### 2.1 Magnetic field lines

Magnetic field lines are the fundamental element in all discussions of magnetic topology. A field line,
sometimes called a line of force, is a space-curve which is everywhere tangent to the local magnetic
field vector . It satisfies the differential equation
(It can be seen that this definition yields , demonstrating
that the parameter is the arc-length along the field line, measured
forward
from the point .) The field line is a curve, and therefore has volume zero. A flux tube may be
constructed by bundling together a group of field lines with net flux . The tube’s net flux is found by
integrating over any surface pierced by the entire tube.
As with any equation of its form, Equation (1) may be solved either forward or backward from any
“initial” point , except from so-called singular points where , since the vector’s
direction would not be defined there (Arnold, 1973). Any volume where may, in
principle, be completely filled with field lines such that a unique field line passes through each
point.

In general circumstances, the only way to find a field line is to integrate Equation (1). A useful shortcut
is available, however, in cases with one symmetry dimension (i.e. in two dimensions). In these special
circumstances a general magnetic field satisfying can be written in terms of a scalar function
called the flux function, and an arbitrary component in the ignorable direction, both depending only on two
coordinates.
When is the ignorable coordinate (planar symmetry), the expression is

and the flux function is the component of the magnetic vector potential. In the case where
is ignorable (azimuthal symmetry) the field takes the form
where the flux function is related to the vector potential as .
In either geometry the flux function has the useful property that it is constant along field lines, since its
derivative

or similarly for . In two-and-a-half dimensional cases, i.e. , a field line equation like
Equation (1) must still be solved within the flux surface constant. It is often the case, however, that
the topology of the flux surface defines the topology of its field lines. Part of the appeal of working with
two-dimensional models is the ability to easily draw a selection of field lines by contouring the flux
function.
The analog of a flux function in three dimensions are the Euler
potentials and ,
which generate the magnetic field
(Sweet, 1950; Dungey, 1953; Stern, 1966; Sturrock and Woodbury, 1967)

The two expressions on the right show that the field is automatically divergence-free and that its vector
potential can be written as or through the gauge transformation as . It
is easily verified that both potentials are constant along field lines, so a given field line may be identified by
the pair of values . Finding potentials to generate a given magnetic field requires the solution
non-linear differential equations. These can prove difficult even for simple cases such as a potential field,
. Indeed, many useful magnetic fields cannot be even written in the form given by
Equation (5) at all. This very powerful method is therefore used far less frequently than are flux
functions.