A line integral is the integral of a multi-variable function along a curve. Line integrals can be categorised into scalar line integrals and vector line integrals. They are usually evaluated using parametric equations, which are then reduced to one or more scalar integrals.
Scalar line integrals
Let’s begin with scalar line integrals. A scalar function of two variables is represented geometrically by a surface on a three-dimensional graph (see diagram above). Consider two points, A and B, on the surface. There are infinite ways to move along the surface between these two points. For simplicity, we have indicated two of these infinite paths using the red and pink curves, which can be projected onto the xy plane for better visualisation (see diagram below).
Due to the many paths available between the two points, the integral of between the points is not well defined. In other words, the integral is path-dependent. For a particular path, the integral of
along this path, which is specified by one of the curves on the
plane, is called a line integral.
Geometrically, the result is the area of the ‘curtain’ extending from that path, between and
on the xy plane, to the curve on the surface (see diagram above), i.e.:
where denotes ‘curve’, which is the specified path linking the two points on the xy plane, and
is the infinitesimal change in arc length along the curve.
Since the specific path is also a function of
and
, we convert
and
into parametric equations consisting of a single parameter and carry out the integration analytically (see 3rd Q&A below for an example).
Question
What if the two points, A and B, are the same?
Answer
In the scenario where the two points are the same, the infinite paths from to
are infinite loops. Similarly, integrating the function via different loops may give different results. Each line integral in this case is represented by:
where the circle on the integral sign denotes a loop or a cyclic process.
Vector line integrals
Let’s now look at vector line integrals, an example of which is the work done by a vector field (see diagram above). For instance, the work done by a variable force on a charged particle moving along some path in an electric field is
As mentioned in the opening paragraph, such a line integral is evaluated by converting it into one or more scalar integrals. We write and
in terms of their two-dimensional Cartesian components:
which can then be evaluated when is specified.
Question
What is the difference between a scalar field and a vector field?
Answer
A scalar field, expressed by a scalar function , associates a scalar with each point in some region of space, while a vector field, expressed by a vector function
associates a vector with each point.
Line integral with respect to coordinates
Line integrals can also be carried out with respect to one of the function variables instead of with respect to the arc length, e.g. the scalar integral . The geometric interpretation of
is the projection of
on the
plane (see above diagram). An example of such a line integral is the work done on an ideal gas in a reversible process:
If the path of the above integral is defined by introducing a constraint, e.g. when , where
is a constant, we get a plane that intersects with the surface at a particular contour (see diagram below). A single curve, an isothermal curve, is then projected on the
plane.
Consequently, the integral reduces to one involving a single variable:
Similarly, if the intersecting plane is or
, the projection on the
plane is a horizontal line (isobaric process) or a vertical line (isochoric process) respectively.
Line integral of differential forms
Line integrals are sometimes written in differential form. Consider the work done by a vector field on a particle moving along some path
. We can write
and
in terms of their components:
and
. So, eq92 becomes
Compared to the general differential equation of two independent variables of the form , the RHS of the third equality of the above equation is the line integral of a differential equation, which in this case is an inexact differential. In other words,
.
In the case of an exact differential , its line integral is equal to the difference in values of the function
at the final point and at the starting point:
This is known as the fundamental theorem of line integral. We call such a function, whose output is independent of the path taken to reach it, a state function.
Question
How do we proof the fundamental theorem of line integral?
Answer
Consider a function with an exact differential of the form
or
where and
.
To carry out the line integral between two points and
, we convert the function into its parametric form, which is to let
and
be some function of
, i.e.
. Correspondingly, we have
and hence
. So,
Using the definition of the chain rule for a multivariable function (see eq14b),
Finally, we have shown in a previous article that the change in internal energy of a system is path-independent for a system containing a perfect gas. We will show in the article on entropy that
is path-independent for any system. This makes
an exact differential and
, a state function. Thus, we can write
If the two points are the same, ,
Therefore, the change in internal energy in a cyclic process is zero: . In general, the line integral of any exact differential involved in a cyclic process is zero.
