In a general topological space X , a path h x to y is a continuous map h:[0,1]→X such that h(0)=x and h(1)=y.
In this deliberately simple formulation, much of the interesting behaviour is necessarily considered at the level of the space as a whole - for instance, X is described as path connected iff every pair of points in X are joined by a path; and this immediately implies the property of connectedness.
Throw geometry and calculus into the mix, and much more can be said. The notion of a path is central to integration in the complex plane, where many powerful results are formulated in terms of path integrals. To do so, however, requires a bewildering array of qualifiers for the type of path being considered.
A first and vital observation to make is that the path is not the curve or contour itself (or, in non-mathematical terms, the ground you would walk upon during your travels along a 'path'), but a function mapping to that curve (a rule for where to walk next). It is also convenient to work on a general interval [a,b] rather than the [0,1] suggested in the topological defintion; but these turn out to be equivalent approaches.
Types of paths
A path given by a continous map γ[a,b]→C may be described as:
- Simple If γ is injective.
This means that the function never takes the same value twice, so someone following the path would never visit the same point twice. This means that the curve created does not cross itself, which turns out to be useful for obtaining a sense of direction.
- Closed If γ(a)=γ(b).
This means the curve starts and ends in the same place, such as a circle.
- Simple closed If γ(s)=γ(t) ⇒ s=t or {s,t}={a,b}; equivalently, γ is a closed path that is injective on (a,b).
A path couldn't be both closed and simple as defined above, as the start and end point is visited twice. So this is the next best thing- a circle is simple closed, but a figure-8 is only closed.
The image Γ of γ on [a,b] (that is, the points traced out in the complex plane by this rule) is named a Jordan curve: if γ was simple, we call it a simple Jordan curve; if γ was closed, we call it a closed Jordan curve; and if γ was simple closed then, unsurprisingly, Γ is called a simple closed Jordan curve. The seemingly-obvious but difficult to prove Jordan curve theorem asserts that for a simple closed Jordan curve Γ, C\Γ is the disjoint union of two domains (open and connected subsets), namely the interior and exterior of Γ.
We allowed ourselves the luxury of calculus, so let's put that to good use. A path is described as:
- Smooth If the derivative γ'(t) exists for all t in [a,b] and is a continuous function on [a,b]
A path integral does not make sense without the ability to differentiate the path
- Regular If the path is smooth with non-zero derivative.
- Piecewise smooth/regular If it is the composition (see later) of finitely many paths, all smooth/regular.
This is handy for building fancy contours such as squares
The image of a piecewise regular path γ is called a contour - as before we may describe it as being simple, closed or simple closed.
New Paths from old
Reparametrisation
I claimed earlier that it didn't matter if we worked on [a,b] or [0,1]. It turns out that one is simply a reparametrisation of the other (depending on which you consider to be the more natural): If γ[a,b]→C is a regular path parametrising Γ, and h:[c,d]→[a,b] is surjective and regular, then β=γoh:[c,d]→C, where o denotes composition, gives a new regular parametrisation of Γ. Intuitively, a route is the same regardless of how fast you walk it - the path describes the curve for a walking pace determined, effectively, by the interval [a,b].
Restriction of a path
Feeling lazy? If γ[a,b]→C is a path and [c,d] a subinterval of [a,b], then β[c,d]→C, β(t)=γ(t) is also a path. Some caution is needed, however, as not all the properties above are preserved- in particular, you're almost certain to break the closed property of a simple closed curve.
Inverse paths
If you can get there, you can get back: given a path γ[a,b]→C the reversed path γ-[a,b]→C, γ-(t)=γ(b+a-t) is also a path. It describes the same curve, but going in the opposite direction; which raises the question of how to orientate a curve.
Path composition
Given two curves such that one begins where the other ends, it is natural to consider one, large path that traces out their combined curve. To this end, let γ[a,b]→C, β[c,d]→C be paths such that γ(b)=β(c). Then we can define γ#β:[a,b+(d-c)]→C by
t→ γ#β(t) = γ(t) for t in [a,b], β(t+c-b) for t in [b,b+(d-c)].
Some examples and a note on orientation
It turns out that the value of a path integral depends on the parametrisation only as far as a change of sign - namely, the integral over the reversed path has the same magnitude and opposite sign to that of the original path. For a simple closed curve, we refer to the orientation, which is the direction around the loop (this can be uniquely determined due to the simplicity). Conventionally, curves are parametrised so as to be traced out in an anti-clockwise direction from the starting point.
A very easy parametrisation is the straight line joining two points. In the real numbers, this wasn't even an issue, but for the complex plane there are infinitely many ways to get from one point to another besides this obvious one. Working with [0,1] turns out to be convenient here- for the straight line from a complex point x to another, y, we can take the path γ(t)=x+t(y-x).
A more difficult result that requires some background with complex geometry is the parametrisation of a circle around a point. For a circle of radius r, centre z, the path γ[0,2π]→C given by γ(t)=z+r*exp(i*t) traces out the desired curve anticlockwise from z+r, the rightmost point.
Mathematical notation, especially when there are several related ideas or meanings, cann often vary. The terms here are what I've encountered during my studies of complex analysis (British university, German lecturer), but in ordinary vector calculus we simply referred to line integrals, and unperson notes that the complex plane integral is often called contour integration, with what I describe as a path being called a curve. I drew a distinction between the function (path) and image (curve) that seems worthwhile, but if this causes confusion with your own sources I apologise! Swap also points out that the term Jordan curve often implicitly means simple closed since these are the most useful ones.