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 $r\sin \theta$.

Resources

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