi }}=-(\sin \theta {\hat {\mathbf {r} }}+\cos \theta {\hat {\mathbf {\theta } }})$

Any vector field $\mathbf {F}$ expressed in spherical coordinates has the form: $\mathbf {F} (\mathbf {q} )=F_{r}(\mathbf {q} ){\hat {\mathbf {r} }}+F_{\theta }(\mathbf {q} ){\hat {\mathbf {\theta } }}+F_{\phi }(\mathbf {q} ){\hat {\phi }}$

Given an arbitrary position $\mathbf {q} =(r,\theta ,\phi )$ that changes with time, the velocity of this position is:

${\frac {d\mathbf {q} }{dt}}={\frac {dr}{dt}}{\hat {\mathbf {r} }}+r{\frac {d\theta }{dt}}{\hat {\mathbf {\theta } }}+r\sin \theta {\frac {d\phi }{dt}}{\hat {\mathbf {\phi } }}$

The coefficient of $r$ for the term $r{\frac {d\theta }{dt}}{\hat {\mathbf {\theta } }}$ arises from the fact that as the latitudinal angle $\theta$ changes, the position $\mathbf {q}$ traverses a great circle at a speed of $r$.

The coefficient of $r\sin \theta$ for the term $r\sin \theta {\frac {d\phi }{dt}}{\hat {\mathbf {\phi } }}$ arises from the fact that as the longitudinal angle $\phi$ changes, the position $\mathbf {q}$ traverses a latitude circle at a speed of $r\sin \theta$.

Operations on vector fields

Line integral

A common technique in physics is to integrate a vector field along a curve, also called determining its line integral. Intuitively this is summing up all vector components in line with the tangents to the curve, expressed as their scalar products. For example, given a particle in a force field (e.g. gravitation), where each vector at some point in space represents the force acting there on the particle, the line integral along a certain path is the work done on the particle, when it travels along this path. Intuitively, it is the sum of the scalar products of the force vector and the small tangent vector in each point along the curve.

The line 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 V and a curve γ, parametrized by t in [a, b] (where a and b are real numbers), the line integral is defined as

$\int _{\gamma }V({\boldsymbol {x}})\cdot \mathrm {d} {\boldsymbol {x}}=\int _{a}^{b}V(\gamma (t))\cdot {\dot {\gamma }}(t)\;\mathrm {d} t.$

Divergence

The divergence of a vector field on Euclidean space is a function (or scalar field). In three-dimensions, the divergence is defined by

$\operatorname {div} \mathbf {F} =\nabla \cdot \mathbf {F} ={\frac {\partial F_{1}}{\partial x}}+{\frac {\partial F_{2}}{\partial y}}+{\frac {\partial F_{3}}{\partial z}},$

with the obvious generalization to arbitrary dimensions. The divergence at a point represents the degree to which a small volume around the point is a source or a sink for the vector flow, a result which is made precise by the divergence theorem.

The divergence can also be defined on a Riemannian manifold, that is, a manifold with a Riemannian metric that measures the length of vectors.

Curl in three dimensions

The curl is an operation which takes a vector field and produces another vector field. The curl is defined only in three dimensions, but some properties of the curl can be captured in higher dimensions with the exterior derivative. In three dimensions, it is defined by

$\operatorname {curl} \,\mathbf {F} =\nabla \times \mathbf {F} =\left({\frac {\partial F_{3}}{\partial y}}-{\frac {\partial F_{2}}{\partial z}}\right)\mathbf {e} _{1}-\left({\frac {\partial F_{3}}{\partial x}}-{\frac {\partial F_{1}}{\partial z}}\right)\mathbf {e} _{2}+\left({\frac {\partial F_{2}}{\partial x}}-{\frac {\partial F_{1}}{\partial y}}\right)\mathbf {e} _{3}.$

The curl measures the density of the angular momentum of the vector flow at a point, that is, the amount to which the flow circulates around a fixed axis. This intuitive description is made precise by Stokes' theorem.

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 R³ is assigned to each point of the unit sphere S² 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 S².

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.

Resources

• Vector Calculus, Wikibooks: Calculus
• Vector Fields, Mathematics LibreTexts
• Introduction to Vector Fields, Khan Academy
• Line Integrals and Vector Fields, Khan Academy

Licensing

Content obtained and/or adapted from:

• Vector field, Wikipedia under a CC BY-SA license
• Vector calculus, Wikibooks: Calculus under a CC BY-SA license